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\pageno=1 \noindent
\centerline{\bf A Group Algebra for Inductive Limit Groups.}
\centerline{\bf Continuity Problems of the Canonical Commutation
Relations.}
\vglue .3in
\centerline{Hendrik Grundling,}
\centerline{ Department of Pure Mathematics, University of
New South Wales,}
\centerline{ P.O. Box 1, Kensington, NSW 2033, Australia.}
\centerline{ email: hendrik@hydra.maths.unsw.edu.au}
\vglue .2in
\itemitem{{\bf Abstract}}{\sl Given an inductive limit group
$G=\lim\limits_{\rightarrow}G_\beta$,
$\beta\in\Gamma$ where each $G_\beta$ is locally compact,
and a continuous two--cocycle $\rho\in Z^2(G,\,{\bf T})$,
we construct a C*--algebra $\cl L.$ for which the
twisted discrete group algebra $C^*_\rho(G_d)$ is imbedded in its
multiplier algebra $M(\cl L.)$,
and the representations
of $\cl L.$ are identified with the
strong operator continuous $\rho\hbox{--representations}$
of $G$. If any of these representations are faithful,
the above imbedding is faithful. When $G$ is locally compact,
$\cl L.$ is precisely $C^*_\rho(G)$, the twisted group
algebra of $G$, and for these reasons we regard $\cl L.$
in the general case
as a twisted group algebra for $G$. Applying this construction to
the CCR-algebra over an infinite dimensional symplectic space $(S,\,B)$,
we realise the regular representations as
the representation space of the C*--algebra $\cl L.$,
and show that pointwise continuous symplectic group actions on
$(S,\, B)$ produce pointwise continuous actions on $\cl L.$,
though not on the CCR--algebra. We also develop the theory to
accommodate and classify ``partially regular'' representations,
i.e. representations which are strong operator continuous on some
subgroup $H$ of $G$ (of suitable type) but not necessarily on $G$,
given that such representations occur in constrained quantum systems.}
\par\noindent
{\bf Keywords:} twisted group algebra, inductive limit group,
discontinuous group representation, CCR--algebra, symplectic action,
quantum field.
\chop {\bf AMS classification:} 81T05, 81R10, 22A25, 22D15,
43A10, 46L60.
\chop{\bf Running headline:} Group Algebra for Inductive Limit Groups
\vfill\eject
\beginsection 1. Introduction.
It is a well known fact of life that the C*--algebra of the CCRs has
a continuity problem in that it admits nonregular representations,
and the natural C*--dynamical systems on it, defined from nondiscrete
symplectic actions, do not have pointwise norm continuity. These
problems have inspired a long line of papers (cf. [1--3] and
references therein) of which this is one. There is however a
simple idea which explains the continuity problems on a structural
level and allows one to circumvent it, using only standard
C*--techniques, and it appears not to have been fully
exploited for the
CCR discontinuity problem yet. Part of the construction seems to be known,
having appeared in various disguises, e.g. in Segal [18],
and at greater length in Kastler [19], using
twisted convolution algebras;-- here we follow that line of thought
through the use of twisted group algebras.
It reduces to the problem of defining a
a suitable `` twisted group C*--algebra'' for
groups which are inductive limits of locally compact groups.
We hasten to remark that whilst the continuity problem of the CCRs
is generally regarded as a blemish on the C*--theory of bosons
(it does not occur for the CAR algebra), we did put it to good use
in [10] (with C.A. Hurst) where it was demonstrated that in the
presence of selfadjoint constraints, linear in the fields, the physical
representations must be nonregular on the nonphysical parts of the
theory but regular on the physical part. This was in order to
avoid continuous spectrum problems for the constraint conditions.
Hence in some contexts it is imperative that there are such
``partially regular'' representations on the field algebra.
In this note we also wish to start a structure theory for these
representations.
We begin with the CCR--algebra. Let $S$ be a separable real linear
topological space with a nondegenerate continuous symplectic form
$B$ given on it. The CCR algebra $\ccr S,B.$ is then the unique (simple)
C*--algebra generated by unitaries ${\set\delta_x,x\in S.}$
which satisfy the Weyl relations:
$$\delta\s x.\delta\s y.=e^{iB(x,\,y)/2}\delta\s x+y.\qquad\forall\,
x,\;y\in S$$
(cf. [4, 5] for the uniqueness proof).
Then ${\|\delta_x-\delta_y\|}=2$ if $x\not=y$ and hence for a
fixed $x\in S$ the maps $\lambda\to\delta\s\lambda x.$, $\lambda\in\r$
are not continuous with respect to the C*--norm.
The regular representations of $\ccr S,B.$ are those representations
$\pi$ for which the one parameter groups $\lambda\to\pi(\delta\s\lambda x.)$
(with fixed $x$) are weak operator continuous for all $x\in S$.
(Hence in a regular representation Stone's theorem can be applied
to obtain selfadjoint generators --the quantum fields for some $S$,$B$--
for these groups).
The regular states of $\ccr S,B.$ are those states with regular
GNS--representations. However the regular states do not comprise
the state space of $\ccr S,B.$
and the question arises about the nature and existence of a
C*--algebra with state space exactly the regular states. We would also
like to have a more algebraic characterisation of regular
representations than the topological one above.
Let $\Sp(S,\,B)$ denote the group of linear symplectic transformations
on $S$ with the topology of pointwise convergence (i.e. given a net
$\{T_\nu\}\subset\Sp(S,\,B)$ then it converges to $T\in\Sp(S,\,B)$
if $T_\nu(x)\to T(x)$ $\forall\,x\in S$).
In practice we are frequently given a continuous group homomorphism
$\gamma:\cl G.\to\Sp(S,\,B)$ where $\cl G.$ is some topological group of
physical symmetries. Such a homomorphism defines an action
$\alpha:\cl G.\to\aut\ccr S,B.$ by $\alpha_g(\delta\s x.):=
\delta\s\gamma_g(x).$ $\forall\,g\in\cl G.,\;x\in S$.
However since ${\|\delta_x-\delta_y\|}=2$ if $x\not=y$
the action $\alpha$ will only be pointwise norm continuous
if $\cl G.$ is a discrete group (assuming $\gamma$ is injective).
This has given rise to the construction of an auxiliary
C*--algebra on which $\alpha$ is norm continuous [1], but here
we consider construction more natural for the CCRs.
Briefly then, in this note we aim to find a C*--algebra with
state space exactly the regular states, and on which the continuous
symplectic actions define pointwise norm continuous actions.
We would also like to characterise regular states and representations
algebraically, and give a framework to deal with partially regular
states and representations. This will be almost trivial when
${\rm dim}\,S<\infty$, and the main task is to extend the
finite dimensional construction to the infinite dimensional case.
We will develop the basic construction in a more general
context than is warranted by the CCR algebra alone, because of the
possible utility for some infinite dimensional groups.
The content of Sect. 2 is well--known, with the only (presumed) novelty
the application to the regularity question of the CCRs.
Readers in a hurry can start with Sect. 3 after skimming
Sect. 2 up to theorem 2.4.
\beginsection 2. The Basic Idea: Finite Dimensional Case.
Assume in this section that
${\rm dim}\,S<\infty$, so $S\cong\r^n\times\r^n$ and the
symplectic form is
$$B((x_1,\,x_2)\,,\,(y_1,\,y_2))=x_1\cdot y_2-x_2\cdot y_1\qquad
\hbox{for}\quad x_i,\,y_i\in\r^n\,.$$
In this case we know by the Stone--Von Neumann theorem that
there is a unique (up to unitary equivalence) irreducible
regular representation
$\pi$ of $\ccr S,B.$.
Denote:
$\sigma(x,\,y):={\exp\f i,2.B(x,\,y)}$ and note that $\sigma$
is a jointly continuous two--cocycle on the space $S$, regarded as
an Abelian group. The regularity condition, that
$\lambda\to\pi(\delta\s\lambda x.)$ is weak operator continuous,
is equivalent to the requirement that the
$\sigma\hbox{--projective}$ representation of
$S$ given by $\pi(x):=\pi(\delta_x)$, $x\in S$
is weak operator continuous.
Nevertheless, there are many
nonregular states and representations and many regular states
of $\ccr S,B.$, although only one regular irreducible
representation. These are best studied through twisted group
algebras, which we next define, following [11, 6].
Let $G$ be a nondiscrete locally compact group with Haar measure $\mu$,
and let $\rho\in Z^2(G,\,{\bf T})$ be a Borel 2--cocycle
with $\rho(e,\, g)=\rho(g,\, e)=1$ $\forall\;\ab g\in G$.
We further assume that the maps $g\to\rho(g,\,x)$ are continuous
for $\mu\hbox{--almost}$ all $x$.
Start with the dual space $C_0(G)^*$, which we know by the Riesz--Markov
theorem to be identified with the Banach space of bounded complex
Baire measures on $G$. Each Baire measure has a unique extension
to a regular Borel measure [21] on $G$. Denote this Banach space of
regular Borel measures by $\cl M.(G)$.
Define twisted convolution and
involution on $\cl M.(G)$ by:
\itemitem{{\bf Def. 2.1}} For $\gamma,\;\nu\in\cl M.(G)$, $f\in C_0(G)$,
define $$\eqalignno{
\int f(t)\,d(\gamma*\nu)(t)&:=\int\!\!\int f(st)\rho(s,\,t)\,d\nu(t)\,
d\gamma(s)\cr
\hbox{and}\qquad\qquad\qquad\int f(t)\,d\gamma^*(t)&:=
\overline{\int\overline{f}(t^{-1})\,\rho(t^{-1},\,t)\,d\gamma(t)}\cr}$$
\noindent then (with a further condition on $\rho$)
these operations make $\cl M.(G)$ into a
Banach *--algebra. There are two distinguished subalgebras of
$\cl M.(G)$: identify $L^1(G)$ as the closed *--ideal of
$\cl M.(G)$ consisting of finite measures which are absolutely
continuous w.r.t. the Haar measure $\mu$ of $G$. Then by writing
$d\gamma(s)=f(s)d\mu(s)$, $f\in L^1(G)$ for these, we obtain
from 2.1:
\item{{\bf 2.2}} for $f,\;g\in L^1(G)$, twisted convolution
and involution is:
$$\eqalignno{f*g(x)&=\int_Gf(y)\,g(y^{-1}x)\,\rho(y,\,y^{-1}x)\,
d\mu(y)\cr
f^*(x)&:=\Delta(x)^{-1}\overline{\rho(x,\,x^{-1})}\,\,\overline{
f(x^{-1})}\;,\cr}$$
where $\Delta$ denotes the modular function.
\noindent Note that $f^{**}=f$ implies that $\rho(x^{-1},\, x)=
\overline{\rho(x,\, x^{-1})}$.
Then $L^1(G)$ becomes a Banach *--algebra, denoted
$L_\rho^1(G)$, the closure of which in the enveloping
C*--norm is the twisted group algebra of $G$, denoted
$C_\rho^*(G)$. Since $G$ is nondiscrete, $C^*_\rho(G)$ is not
unital. The other distinguished subalgebra of $\cl M.(G)$
is the Banach *--algebra generated by the Dirac measures
${\set\delta_x,x\in G.}$, which clearly is just $L^1(G_d)
=\ell^1(G)$, and where $G_d$ denotes $G$ with the discrete
topology.
(The notation $\delta_x$ here extends the previous notation for the
generating unitaries of the CCR--algebra, as we shall shortly
see). The multiplication in $\cl M.(G)$ specializes to:
$$\leqalignno{
(\delta\s y.*f)(x)&=f(y^{-1}x)\,\rho(y,\,y^{-1}x)
\;,&\hbox{{2.3}}\cr
(f*\delta\s y.)(x)&=f(xy^{-1})\,\rho(xy^{-1},\,y)\,\Delta(y)^{-1}
\;,\qquad\forall\,f\in L_\rho^1(G),\;\;x,\,y\in G\cr
\hbox{and}\qquad\qquad\delta\s x.*\delta\s y.&=\rho(x,\,y)\,\delta\s xy.
\;,\qquad\quad\delta\s x.^*=\overline{\rho(x^{-1},\,x)}\,\,\delta\s x^{-1}.\;.
\cr}$$
Observe that
$$\eqalignno{[\rho(x,\, x^{-1})]^2=(\delta_x\delta\s x^{-1}.)^2&=
\delta_x(\delta\s x^{-1}.\delta_x)\delta\s x^{-1}.=\rho(x^{-1},\, x)\,
\rho(x,\, x^{-1})\cr
\hbox{so}\qquad\qquad\rho(x,\,x^{-1})=&\rho(x^{-1},\, x)=
\overline{\rho(x,\, x^{-1})}=\pm 1\; .\cr}$$
This is the condition which we will henceforth assume
for $\rho$ to ensure $\cl M.(G)$ is a *--Banach algebra.
Let $H\subset G$ be a proper closed locally compact subgroup
of $G$. Then its measures $\cl M.(H)$ are embedded in
$\cl M.(G)$ as those measures on $G$ with support in $H$,
and the multiplication operation in $\cl M.(G)$ also
produces an action of $\cl M.(H)$ on $L^1_\rho(G)$.
In particular it gives an $L^1\hbox{--continuous}$ embedding
of $L_\rho^1(H)$ in $\cl M.(G)$ (the restriction of $\rho$
to $H$ is still denoted by $\rho$) such that $L_\rho^1(H)$
is still in the relative multiplier of $L^1_\rho(G)$.
%The following two facts are well known, and the proofs are included
%here because the author could not locate it in the literature.
%The symbol $M(\cl A.)$ denotes the (abstract) multiplier algebra
%of the C*--algebra $\cl A.$. DO A LEMMA
Both of the $L^1\hbox{--embeddings}$ above will extend to their
enveloping C*--algebras, using the universal property of these
cf. Dixmier 1.3.7 [8], and since $L^1_\rho(G)$ contains an
approximate identity for $\cl M.(G)$, we actually obtain
C*--continuous embeddings in the multiplier algebra of
$C^*_\rho(G)$, i.e.
$$C^*_\rho(G_d)\subset M(C^*_\rho(G)),\qquad\hbox{and}\qquad\quad
C^*_\rho(H)\subset M(C^*_\rho(G))\;.$$
We will always assume that $G$ is nondiscrete, in which case
$C^*_\rho(G)$ does not have an identity. Further information
about $C^*_\rho(G)$ is in [6], and the main aspect of interest
here is the universal property that there is a bijection between
the set of nondegenerate representations of $C^*_\rho(G)$
and the set of weak operator continuous $\rho\hbox{--projective}$
representations of $G$ (cf. Theorem 2.4 below).
In particular, if $G=S$, $\rho=\sigma$, then $C^*_\sigma(S_d)=
\ccr S,B.$, indeed definition 2.1 reduces precisely to the
definition given by Manuceau [7] for the CCR--algebra.
For general locally compact groups $G$ we will call those representations
of $C^*_\rho(G_d)$
which come from weak operator continuous $\rho\hbox{--representations}$
of $G$ {\it regular} representations, the others are {\it
nonregular} and a state is regular or not according to whether
its GNS--representation is. This agrees with the usual physics terminology
on $\ccr S,B.$, ${\rm dim}\,S<\infty$, and should not be confused
with the left or right regular representations of $G$.
\def\cg{{C^*_\rho(G)}}
\def\crd{{C^*_\rho(G_d)}}
We will also call a state or representation of $M(\cg)$ (non)regular
if its restriction to $\crd$ is (non)regular.
\def\rep #1,{{\rm Rep}\,#1} \def\repr #1,{{\rm Rep}_\rho #1}
\def\repro #1,{{\rm Rep}_\rho^0 #1}
Denote by $\rep\cl A.,$ the set of nondegenerate representations of an
algebra $\cl A.$ and by $\repr G,$ the set of nondegenerate
weak operator continuous unitary $\rho\hbox{--representations}$
of a topological group $G$. If we relax the weak operator continuity,
we use the notation $\repro G,$ (=$\repr G_d,$).
\def\wt{\widetilde}
Say a representation $\pi\in\repro G,$ is Borel if it is a Borel
map $\pi:G\to U(\cl H._\pi)$ where $U(\cl H._\pi)$ is equipped with
the strong operator topology.
We allow nonseparable representation spaces, hence Borel representations
need not be strong operator continuous.
For a Borel representation $\pi$
we can use the Bochner integrals
$$\pi(\nu)=\int_G\pi(g)\, d\nu(g)\in\cl B.(\cl H._\pi)
\qquad\forall\;\nu\in\cl M.(G)\;.$$
In the next theorem we give the well--known structure of
representations on $M(\cg)$ (we include the proof of
2.4(1) for completeness, cf. Dixmier 13.3.1 [8] and
Kastler [19]), but being explicit in our inclusion of
nonregular representations in the analysis.
Each $\rho\hbox{--representation}$ $\pi$ of $G$ defines
a representation of $\crd$, and we define a ``canonical
extension'' for $\pi$ which will be central to the
developments in Sect. 3.
\thrm Theorem 2.4." Let $G$ be a nondiscrete locally compact group,
$\rho$ as above,
\item{$(1)$} let $\pi\in\repro G,$ be a Borel representation, then the formula
\chop $\widetilde\pi(\nu)=\int\limits_G\pi(g)\,d\nu(g)$, $\nu\in\cl M.(G)$
defines a continuous *--representation $\wt\pi\in\rep\cl M.(G),$
on $\cl H._\pi$ such that
\itemitem{$(\rn1)$} $\wt\pi(\delta_g)=\pi(g)$ $\forall\, g\in G$,
\itemitem{$(\rn2)$} $\wt\pi\in\rep\cl M.(G),$ is irreducible iff
$\pi\in\repro G,$ is irreducible,
\itemitem{$(\rn3)$} $\wt\pi$ is nondegenerate
on $L_\rho^1(G)$ iff $\pi\in\repro G,$
is strong operator continuous.
\item{$(2)$} Any $\pi\in\rep\cg,$ extends uniquely on the same Hilbert
space to a representation $\hat\pi$ of $M(\cg)$ preserving irreducibility, and
it is regular on $C^*_\rho(G_d)$.
\item{$(3\;\rn1)$} Let $\cl E.:=C^*\big(\cg\cup\crd\big)$. Then a
representation $\pi\in\rep\cl E.,$ has a unique decomposition
$\pi=\pi_1\oplus\pi_2$ such that $\pi_i$ are both nondegenerate and
$\cg\subseteq\ker\pi_2$ and $\pi_1(\cg)$ is strongly dense in
$\pi_1(\cl E.)$. Then
$$\pi_1(f)=\int_G\pi_1(\delta_g)\, f(g)\, d\mu(g)\qquad\forall\;
f\in L^1_\rho(G)\;,$$
where $\mu$ is the Haar measure on $G$.
\item{$(3\;\rn2)$} A representation $\pi\in\rep\crd,$ has an extension
on the same Hilbert space to a representation $\check\pi\in\rep\cl E.,$
with $\cg\subseteq\ker\check\pi$.
This extension is unique if $\pi$ is irreducible and nonregular.
\item{$(3\;\rn3)$} A representation $\pi\in\rep\crd,$ has a
{\bf canonical extension} $\wt\pi$ to $\cl E.$ on the same space
$\cl H._\pi$, in the sense that
for any other extension $\pi'$ to $\cl E.$ on
$\cl H._\pi$ we have $\cl H.\s{\wt\pi_1}.\supseteq\cl H.\s{\pi'_1}.$
where $\wt\pi=\wt\pi_1\oplus\wt\pi_2$ and $\pi'=\pi'_1\oplus\pi'_2$
are the unique decompositions of $(3\;\rn1)$.
This defines a decomposition $\pi=\pi_R\oplus\pi_N$ where
$\pi_R:=\wt\pi_1\,\big|\,\crd$ is the {\bf regular part} of $\pi$,
and $\pi_N:=\wt\pi_2\,\big|\,\crd$ is the {\bf irregular part.}
\item{$(3\;\rn4)$} A representation $\pi\in\rep\crd,$ is regular iff
its canonical extension $\wt\pi$ is nondegenerate on $\cg$,
i.e. $\wt\pi_2=0$.
\item{$(4)$}
There are bijections between the following sets:
$$\eqalignno{\cl P.&:=\set\pi\in{\rep C^*_\rho(G_d),},\pi\hbox{$\,$ is
regular}.\cr
{\rm Rep}\,C^*_\rho(G)&:=\left\{\hbox{ nondegenerate representations
of }\,C^*_\rho(G)\;\right\}\cr
{\rm Rep}_\rho(G)&:=\left\{\hbox{weak operator continuous
$\rho\hbox{--representations}$ of}\;G\,\right\}\cr}$$
which preserve irreducibility, and these are obtained from the
extensions above, so
we have a bijection $t:\rep\cg,\to\cl P.$ given by
$t(\pi)=\hat\pi\uprightharpoon\crd$ where $\hat\pi$ is the
unique extension of (2), and $t^{-1}(\pi)=\wt\pi\uprightharpoon\cg$
where now $\wt\pi$ is the canonical extension of $(3\;\rn3)$."
\def\t{\widetilde}
{\bf (1)} Since $\nu\in\cl M.(G)$,
$$\|\t\pi(\nu)\|\leq\sup_g\|\pi(g)\|\int d|\nu(g)|=\|\nu\|_1<\infty$$
and hence $\t\pi(\nu)\in\cl B.(\cl H._\pi)$ and the map
$\nu\to\t\pi(\nu)$ is $L^1\hbox{--continuous}$.
$$\eqalignno{\t\pi(\gamma*\nu)&=\int\pi(g)\, d(\gamma*\nu)(g)
=\int\int\pi(sg)\,\rho(s,\,g)\,d\nu(g)\,d\gamma(s) \cr
&=\int\int\pi(s)\cdot\pi(g)\,d\nu(g)\, d\gamma(s)
=\t\pi(\gamma)\cdot\t\pi(\nu) \cr}$$
for all $\gamma,\;\nu\in\cl M.(G)$, $\pi\in\repr G,$.
Moreover for all $\nu\in\cl M.(G)$ and $\eta,\;\ab\xi\in
\cl H._\pi$ we have
$$\eqalignno{(\eta,\;\t\pi(\nu^*)\xi)&=\int\big(\eta,\,\pi(g)\xi\big)
\, d\nu^*(g)=\overline{\int\overline{(\eta,\,\pi(g^{-1})\xi)}\,
\rho(g^{-1},\,g)\,d\nu(g)} \cr
&=\overline{\int\big(\rho(g,\,g^{-1})\,\pi(g)^*\xi,\;\eta\big)\,
\rho(g,\,g^{-1})\,d\nu(g)} \cr
&=\overline{\int\big(\xi,\,\pi(g)\eta\big)\,d\nu(g)}
=\big(\t\pi(\nu)\eta,\,\xi\big) \cr}$$
where we made use of $\pi(g^{-1})=\rho(g,\,g^{-1})\pi(g)^*$
and $\rho(g,\,g^{-1})=\overline{\rho(g^{-1},\,g)}$.
Thus $\t\pi(\nu^*)=\t\pi(\nu)^*$, and so $\t\pi$ is a continuous
*--representation of $\cl M.(G)$.\chop
$(\rn1)$ Clearly $\t\pi(\delta_g)=\int\pi(h)\,d\delta\s g.(h)=\pi(g)$.
\chop
$(\rn2)$ That $\t\pi(\nu)$ on $\cl M.(G)$ is irreducible
iff $\pi$ is on $G$, follows from $(\rn1)$.\chop
$(\rn3)$ If $\t\pi$ is nondegenerate on $L^1_\rho(G)$, then the equation
\chop ${}\qquad\qquad\pi(g)\,\t\pi(f)=\t\pi(\delta\s g.* f)\quad
\forall\, f\in L^1_\rho(G)\,,\; g\in G$\chop
determines the operator $\pi(g)$ uniquely on $\cl H._\pi$, and it is easily
seen to be unitary.
Now to show that $\pi$ is strong operator continuous,
$$\eqalignno{\left\|\big(\pi(g)-\pi(e)\big)\t\pi(f)\,\xi\right\|
&=\left\|\t\pi(\delta\s g.*f-f)\xi\right\| \cr
&\leq\left\|\t\pi(\delta\s g.*f-f)\right\|\cdot\|\xi\| \cr
&\leq\left\|\delta\s g.*f-f\right\|_{L^1}\cdot\|\xi\| &(*) \cr}$$
using the $L^1\hbox{--continuity}$ of $\t\pi$. But
$$\eqalignno{\|\delta\s g.*f-f\|_{L^1}&=\int_G\big|f(g^{-1}x)\,
\rho(g,\,g^{-1}x)-f(x)\big|\,d\mu(x) \cr
&=\int_G\big|f(x)\,\rho(g,\,x)-f(gx)\big|\,d\mu(x)\;\; .\cr}$$
Now for all $f\in C_c(G)$ (the continuous functions of compact support)
we have:
$$\eqalignno{\big|f(x)\,\rho(g,\,x)-f(gx)\big|&\leq
\big|f(x)\big(\rho(g,\,x)-1\big)-\big(f(gx)-f(x)\big)\big| \cr
&\leq\big|f(x)\big|\cdot\big|\rho(g,\,x)-1\big|+
\big|f(gx)-f(x)\big| \cr
&\longrightarrow \, 0 \quad\hbox{as}\quad g\to e \cr}$$
for $\mu\hbox{--almost}$all $x$, using the assumed continuity
property of $\rho$. Thus,
invoking the locally compactness of $G$ to apply the Lebesgue
dominated convergence theorem, we conclude that
${\|\delta\s g.*f-f\|_{L^1}}\to 0$ as $g\to e$ for all $f\in C_c(G)$.
So since $\t\pi\big(C_c(G)\big)\cl H._\pi$ is dense in
$\cl H._\pi$ and $\pi(g)-1$ is bounded,
we obtain from the inequality $(*)$ that $\pi$
is strong operator continuous on $G$.\chop
Conversely, let $\pi\in\repro G,$ be strong operator continuous.
Let $J$ be a set of decreasing neighbourhoods of $e\in G$,
ordered by inclusion, with intersection $\{e\}$
and let $u_i\in L_+^1(G)$ for $i\in J$ be a function with
${\rm ess}\;{\rm supp}(u_i)\subset i$ and $\int u_i(g)\, d\mu(g)=1$.
By strong operator continuity of $\pi$, we see that for each
$\xi\in\cl H._\pi$, $\varepsilon>0$, there is an $i\in J$
such that ${\big\|\big(\pi(g)-1\big)\xi\big\|}<\varepsilon$ for all
$g\in i$. Thus
$$\eqalignno{\big\|\big(\t\pi(u_i)-1\big)\,\xi\big\|&=
\big\|\int u_i(g)\,\big(\pi(g)-1\big)\xi\, d\mu(g)\big\| \cr
&\leq\sup_{g\in i}\big\|(\pi(g)-1)\xi\big\|\,\int u_i(g)\,
d\mu(g)<\varepsilon\; .\cr}$$
So $\t\pi(u_i)\to 1$ in the strong operator topology and thus
${\t\pi\,\Big|\,L^1_\rho(G)}$ is nondegenerate.\chop
{\bf (2)} Since $\cg$ is a closed two sided ideal of $M(\cg)$,
the assertion follows directly from Dixmier 2.11.1 [8] and (1) above.
That it is regular on $\crd$ follows from the first part of the proof of
$(1\;\rn3)$.
\chop
{\bf (3\rn1)} Since $\cg$ is an ideal of $\cl E.$,
the first part follows from Dixmier 2.11.1 [8]. To see that
$$\pi_1(f)=\int_G\pi_1(\delta_g)\, f(g)\, d\mu(g)\qquad\forall\; f\in
L_\rho^1(G)\; :$$
observe that since $\pi_1$ is nondegenerate on $\cg$, and we know that
every nondegenerate representation of $\cg$ comes from a regular
$\rho\hbox{-representation}$ of $G$ via an integral as in the formula,
there must be a regular representation $\pi_0$ of $\crd$ on
$\cl H._{\pi_1}$ such that
$$\pi_1(f)=\int_G\pi_0(\delta_g)\, f(g)\, d\mu(g)\qquad\forall\, f\in
L_\rho^1(G)\;.\eqno{(*)}$$
Thus $\pi_1=\pi_0$ a.e. on $G$. From the uniqueness of extensions
of nondegenerate representations from $\cg$ to $\cl E.$ on the
same space, we conclude that $\pi_1=\pi_0$.\chop
{\bf (3\rn2)} To see that an extension $\check\pi$ exists, observe that
$$\cl E.=\overline{L\big(\cg\cup\crd\big)}=\overline{L\big(L_\rho^1(G)
\cup L_\rho^1(G_d)\big)}\,,$$
so since $G$ is nondiscrete,
$$L_\rho^1(G)\cap L_\rho^1(G_d)=\{0\}\;,$$
we have
$$L\big(L_\rho^1(G)\cup L_\rho^1(G_d)\big)\big/L^1_\rho(G)\cong L_\rho^1(G_d)$$
hence any $\pi\in\rep L_\rho^1(G_d),=\rep\crd,$ has a lifting
$\check\pi$ on the same space to ${L(L^1_\rho(G)\cup L_\rho^1(G_d))}$
for which $L_\rho^1(G)\subset\ker\check\pi$.
By continuity, extend $\check\pi$ to $\cl E.$ to obtain an
extension $\check\pi$ of $\pi\in\rep\crd,$ to $\cl E.$ on the
same space, and with $\cg\subset\ker\check\pi$.\chop
Given a $\pi\in\rep\crd,$ which is irreducible and nonregular
observe that any extension $\wt\pi$ of $\pi$ to $\cl E.$
on the same space will also be irreducible. Thus the decomposition
of 3(\rn1) becomes either $\wt\pi=\wt\pi_1$ or $\wt\pi=\wt\pi_2$.
If it is nonregular, then $\wt\pi$ must be degenerate on
$\cg$, so the first alternative is impossible, so
$\wt\pi=\wt\pi_2$ which vanishes on $\cg$. Thus $\wt\pi$
can only be the lifting constructed above.\chop
{\bf (3\rn3)} Let $\pi\in\rep\crd,$ and define
$$\cl H._B:=\set\psi\in\cl H._\pi,g\to\pi(\delta_g)\psi\quad
\hbox{is Borel from $G$ to $\cl H.$}.$$
which is a closed linear space since sums and limits of sequences
of Borel maps are Borel. Observe that $\cl H._B$ is
preserved under $\pi\big(\crd)\big)$, using continuity assumption
of $\rho$. Thus $\pi\uprightharpoon\cl H._B$ is a subrepresentation,
Borel and nondegenerate since $\pi$ is nondegenerate and $\un\in\crd$.
So using 2.4(\rn1), the representation $\pi\uprightharpoon\cl H._B$
extends on $\cl H._B$ to a representation $\wt\pi$ of $\cl M.(G)$
which is given on $L_\rho^1(G)$ by:
$$\wt\pi(f)\,\psi=\int_G\pi(\delta_g)\psi\, f(g)\,d\mu(g)\qquad
\forall\,\psi\in\cl H._B\,,\; f\in L_\rho^1(G)\,.$$
On $\cl H._B^\perp$, set $\wt\pi(L_\rho^1(G))=0$
to obtain the extension $\wt\pi$ on $\cl H.$.\chop
The statement that the regular part of $\wt\pi$ is
maximal, is clear by combining $(3\rn1)$ and $(1\rn3)$.\chop
{\bf (3\rn4)} Clearly if $\pi$ is regular, then $\cl H._B
=\cl H._\pi$ and so by nondegeneracy of $\pi$, $\cl H._B=
\cl H._{\wt\pi_1}=\cl H._\pi$, so $\wt\pi_2=0$.
\chop Conversely, if $\wt\pi_2=0$, then $\wt\pi=\wt\pi_1$
is nondegenerate on $\cg$ and so by part (2) is uniquely
determined by its values on $\cg$ and so by parts (1) and
(3\rn1) must be regular.\chop
{\bf (4)} This is well known, and easily seen from parts
(1 \rn3) and (3), using Dixmier 2.7.4, though explicitly
it is instructive to check that the map $t:\rep\cg,\to
\cl P.$ by $t(\pi)=\hat\pi\,\big|\,\crd$
is the inverse of $t^{-1}(\pi)=\wt\pi\,\big|\,\cg$:
for any $\pi\in\cl P.$ we have:
$$\eqalignno{\hat{\wt\pi}(\delta_h)\,\wt\pi(f)&:=
\wt\pi(\delta_h*f)=\int_G(\delta_h*f)(g)\,\pi(\delta_g)\,
d\mu(g)\cr
&=\int_Gf(h^{-1}g)\,\rho(h,\,h^{-1}g)\,\pi(\delta_g)\, d\mu(g)
\qquad\hbox{using 2.3}\cr
&=\int_Gf(g)\,\rho(h,\, g)\,\pi(\delta\s hg.)\, d\mu(g)\cr
&=\pi(\delta_h)\,\int_Gf(g)\,\pi(\delta_g)\, d\mu(g)
=\pi(\delta_h)\cdot\wt\pi(f)\cr}$$
for all $f\in L^1_\rho(G)$, hence $\hat{\wt\pi}\,\big|\,\crd=\pi$.
\item{\bf Remarks:} {$(\rn1)$}
Theorem 2.4 gives an algebraic characterization of
regular representations which can replace the
topological one for locally compact groups.
Observe that the representation space $\cl H.\s\wt\pi_1.$ of the regular part
$\wt\pi_1$ of the canonical extension $\wt\pi$ can be characterised by
$$\cl H._R:=\set\psi\in\cl H._\pi,g\to\pi(\delta_g)\psi \;\;
\hbox{continuous from $G$ to $\cl H.$}.\subseteq\cl H._B\,.$$
Hence we have on $\crd$ an intrinsic method for obtaining
the decomposition $\pi=\pi_R\oplus\pi_N$.
[Proof that $\cl H._R=\cl H.\s\wt\pi_1.$. It is obvious that
$\cl H.\s\wt\pi_1.\subseteq\cl H._R$. Conversely, let $\psi\in\cl H._R$,
then since $\cl H._R\subset\cl H._B$, the integrals
$$\wt\pi(f)\,\psi=\int_Gf(g)\,\pi(\delta_g)\psi\, d\mu(g)$$
are defined for all $f\in L_\rho^1(G)$, and must be nonzero
for some, because the function $g\to(\psi,\,\pi(\delta_g)\psi)$
is continuous and nonzero. Hence $\psi$ must have a component in the
essential subspace of $\wt\pi(L_\rho^1(G))$
which is $\cl H.\s\wt\pi_1.$.
To see that this component is all of $\psi$, assume the contrary,
i.e. $\psi=\psi_1+\psi_0$, $\psi_1\in\cl H.\s\wt\pi_1.$ and
$\psi_0\perp\cl H.\s\wt\pi_1.$. Then since $\pi(\delta_g)$
is continuous in $g$ on both $\psi$ and $\psi_1$, it must
be continuous on $\psi_0$ on which some $\pi(f)$ must therefore
be nonzero, which is a contradiction. Thus $\psi\in\cl H.\s\wt\pi_1.$.]
\item{$(\rn2)$} Note from 2.4(\rn2) and (\rn4) that a regular
representation of $\crd$ has more than one extension to $\cl E.$.
The author owes the extension $\check\pi$ of a representation
$\pi$ and this observation to R. Schaflitzel.
\item{$(\rn3)$} Since $\cg$ has as representation space only the regular
representations, it appears to be a more convenient algebra
than $\crd$ (e.g. the CCR--algebra $\ccr S,B.$) if we only want
to admit these representations into the theory.
A further improvement of $\cg$ over $\crd$ is that $\cg$
is separable if $G$ is.
The question
considered in [2], of the existence of direct integral decompositions
of regular representations into
irreducible regular representations is obviously answered
in the affirmative for the locally compact case, since these
are now realized as the representation space
of a separable C*--algebra.
However in the present finite dimensional context, for the CCRs, the last
statement is already obtainable by the Stone--Von Neumann uniqueness theorem.
\item{$(\rn4)$} In the case of the CCRs, where $G=S$, $\rho=\sigma$
nondegenerate, the
algebra $C_\sigma^*(S)$ has been called the
algebra of ``regular'' or ``smooth''
observables, cf. Segal [16, 18], and is well known to be isomorphic
to the algebra of compact operators $\cl K.$ on a separable Hilbert space,
[16], and Folland theorem 1.30 [17]. Thus theorem 2.4(4)
yields an immediate (and known) proof of the Stone--von Neumann theorem,
using the fact that $\cl K.$ has only one unitary equivalence class of
irreducible representations. Moreover, $M(C^*_\sigma(S))\cong M(\cl K.)
=\cl B.(\cl H.)$, so if $\varphi:\cl B.(\cl H.)\to\cl B.(\cl H.)/\cl K.$
is the canonical factorization map to the Calkin algebra,
then $\varphi\circ\pi$ produces a faithful embedding of
the CCR--algebra into the Calkin algebra where $\pi$ can
be any irreducible regular representation of $\ccr S,B.$.
(It is faithful because
the CCR algebra is simple-- for nondegenerate $B$).
Thus in an irreducible regular representation $\pi$ of $\ccr S,B.$, the
essential spectrum of any $A\in\pi\big(\ccr S,B.\big)$ is the same
as its ordinary spectrum, and also, the Fredholm index theory
on $\pi\big(\ccr S,B.\big)$ simplifies.
\chop
Furthermore, since the states of $\cl K.(\cl H.)\cong\cg$ are given by
normalised ``density matrices'' i.e. trace--class operators $D$ by
$$\omega(A)={\rm Tr}\,(DA)\qquad\quad\forall\, A\in\cl K.(\cl H.)\,,$$
this shows the well--known fact that for finite dimensional
quantum mechanics, the regular states are all given by
density matrices.
Thus the inclusion $\ccr S,B.\subset M\big(C^*_\sigma(S)\big)$ expresses
in compact form several important facts of quantum mechanics.
\item{$(\rn5)$} If $\pi\in\rep\crd,$ is Borel w.r.t. the map
$g\to\pi(\delta_g)$, then the canonical extension $\wt\pi$
of Theorem 2.4(3) is exactly the restriction to $\cl E.$
of the representation of $\cl M.(G)$ constructed in part 2.4(1)
by the formula $\wt\pi(\nu)={\int\pi(\delta_g)\, d\nu(g)}$.
Now by the von Neumann theorem a representation of a one--parameter
group on a separable Hilbert space is Borel iff it is
strong operator continuous [21]. Hence Borel representations
of $\r$ can only be nonregular on nonseparable Hilbert spaces.
If $\pi\in\rep{\ccr S,B.},$ is Borel but not regular, there
is an $x\in S$ for which the one--parameter group
$\lambda\to\pi(\delta\s\lambda x.)$ is not strong operator
continuous, and so $\cl H._\pi$ must be nonseparable.
One example of such a nonregular representation, is the
GNS--representation of the tracial state of the CCRs, i.e.
the state $\omega_0$ on $\ccr S,B.$ defined by
$\omega_0(\delta_x)=0$ if $x\not=0$, and $\omega_0(\delta_0)=1$.
This state $\omega_0$ is pure (and the unique trace) iff $B$ is nondegenerate.
A more interesting class of such states (and the motivation for
this paper) is the Dirac states for a linear Bosonic system
with linear selfadjoint constraints, cf. [10].
Given a CCR-algebra $\ccr S,B.$ and a specified constraint subspace
$\cl C.\subset S$ for which $B(\cl C.,\,\cl C.)=0$,
such states by definition satisfy
$\omega(\delta_x)=1$ for all $x\in\cl C.$.
This then implies that for these states we have
$$\omega(\delta_y)=0\qquad\hbox{for all}\;\; y\in S\;\;\hbox{with}\;\;
B(y,\cl C.)\not=0\,.$$
and hence $\lambda\to\omega(\delta\s \lambda y.)$ is discontinuous
at $0$ for these $y$, so nonregular.
For those Dirac states which are regular on the subspace
${\bf p}:=\set x\in S,{B(x,\,\cl C.)=0}.$
we find that their GNS--representations are Borel, nonregular
outside ${\bf p}$, and have nonseparable representation space.
So when constraints are present, density matrices are
not enough for finite dimensional quantum mechanics.
Such states have also been studied recently by Acerbi, Morchio
and Strocchi [22]. Another kind of nonregular state
is the momentum states corresponding to physicist's
plane waves, and these were introduced on the CCR algebra by
Fannes and Verbeure [24].
Nonregular representations are also central in
Narnhofer and Thirring [23], and are included in the analyses
by Manuceau, Sirugue, Testard and Verbeure [27] and
Goderis, Verbeure and Vets [28].
\item{}
In the parlance of 2.4(1\rn3), a Borel representation which has
degenerate canonical extension to $C^*_\sigma(S)$
must have a nonseparable representation space.\chop
The Dirac states suggest that what is needed is a structure
theory for ``partially regular'' representations, and
indeed, in this case for $S$ finite dimensional,
that simply is the structure
that if $U\subset S$ is a closed linear subspace,
then $C^*_\sigma(U)\subset M(C^*_\sigma(S))$, and a representation
can be nondegenerate on $C^*_\sigma(U)$, but degenerate on
$C^*_\sigma(S)$ (though not conversely).
\item{$(\rn6)$} We call a decomposition $\pi=\pi_1\oplus\pi_2$ as
in 2.4(3\rn1) an ideal decomposition.
\item{\bf Def.} A nondegenerate representation $\pi$ on $\crd$
(resp. $\cl E.$) is called \chop {\it irregular} if $\cg\subseteq
\ker\t\pi$ where $\t\pi$ is the canonical extension on the same
space of $\pi$ to $\cl E.$. A state $\omega\in\wp(\crd)$ is
irregular if its GNS--representation $\pi_\omega$ is,
($\wp(\cdot)$ denotes the state space of its argument).
\noindent Observe that an irregular representation
may still be regular w.r.t. some closed subgroup $H\subset G$.
This suggest that we call an irregular representation
{\it absolutely irregular} if it is irregular w.r.t.
all closed locally compact subgroups $H$ of $G$.
One example of such, is the GNS--representation of the
tracial state above.
Using the canonical extension of representations above,
we can define the following canonical (though nonunique) extension
of a state $\omega\in\wp(\crd)$ to $\cl E.$:
$$\t\omega(A):=\big(\Omega_\omega,\,\t\pi_\omega(A)\Omega_\omega
\big)\qquad\forall\,A\in\cl E.$$
where $(\pi_\omega,\,\Omega_\omega,\,\cl H._\omega)$ is the
GNS--representation for $\omega$. Clearly $\pi\s\t\omega.$
is unitarily equivalent to $\t\pi_\omega$.
Now because $\ker\pi\s\t\omega.$ is a maximal two sided ideal
in $\ker\t\omega$, and $\cg$ is an ideal of $\cl E.$,
we see that a state $\omega$ is irregular iff
$\cg\subset\ker\t\omega$.
Introduce the following notation:\chop
let $\cl P.^\perp$ (resp. $\wp\s{\cl P.}.$, $\wp_{\cl P.}^\perp$)
denote the set of irregular representations
(resp. regular states, irregular states) of $\crd$.
Now Theorem 2.4(3) seems to exhibit asymmetry between the
regular and irregular parts of a representation, but it
is possible to restore symmetry by enlarging $\cl E.$ as
follows.
Consider the universal enveloping von Neumann algebra
$\cl E.''\supset\cl E.$. Since $\cg$ is an ideal of
$\cl E.$, there is a central projection $P\in\cl E.''\cap
\cl E.'$ such that $P\cl E.''=\cg''$. Define
$\t\cl E.:=C^*(\cl E.\cup \{P\})\subset\cl E.''$,
then any representation on $\cl E.$ extends uniquely on the
same space to $\cl E.''$, hence to $\t\cl E.$.
If $\pi\in\rep\cl E.,$ is regular, then $\pi(P)=\un$, and if
$\pi$ is irregular then $\pi(P)=0$. Now we have the two
ideals $\t\cl E._1:=P\t\cl E.$ and $\t\cl E._2:=(\un-P)\t\cl E.$
in $\t\cl E.$ such that $AB=0=BA$ if $A\in\t\cl E._1$,
$B\in\t\cl E._2$, so $\t\cl E.=\t\cl E._1\oplus\t\cl E._2
=\overline{L(\t\cl E._1\cup\t\cl E._2)}$. Note that
$\cg\subset\t\cl E._1$, but since $\t\cl E._1$ is unital
with unit $P$, these algebras are not equal. Thus any
$\pi\in\rep\t\cl E.$ has a unique decomposition
$\pi=\pi_1\oplus\pi_2$ where each $\pi_i\in\rep\t\cl E._i,$
and clearly, $\pi_1$ (resp. $\pi_2$) is the regular
(resp. irregular) part of $\pi$, since $P\in\t\cl E._1$
and $(\un-P)\in\t\cl E._2$. Thus in $\t\cl E.$ the characterisation
of regular and irregular representations is symmetrical.
For $A=f\in L^1_\rho(G)$, we obtain from 2.4 for the canonical
extension:
$$\t\omega(f)=\int f(g)\,\omega_1(\delta\s g.)\, d\mu(g)$$
where $\omega_1(A):=(\Omega_\omega,\,\wt\pi_1(A)\,\Omega_\omega)$,
and since $\cl E.=\overline{L(\cg\cup\crd)}$, this formula
uniquely determines the extension $\t\omega$, and $\t\omega$
is pure if $\omega$ is.
\thrm Theorem 2.5." Let $\omega\in\wp(\crd)$ with canonical
extension $\t\omega$ to $\t\cl E.$.
\item{(1)} Then there is a unique decomposition
$\t\omega=\lambda\omega_1+(1-\lambda)\omega_2$, $\lambda
\in[0,\,1]$, where the state $\omega_1$ (resp. $\omega_2$)
is regular (resp. irregular), or equivalently,
$\t\cl E._1\subset\ker\omega_2$ and $\t\cl E._2\subset
\ker\omega_1$. Furthermore, on $\cl E.$ the state $\omega_1$
is characterised by $\omega\,\big|\,\cg$ being a state and
$\omega_2$ by $\cg\subset\ker\omega_2$. We also have
$\lambda\omega_1(A)=\t\omega(PA)$ and $(1-\lambda)\omega_2(A)=
\t\omega{((\un-P)A)}$ for all $A\in\crd$. Pure states are
either regular or irregular.
\item{(2)} The canonical extension defines two injections:
$$\kappa:\wp(\crd)\to\wp(\cl E.)\quad\hbox{and}\quad
\t\kappa:\wp(\crd)\to\wp(\t\cl E.)$$
which are w*--continuous, preserve pure states, and satisfy:
$$\eqalignno{\kappa(\wp_\cl P.)&=\set\omega\in\wp(\cl E.),
\omega\;\;\hbox{is a state on}\;\;\cg.=\wp(\cg) \cr
\kappa(\wp^\perp_\cl P.)&=\set\omega\in\wp(\cl E.),\cg\subset\ker\omega.\cr
\t\kappa(\wp_\cl P.)&\subset\set\omega\in\wp(\cl E.),
{\t\cl E._2\subset\ker\omega}.=\wp(\t\cl E._1) \cr
\t\kappa(\wp^\perp_\cl P.)&\subset\set\omega\in\wp(\cl E.),
{\t\cl E._1\subset\ker\omega}.=\wp(\t\cl E._2)\;.\cr}$$
\item{Remark:} Thus a regular state on $\crd$ is characterised by
having a canonical extension which is a state on $\cg$
(equivalently, which vanishes on $\t\cl E._2$). "
{\bf (1)} The first two statements are direct applications of
Dixmier
2.11.7 [8], which states that: \chop
If $\cl I.$ is a closed two--sided ideal of the C*--algebra $\cl A.$
and if $f$ is a positive form on $\cl A.$, then there is a unique
decomposition $f=f_1+f_2$ where $f_i$ are positive forms on
$\cl A.$ with $\|f_1\|={\|f_1\uprightharpoon\cl I.\|}$
and $f_2(\cl I.)=0$ and the pair $(\pi_f,\,\xi_f)$ can be identified
with ${(\pi_{f_1}\oplus\pi_{f_2},\,\xi_{f_1}+\xi_{f_2})}$ where
$\cl I.\subset{\rm Ker}\,\pi_{f_2}$ and $\pi_{f_1}\uprightharpoon
\cl I.$ is nondegenerate. Then the property $\|f_1\|={\|f_1
\uprightharpoon\cl I.\|}$ shows that there is a unique
extension of a state on $\cl I.$ to a state on $\cl A.$. The
rest follows from this and the fact that
$\t\cl E.=\t\cl E._1\oplus\t\cl E._2$ and $\cg$ is an ideal
of $\cl E.$.\chop
Also recall that $\t\cl E._i\subset\ker\omega_j$ iff
$\t\cl E._i\subset\ker\pi\s\omega_j.$, $i\not= j$ and
$\cg\subset\ker\omega_2$ iff $\cg\subset\ker\pi\s\omega_2.$
by Dixmier 2.4.10. Let $A\in\t\cl E.$, so $A=A_1+A_2$,
$A_i\in\t\cl E._i$, $PA=A_1$, $(\un-P)A=A_2$. Hence\chop
$\t\omega(A)=\lambda\omega_1(A)+(1-\lambda)\omega_2(A)=
\lambda\omega_1(A_1)+(1-\lambda)\omega_2(A_2)$\chop
and so $\t\omega(PA)=\t\omega(A_1)=\lambda\omega_1(A_1)
=\lambda\omega_1(A)$ and similarly\chop
$\t\omega((\un-P)A)=(1-\lambda)\omega_2(A)$.\chop
{\bf (2)} That $\kappa$ and $\t\kappa$ are injective
follows from the uniqueness of the expression for
$\t\omega$. That it preserves pure states follows from the fact
that $\t\pi$ is irreducible if $\pi$ is (cf. 2.4(3)),
and the uniqueness of the equivalence class of the GNS--representation
of $\t\omega$. To see that $\kappa$ and $\t\kappa$ are
w*--continuous, let $\{\omega_\beta\}$ be a net in
$\wp(\crd)$ w*--converging to $\omega$, i.e.
$\big|\omega_\beta(A)-\omega(A)\big|\to 0$ as $\beta\to\infty$
for all $A\in\crd$. Then we need to show that
$\big|\kappa(\omega_\beta)(A)-\kappa(\omega)(A)\big|\to 0$
as $\beta\to\infty$ for all $A\in\cl E.$.
Now $\cl E.={\overline{L(\crd\cup\cg)}}$, and since we already
have convergence on $\crd$, we only need to show convergence
on $\cg$. For all $f\in L^1_\rho(G)$:
$$\eqalignno{\big|\kappa(\omega_\beta)(f)-\kappa &(\omega)(f)\big|
=\big|\int f(g)\,\big(\omega_\beta(\delta_g)-\omega(\delta_g)\big)
\, d\mu(g)\,\big| \cr
&\leq\int|f(g)|\cdot\big|\omega_\beta(\delta_g)-\omega(\delta_g)\big|
\, d\mu(g)\longrightarrow 0 \cr}$$
as $\beta\to\infty$, using the Lebesgue dominated convergence
theorem together with ${|\omega_\beta(\delta_g)-\omega(\delta_g)|}
\to 0$ as $\beta\to\infty$, and\chop
$\big|\omega_\beta(\delta_g)-\omega(\delta_g)\big|\leq
\|\delta_g\|+\|\delta_g\|=2$ which is $L^1$ for the measure
${|f(g)|\, d\mu(g)}$. \chop
As for $\t\kappa$, observe that $\t\omega(P)=\lambda $ (same
$\lambda$ as in part 1), so w*--continuity follows from that
on $\cg$. Any state on $\cg$ extends uniquely to a state on
$\cl E.$, so any $\omega\in\wp(\cl E.)$ which is a state on
$\cg$ is uniquely determined by its values on $\cg$, and this establishes
the equality between $\wp(\cg)$ and the set of states on $\cl E.$
restricting to states on $\cg$ (cf. Dixmier 2.11.8 also).
Now from part (1) we have inclusions $\kappa(\wp_\cl P.)\subseteq
\wp(\cg)$, $\kappa(\wp_\cl P.^\perp)\subseteq\set\omega\in
\wp(\cl E.),\cg\subset\ker\omega.$, $\t\kappa(\wp_\cl P.)\subseteq
\wp(\t\cl E._1)$, $\t\kappa(\wp_\cl P.^\perp)\subseteq
\wp(\t\cl E._2)$. For the reverse inclusion of the first inclusion,
recall that every nondegenerate representation of $\cg$ is obtained
from one on $\crd$ via the relation $\pi(f)={\int f(g)\,\pi(\delta_g)
\, d\mu(g)}$. In particular, this is true for cyclic representations,
and hence every state on $\cg$ is obtained from one on $\crd$
via the canonical extension. Thus $\kappa(\wp_\cl P.)=
\wp(\cg)$. For reverse inclusion of the second, let
$\omega\in\wp(\cl E.)$ with $\cg\subset\ker\omega$, then
since $\cl E.={\overline{L(\cg\cup\crd)}}$, $\omega$
is uniquely determined by its values on $\crd$, so $\omega$
is in the range of $\kappa$.
\medskip Next we wish to show that the problem of pointwise norm
discontinuous actions on $\ccr S,B.=C^*_\sigma(S_d)$ defined from
symplectic actions $\beta:\cl G.\to\Sp(S,\,B)$ where $\cl G.$
is a nondiscrete topological group, can also be circumvented
by working with the algebra $C_\sigma^*(S)$ instead of
$\ccr S,B.$.
Let $\theta\in{\rm Aut}_\rho G:=\set\theta\in\aut G,\rho\big(\theta(g),\,
\theta(h)\big)=\rho(g,\,h)\;\forall\, g,\;h\in G.$
be a Borel automorphism where $G$ and
$\rho$ are as before. Denote the Haar measure of $G$ by $\mu$.
Then $\mu\circ\theta$ is again a left invariant measure of $G$
(on the same $\sigma\hbox{--algebra}$), hence by uniqueness of the
Haar measure there is a constant $C_\theta\in\r_+$ such that
$\mu\circ\theta=C_\theta\,\mu$. Denote
${\rm Aut}_\rho^BG:=\set\theta\in{\rm Aut}_\rho G, \theta\;\;\hbox{
a Borel automorphism}.$.
We endow ${\rm Aut}_\rho^BG$ with the topology of pointwise
sequential convergence, i.e. a sequence $\{\theta_n\}\subset
{\rm Aut}_\rho^BG$ converges to $\theta\in\aut G$ if
$\theta_n(x)\to\theta(x)$ for each $x\in G$.
If one assumes the stronger assumption of
joint continuity of $\rho$, we see that ${\rm Aut}_\rho^BG$
is closed in the topology above.
\def\wt{{\widetilde\theta}}
\thrm Theorem 2.6." $(\rn1)$ Each $\theta\in{\rm Aut}_\rho^BG$ defines
an automorphism $\widetilde\theta\in\aut M(\cg)$ which preserves
$\cg$ and $\crd$ (hence $\cl E.$) and is given on $L^1_\rho(G)$ by
$(\widetilde\theta f)(t)=f(\theta^{-1}t)\cdot C_{\theta^{-1}}$
and on $\crd$ by $\widetilde\theta(\delta_t)=\delta\s\theta t.$
$\forall\,t\in G$.\chop
$(\rn2)$ Endow ${\rm Aut}_\rho^BG$ with the
topology above,
and give $\aut\cg$
the pointwise norm topology. Then the map
${\widetilde{}:{\rm Aut}_\rho^BG\to\aut\cg}$ is continuous.
(Hence any continuous homomorphism $\beta:\cl G.\to{\rm Aut}_\rho^BG$
produces a pointwise norm continuous action $\widetilde\beta:
\cl G.\to\aut\cg$)."
$(\rn1)$ First we show that the definition $(\widetilde\theta f)(t):=
C\s\theta^{-1}.f(\theta^{-1}t)$ produces an automorphism of
$L^1_\rho(G)$. It is only necessary to check that $\widetilde\theta$
is a homomorphism under the operations of 2.2, as it is obviously linear,
invertible and preserves the $L^1\hbox{--norm}$. Since
$$\displaylines{(\mu\circ\theta)(Rx)=\mu\big(\theta(R)\theta(x)\big)
=C_\theta\Delta(\theta(x))\,\mu(R)\cr
=C_\theta\mu(Rx)=C_\theta\Delta(x)\,\mu(R)\cr}$$
for all Borel sets $R$, we see $\Delta(\theta(x))=\Delta(x)$. Now
$$\eqalignno{\big(\widetilde\theta(f^*)\big)(x)&=C\s\theta^{-1}.
f^*(\theta^{-1}x)\cr
&=C\s\theta^{-1}.\Delta(\theta^{-1}x)^{-1}\,
\overline{\rho(\theta^{-1}(x),\,\theta^{-1}(x^{-1}))}\,\,
\overline{f(\theta^{-1}(x^{-1}))}\cr
&=C\s\theta^{-1}.\delta(x)^{-1}\overline{\rho(x,\,x^{-1})}\,\,
\overline{f(\theta^{-1}(x^{-1}))}\cr
&=\Delta(x)^{-1}\,\overline{\rho(x,\,x^{-1})}\,\,\overline{
(\widetilde\theta f)(x^{-1})}=(\widetilde\theta(f))^*(x)\qquad\hbox{and}\cr
\big(\widetilde\theta(f*g)\big)(x)&=C\s\theta^{-1}.(f*g)(\theta^{-1}(x))\cr
&=C\s\theta^{-1}.\int_Gf(y)\,g\big(y^{-1}\theta^{-1}(x)\big)\,
\rho(y,\,y^{-1}\theta^{-1}(x))\,d\mu(y)\cr
&=C\s\theta^{-1}.\int_Gf(\theta^{-1}(y))\,g(\theta^{-1}(y^{-1}x))\,
\rho(y,\,y^{-1}x)\,d\mu(\theta^{-1}(y))\cr
&=C^2\s\theta^{-1}.\int_Gf(\theta^{-1}(y))\,g(\theta^{-1}(y^{-1}x)\,
\rho(y,\,y^{-1}x)\,d\mu(y)\cr
&=\int_G(\widetilde\theta f)(y)\,\,(\widetilde\theta g)(y^{-1}x)\,\,
\rho(y,\,y^{-1}x)\,d\mu(y)=\big[(\widetilde\theta f)*(\widetilde\theta g)
\big](x)\cr}$$
for all $f,\;g\in L^1_\rho(G)$, $x\in G$, $\theta\in{\rm Aut}_\rho^BG$.
Hence $\widetilde\theta\in\aut L^1_\rho(G)$. Then it follows from
Dixmier 1.3.7 [8] that $\widetilde\theta$ is also continuous in the
C*--norm, hence it extends uniquely to an automorphism
$\widetilde\theta\in\aut\cg$, and also by the usual arguments to
$M(\cg)$. We check that $\widetilde\theta$ preserves $\crd$ and that
it is given by the formulae in $(\rn1)$. By 2.3:
$$\eqalignno{\wt(\delta_y*f)(x)&=C\s\theta^{-1}.(\delta_y*f)
(\theta^{-1}x)=C\s\theta^{-1}.\,f(y^{-1}\theta^{-1}(x))\,
\rho(y,\,y^{-1}\theta^{-1}(x))\cr
&=(\wt f)(\theta(y^{-1})x)\,\rho(\theta(y),\,\theta(y^{-1})x)
=\big[\delta\s\theta y.*\wt(f)\big](x)\cr}$$
and hence $\wt(\delta_y)=\delta\s\theta y.$ $\forall\,y\in G$
which proves that $\wt$ preserves $\crd$.
\chop\chop
$(\rn2)$ Under the canonical bijection between regular representations
\chop $\pi\in\cl P.$ on $\crd$ and nondegenerate representations of
$\cg$, we have $\pi(f)=\int\limits_G f(x)\,\pi(\delta_x)\,d\mu(x)$
for all $f\in L^1_\rho(G)$, and all nondegenerate representations
of $L^1_\rho(G)$ are of this form. Let $\{\theta_n\}\subset
{\rm Aut}_\rho^BG$ be a sequence converging pointwise to
$\theta\in{\rm Aut}_\rho^BG$, then if $\pi$ and $\Omega$ are the
GNS--representation and cyclic vector respectively of a regular state
$\omega$,
$$\eqalignno{\big\|\pi(\wt_n&(f)-\wt(f))\Omega\big\|
=\Big\|\int_G\left[(\wt_n f)(x)-(\wt f)(x)\right]\,\pi(\delta_x)
\,d\mu(x)\,\Omega\Big\|\cr
&=\Big\|\int_G\left[C\s\theta_n^{-1}.f(\theta_n^{-1}(x)
)-C\s\theta^{-1}.f(\theta^{-1}(x))\right]\,\pi(\delta_x)\,d\mu(x)\,\Omega
\Big\|\cr
&=\Big\|\Big[\int_Gf(\theta_n^{-1}(x))\,\pi(\delta_x)\,d\mu(
\theta_n^{-1}(x))-
\int_Gf(\theta^{-1}(x))\,\pi(\delta_x)\,d\mu(\theta^{-1}(x))\Big]\Omega
\Big\|\cr
&=\Big\|\int_G\big(\pi(\delta\s\theta_n(x).)
-\pi(\delta\s\theta(x).)\big)\,f(x)\,d\mu(x)\,\Omega\Big\|\cr
&\leq\int_G|f(x)|\cdot\big\|\big(\pi(\delta\s\theta_n(x).)
-\pi(\delta\s\theta(x).)\big)\Omega\big\|\,d\mu(x)\cr
&\leq\|f\|_1\cdot{\rm sup}\set\big\|(\pi(\delta\s\theta_n(x).)
-\pi(\delta\s\theta(x).))\Omega\big\|,x\in G.\longrightarrow 0\cr}$$
as $n\to\infty$ using the strong operator continuity of
$\pi(\delta_x)$. Hence
$$\omega\left(\big(\wt_n(f)-\wt(f)\big)^*\big(\wt_n(f)
-\wt(f)\big)\right)\longrightarrow 0\qquad\hbox{as}\qquad n\to
\infty$$
for all $f\in L^1_\rho(G)$, $\omega\in\wp(\cg)$ using 2.5
(where $\wp$ denotes the state space of its argument).
However by Dixmier 2.7.1 [8], this implies that
${\|\wt_n(f)-\wt(f)\|}\to 0$ as $n\to 0$ in the C*--norm
of $\cg$, which completes the proof of $(\rn2)$.
\medskip\parindent=.6 true in
\itemitem{{\bf Corollary 2.6}} If $G=S=\r^n\times\r^n$, endow
$\Sp(S,\,B)$ with the matrix topology it inherits from
its inclusion in ${\rm M}^{2n}(\r)$, then any continuous
homomorphism $\beta:\cl G.\to\Sp(S,\,B)$ produces a
pointwise norm continuous action $\widetilde\beta:\cl G.
\to\aut C_\sigma^*(S)$.\hfill\tomb\parindent=20pt
\noindent Next we wish to extend this structure to deal with the
infinite dimensional case, and also to include partially
regular representations into the framework.
\def\rep#1,{{{\rm Rep}\,#1}}
\def\wt{\widetilde}
\def\ilim{\displaystyle{\mathop{{\rm lim}}_{\mathord{\longrightarrow}}}\,}
\beginsection 3. The Infinite Dimensional Case; Regularity of Representations.
If we relax the requirement that ${\rm dim}\,S<\infty$, then
$S$ is not locally compact and the structure of the preceding
section fails. However, there are many ways of writing the symplectic
space $S$ as an inductive limit of finite dimensional subspaces,
and for each of these finite dimensional spaces we can still use the
preceding theory. To exploit this, we assume that $G$ is written as
an inductive limit of closed nondiscrete locally compact subgroups,
$G=\ilim G_\beta$, $\beta\in\Gamma$ where $\Gamma$ is a directed set
and each $G_\beta$ is locally compact.
This covers the cases where $G$ is $S$ (a separable symplectic space),
$\Sp(\infty),\ab\;U(\infty),\ab\;O(\infty)$. Let $\rho\in
{Z^2(G,\,{\bf T})}$ be a two--cocycle such that the restriction of
$\rho$ to any $G_\beta$ is Borel, and on each $G_\beta$ the maps
$g\to\rho(g,\,x)$ are continuous for
all $x$. Consider the monomorphisms
of the inductive limit as inclusions. Where different inductive
systems are available, we choose the finest nondiscrete one,
e.g. for $S$ the inductive system of all its finite dimensional
subspaces. Henceforth the choice of inductive system remains fixed.
\def\cg#1.{{C^*_\rho(G_{#1})}}
The aim of this section is to obtain an adequate group C*--algebra
for an inductive limit group which is not discrete.
More precisely, we want a non--unital C*--algebra $\cl L.$
such that:
\item{(\rn1)} its representation space consists of precisely the
representations of $G$ which are weak--operator continuous,
\item{(\rn2)} there is a homomorphism $\varphi:\cg d.\to M(\cl L.)$
such that given a representation $\pi\in\rep\cl L.,$ with unique
extension $\hat\pi$ to $M(\cl L.)$, then
$$\overline\pi(A):=\hat\pi(\varphi(A))\qquad\forall\, A\in \cg d.$$
defines a regular representation $\overline\pi$ on $\cg d.$, and conversely,
every regular representation is obtained this way.
(Thus if there are faithful regular representations, $\varphi$
is injective).
\item{(\rn3)} when $G$ is locally compact, $\cl L.=\cg .$.
\noindent We do not address here the existence question of
regular $\rho\hbox{--representations}$ for non--locally
compact groups.
\vfill\eject
Recall that we have for $G_\beta$ the
(isometric) imbeddings $\cg\beta d.\subset
M(\cg\beta.)$ and $\cg\delta.\subset M(\cg\beta.)$ if $\delta\leq\beta
\in\Gamma$ where $G_{\beta d}$ denotes $G_\beta$ with the discrete
topology. Simplify notation: $\cl N._\beta:=\cg\beta.$, $\cl N._\beta^d
:=\cg\beta d.$, $\cl E._\beta:=C^*(\cl N._\beta\cup\cl N._\beta^d)$.
and we denote the imbeddings above by:
${i_{\delta\beta}:\cl N._\delta\to M(\cl N._\beta)}$,
${j_{\delta\beta}:\cl N._\delta^d\to M(\cl N._\beta)}$ $\forall\,\delta
\leq\beta\in\Gamma$. Clearly $\cl N._\delta^d\subseteq\cl N._\beta^d$
if $\delta\leq\beta$. Define:
$$\eqalignno{\cl A._\beta&:=C^*\big(\set i_{\delta\beta}(\cl N._\delta),
\delta\leq\beta.\big)\subset M(\cl N._\beta)\cr
\cl M._\beta&:=C^*\left(\cl A._\beta\cup\cl N._\beta^d\right)
\subset M(\cl N._\beta)\cr}$$
\thrm Theorem 3.1." Assume the preceding structures and notation. Then\chop
$(\rn1)$ there are injective homomorphisms:
$$\varphi_{\delta\beta}:\cl A._\delta\to\cl A._\beta\;,\qquad
\psi_{\delta\beta}:\cl M._\delta\to\cl M._\beta\quad
\forall\,\delta\leq\beta$$
such that $\psi\s\delta\beta.\,\big|\,\cl A._\delta=\varphi\s\delta\beta.$,
$\psi_{\delta\beta}$ is unital and
$$\varphi\s\beta\gamma.\circ\varphi\s\delta\beta.=\varphi\s\delta\gamma.\;,
\qquad\psi\s\beta\gamma.\circ\psi\s\delta\beta.=\psi\s\delta\gamma.\quad
\forall\,\delta\leq\beta\leq\gamma\in\Gamma\,.$$
$(\rn2)$ Using the homomorphisms above, the inductive limit
C*--algebra $\cl M.:=\ilim\cl M._\beta$ exists and contains
$\cl A.:=\ilim\cl A._\beta$.
Moreover, if no $G_\beta$ is discrete, $\un\in\cl M.\backslash\cl A.$."
$(\rn1)$ Now $\cl A._\delta\subset\cl M._\delta\subset M(\cl N._\delta)
\subset\cl N._\delta''$, and we have the imbedding
$i\s\delta\beta.:\cl N._\delta\to M(\cl N._\beta)\subset\cl N._\beta''$,
$\delta\leq\beta$ which extends canonically to the multiplier algebra:
$$i\s\delta\beta.:M(\cl N._\delta)\to M(i\s\delta\beta.(\cl N._\delta))
\subset[i\s\delta\beta.(\cl N._\delta)]''\subset\cl N._\beta''$$
for $\delta\leq\beta$. Hence if we take $\psi\s\delta\beta.=i\s\delta\beta.\,
\big|\,\cl M._\delta$ and $\varphi\s\delta\beta.={\psi\s\delta\beta.\,\big|
\,\cl A._\delta}$ it is only necessary to show that
$i\s\beta\gamma.\circ i\s\delta\beta.=i\s\delta\gamma.$ and
$i\s\delta\beta.(\cl M._\delta)\subset M(\cl N._\beta)$ to prove
3.1(\rn1). Both of these will be proven if we can show that
$$\big[i\s\beta\gamma.(i\s\delta\beta.(f)h)g\big](x)=
\big[i\s\delta\gamma.(f)\,i\s\beta\gamma.(h)\,g\big](x)$$
for all $\delta\leq\beta\leq\gamma$, $x\in G_\gamma$, $g\in L^1(G_\gamma)$,
$h\in L^1(G_\beta)$, $f\in L^1(G_\delta)$ to ensure that
$i\s\delta\gamma.(\cl A._\delta)\subset M(\cl N._\gamma)$
together with an even simpler calculation to show that
$i\s\delta\gamma.(\cl M._\delta)\subset M(\cl N._\gamma)$.
We only do the first calculation:
$$\eqalignno{\big[i\s\beta\gamma.(i\s\delta\beta.(f)&\,h)g\big](x)=
\int_{G_\gamma}[i\s\delta\beta.(f)\,h](y)\,g(y^{-1}x)\,
\rho(y,\,y^{-1}x)\,d\mu_\gamma(y)\cr
&=\int_{G_\gamma}\!\int_{G_\delta}f(z)\,h(z^{-1}y)\,\rho(z,\,z^{-1}y)\,
d\mu_\delta(z)\,\,g(y^{-1}x)\,\rho(y,\,y^{-1}x)\,d\mu_\gamma(y)\cr &
&-(*)\cr
\big[i\s\delta\gamma.(f)\,i\s\beta\gamma.(h)&\,g\big](x)
=\int_{G_\delta}f(z)\,[i\s\beta\gamma.(h)\,g](z^{-1}x)\,\rho(z,\,z^{-1}x)\,
d\mu_\delta(z)\cr
&=\int_{G_\delta}f(z)\int_{G_\gamma}h(y)\,g(y^{-1}z^{-1}x)\,
\rho(y,\,y^{-1}z^{-1}x)\,d\mu_\gamma(y)\,\rho(z,\,z^{-1}x)\,
d\mu_\delta(z)\cr
&=\int_{G_\delta}f(z)\int_{G_\gamma}h(z^{-1}y)\,g(y^{-1}x)\,
\rho(z^{-1}y,\,y^{-1}x)\,d\mu_\gamma(y)\,\rho(z,\,z^{-1}x)\,
d\mu_\delta(z)\cr}$$
and this is equal to $(*)$ by virtue of Fubini's theorem and the
two cocycle identity:${\rho(a,\,b)\,\rho(ab,\,c)}
={\rho(a,\,bc)\,\rho(b,\,c)}$.\chop
$(\rn2)$ The existence and inclusion of the inductive limits
follow from $(\rn1)$ and Takeda's criterion [12].
The statement about the identity follows from
$\un\in\cl N._\beta^d$ $\forall\,\beta\in\Gamma$ and the lemma
3.1.3 of Blackadar [13] using $\un\not\in\cl A._\beta\;\forall\,\beta$
(since $\cl A._\beta\subset\ker\pi$ for any irreducible nonregular
representation $\pi$ of $\cl N._\beta^d$).
\itemitem{{\bf Remarks}} $(\rn1)$ Clearly $\ilim\cl N._\beta^d=
C^*_\rho(G_d)\subset\cl M.$ and hence in the case when
$G=S$, we see that $\ccr S,B.\subset\cl M.$ if we set
$\rho=\sigma=\exp\f i,2.B(\cdot,\cdot)$.\chop
$(\rn2)$ For the case of the CCRs, the algebra $\cl A.$ was considered
by Segal [16]
as a CCR field algebra.
The algebra $\cl A.$ is not a suitable infinite
dimensional analogue of the algebra $C^*_\rho(G)$ of the finite
dimensional case, because it has many representations which
are not regular.
The auxiliary C*--algebra of
Schaflitzel [2] is the algebra generated by
the identity and
the algebras $C^*_\sigma(S_\alpha)\subset\cl M.$
where $S_\alpha$ ranges over all
one--dimensional subspaces of $S$,
and it is the one proposed by Kastler [19] as
the proper field algebra for the CCRs.
More recently, Rieffel has defined a large C*--algebra for
the CCRs, containing $\ccr S,B.$ [25].
\chop
$(\rn3)$ Since $\set\cl N._\beta,\beta\in\Gamma.$ generates
$\cl A.$, a state or representation of $\cl A.$ is uniquely
determined by its restriction to these.\chop
$(\rn4)$ Since $\un_{\cl M.}\in C^*_\rho(G_{\beta d})=\cl N._\beta^d$
$\forall\,\beta\in\Gamma$, any nondegenerate representation of
$\cl M.$ restricts to a nondegenerate representation of
$\cl N._\beta^d$.\chop
$(\rn5)$ The measure algebras $\cl M.(G_\alpha)$ also form
an inductive system in the obvious way (if $\alpha<\beta$, imbed
$\mu\in\cl M.(G_\alpha)$ in $\cl M.(G_\beta)$ as a measure
on $G_\beta$ with support in $G_\alpha$), and so one can define
the inductive limit algebra $\cl M._\rho(G):=\ilim\cl M.(G_\alpha)$.
Kastler [19] used this twisted measure algebra
as a ``big field algebra,'' since it is a natural setting
in which to examine representation questions for $G$.
In the present case of $G$ being non--locally compact,
there is clearly no $L_\rho^1(G)$ algebra in $\cl M._\rho(G)$,
although the discrete algebra $\ell_\rho^1(G)=L_\rho^1(G_d)$
is still in $\cl M._\rho(G)$, as are all $L_\rho^1(G_\alpha)$,
$\alpha\in\Gamma$. As in the locally compact case, given a representation
$\pi\in{\rm Rep}_\rho^0G$, Borel on each $G_\alpha$,
we can still define a representation $\wt\pi$ of $\cl M._\rho(G)$
by
$$\wt\pi(\nu)=\int_G\pi(g)\, d\nu(g)\qquad\forall\,\nu\in\cl M.(G_\alpha)\,,
\;\;\alpha\in\Gamma$$
which restricts to $\pi(g)$ on each $\delta_g\in L_\rho^1(G_d)$.
That the representations defined on each $C^*_\rho(G_\alpha)$
by this formula piece together into a well--defined representation
for $\cl M._\rho(G)$, follows from the fact that it is
well--defined for each $\cl M._\alpha$, and then invoking the
inductive limit structure of $\cl M._\rho(G)$.
Thus a Borel representation on $\cg d.$ has an extension
$\wt\pi$ on the same space to $\cl M._\rho(G)$, hence to
$\cl M.$, using the fact that $\cl M.$ is generated
by $L_\rho^1(G_\alpha)$, $\alpha\in\Gamma$ and $\cg d.$.
In this non--locally compact framework, we will call
this $\wt\pi$ the {\it canonical extension} of $\pi$.
It also extends uniquely to $\cl M.''$.
\chop
Next we would like to define a canonical extension for a
representation $\pi\in\rep\cg d.,$ which is not Borel on all
$G_\alpha$. Proceed as in Theorem 2.4(3); first define
$$\cl H._B:=\set\psi\in\cl H._\pi,g\in G_\alpha\to\pi(\delta_g)\psi
\in{\cl H._\pi}\;\;
\hbox{is Borel $\forall\,\alpha$}.\,.$$
Since sums and sequential limits of Borel functions are Borel,
$\cl H._B$ is a closed linear space which is clearly
preserved by $\cg d.$.
Thus $\pi$ restricts on $\cl H._B$ to a Borel representation
of $\cg d.$, hence has a canonical extension on $\cl H._B$
to $\cl M.$.\chop
On $(\cl H._B)^\perp$ we set $\wt\pi(f)\,\psi=0$ $\forall\, f\in
L_\rho^1(G_\alpha)$, $\alpha\in\Gamma$, $\psi\in(\cl H._B)^\perp$.
This clearly defines a representation of $\cl M.$ on $\cl H.$,
and it is also called {\it canonical}.\chop
(Note that the restriction of this canonical extension of $\pi$
to $\cg\alpha.$ need not be the canonical extension of
$\pi\,\big|\,\cg\alpha d.$ in Theorem 2.4 (3\rn2), because
$\cl H._B$ may be smaller here, due to conditions on
$\psi\in\cl H._B$ from $G_\beta$, $\beta\not=\alpha$).
\def\t{\widetilde} \def\rep{{\rm Rep}\,}
Using the canonical extension of representations,
we also have a canonical
extension of a state $\omega$ on $\cg d.$ to $\cl M.$, given by
$$\t\omega(A)=(\Omega_\omega,\,\t\pi_\omega(A)\Omega_\omega)\quad
\forall\, A\in\cl M..$$
However, since $\un\in\cg d.$, we know from the Stone--Weierstrass
theorem that this extension of $\omega$ must be nonunique
for some $\omega$.
\thrm Theorem 3.2."Given the hypotheses of theorem 3.1,\chop
$(\rn1)$ A nondegenerate representation
$\pi:\cl A.\to\cl B.(\cl H.)$ extends uniquely to a
representation $\widetilde\pi:\cl M.\to\cl B.(\cl H.)$ (on the
same space).
\chop
$(\rn2)$ The state $\omega_0$ of $\cg d.$ defined by
$$\omega_0(\delta_x)=\cases{1& if $x=e$\cr 0& if $x\not=e$\cr}$$
is absolutely irregular, in the sense that the canonical
extension of its GNS--representation vanishes on all
$\cl N._\alpha=\cg\alpha.$,\chop $\alpha\in\Gamma$.
\chop
$(\rn3)$ $\cl A.$ is contained in a proper closed two--sided ideal
of $\cl M.$."
{\bf (\rn1) }
Given a nondegenerate representation $\pi:\cl A.\to\cl B.(\cl H.)$,
denote $\cl H._\beta:=\big[\pi(\cl N._\beta)\,\cl H.
\big]$ for $\beta\in\Gamma$ (we use $[\cdot]$ to denote the closed
linear space of its argument). Then $\pi:\cl N._\beta\to\cl B.(\cl H._\beta)$
is a nondegenerate representation, hence
extends uniquely
to a representation $\pi:M(\cl N._\beta)\to\cl B.(\cl H._\beta)$
hence to $\pi:\cl N._\beta^d\to\cl B.(\cl H._\beta)$.
That is, $\pi\,\big|\,\cl N._\beta$ acting on $\cl H.={[\pi(\cl N._\beta)
\cl H.]\oplus[\pi(\cl N._\beta)\cl H.]^\perp}$ determines a unique (degenerate)
representation $\pi$ of $\cl N._\beta^d$ on $\cl H.$ with the same
essential subspace $[\pi(\cl N._\beta)\cl H.]$.
Since $\cl N._\beta^d\subseteq\cl N._\alpha^d$ if $\beta\leq\alpha$,
it is only necessary to check that these representations piece
together to produce a representation of $C^*_\rho(G_d)$.
Given $\beta\leq\alpha$, $\cl N._\beta\subset M(\cl N._\alpha)$
we need to show that the $\t\pi$ obtained from $\cl N._\alpha$
on $\cl N._\alpha^d$ agrees on $\cl N.^d_\beta\subset\cl N._\alpha^d$
with the $\t{\t\pi}$ obtained on $\cl N._\alpha^d$ from
$\pi$ on $\cl N._\beta$.
Let $\{u_\gamma\}$ be an approximate identity in $\cl N._\alpha$,
then for all $\psi\in\cl H.$, $A\in\cl N._\alpha^d$:
$$\t\pi(A)\,\psi=\lim_\gamma\pi(Au_\gamma)\,\psi=
\lim_\gamma\t\pi(A)\,\pi(u_\gamma)\,\psi$$
and likewise, if $\{v_\gamma\}$ is an approximate identity
in $\cl N._\beta$, then for all $A\in\cl N._\beta^d$, $\psi\in\cl H.$
we have $\t{\t\pi}(A)\,\psi=\lim\limits_\gamma\t{\t\pi}(A)
\pi(v_\gamma)\,\psi$.
Moreover $\cl N._\beta\subset M(\cl N._\alpha)$ so $\pi$ extends
uniquely on its essential subspace from $\cl N._\alpha$ to
$\cl N._\beta$. So for $A\in\cl N._\beta^d$,
we find that for all $\psi\in\cl H.$:
$$\eqalignno{\t\pi(A)\,\psi&=\lim_\nu\pi(Au_\nu)\,\psi
=\lim_\nu\,\lim_\gamma\pi(Au_\nu)\,\pi(v_\gamma)\,\psi \cr
&=\lim_\nu\,\lim_\gamma\pi(Au_\nu v_\gamma)\psi
=\lim_\nu\,\lim_\gamma\t{\t\pi}(A)\,\pi(u_\nu v_\gamma)\,\psi \cr
&=\t{\t\pi}(A)\,\psi\;.\cr}$$
Thus $\pi$ is uniquely determined on $\cl M._\beta=C^*(\cl A._\beta
\cup\cl N._\beta^d)$, respects the monomorphisms of the inductive
limit, and so determines uniquely a nondegenerate representation
$\widetilde\pi$ of $\cl M.=\ilim\cl M._\beta$ on $\cl H.=\ilim\cl H._\beta$.
\chop
{\bf (\rn2)} That the formula defines a state is standard.
Given the GNS--representation $(\pi\s\omega_0.,\,\Omega\s\omega_0.,\,
\cl H.\s\omega_0.)$ of $\omega_0$,
observe that the set\chop $\set\pi\s\omega_0.(\delta_x)\Omega\s\omega_0.,
x\in G.$ is an orthonormal basis of $\cl H.\s\omega_0.$
(so it is nonseparable), and that each basis vector
$\psi_x:=\pi\s\omega_0.(\delta_x)\Omega\s\omega_0.$
is cyclic for $\pi\s\omega_0.(\cg d.)$.
Let $\{y_\gamma\}\subset G$ be any net converging to $e$.
Then for each pair $x,\, z\in G$ we have
$$\displaylines{(\psi_z,\,\pi\s\omega_0.(\delta\s y_\gamma.)\psi_x)=0\qquad
\forall\, y_\gamma\not=zx^{-1}\,,\cr
\hbox{and thus:}\qquad\qquad\qquad
\lim_\gamma\pi\s\omega_0.(\delta\s y_\gamma.)\,\psi_x=0\quad
\forall\, x\in G\,, \qquad\qquad\quad{(1)}\cr}$$
using coincidence of weak and strong operator topologies on
the unitaries. Now recall that for each $G_\alpha$ we
have a decomposition
$$\pi\s\omega_0.\,\big|\,\cg\alpha d.=\pi^\alpha_R\oplus\pi^\alpha_N$$
into $G_\alpha\hbox{--regular}$ and irregular parts.
So for each $x\in G$, write $\psi_x=\psi^\alpha_R+\psi^\alpha_N$
according to this decomposition, and we know that for any net
$\{y_\gamma\}\subset G_\alpha$ converging to $e$ we
must have $\lim\limits_\gamma\pi\s\omega_0.(\delta\s y_\gamma.)\,\psi_R^\alpha
=\psi_R^\alpha$. Hence
$$\eqalignno{-\psi_R^\alpha&=\lim_\gamma\pi\s\omega_0.(\delta\s y_\gamma.)
(\psi_x-\psi_R^\alpha)&\hbox{(using $(1)$)}\cr
&=\lim_\gamma\pi\s\omega_0.(\delta\s y_\gamma.)\,\psi_N^\alpha
&(2)\cr}$$
(thus also establishing that the last limit exists).
However $\pi\s\omega_0.(\delta\s G_\alpha.)$ preserves $\cl H._N^\alpha$,
hence we have in $(2)$ an equality of orthogonal vectors, which
therefore must be zero. Thus $\psi_R^\alpha=0$, so
$\psi_x$ has no $G_\alpha\hbox{--regular}$ part, i.e.
is orthogonal to the space
$$\eqalignno{\cl H._R^\alpha&:=\set\psi\in\cl H.\s\omega_0.,
g\in G_\alpha\to{\pi\s\omega_0.(\delta_g)\,\psi\in\cl H.\s\omega_0.}
\hbox{$\quad$ is continuous}.\cr
&\supseteq\set\psi\in\cl H.\s\omega_0.,g\in G\to{\pi\s\omega_0.(\delta_g)
\,\psi\quad}\hbox{is continuous}.\cr
&=\bigcap_{\alpha\in\Gamma}\cl H._R^\alpha=\cl H._R\;.\cr}$$
Thus the whole basis $\set\psi_x,x\in G.$ is orthogonal to $\cl H._R^\alpha$,
and so $\cl H._R^\alpha=\{0\}$ for all $\alpha\in\Gamma$,
i.e. $\pi\s\omega_0.\,\big|\,\cg\alpha d.$ is $G_\alpha\hbox{--irregular}$,
so $\cg\alpha.\subset\ker\wt\pi\s\omega_0.$ for all $\alpha$.
Since $\pi\s\omega_0.$ is a Borel representation, $\cl H._B=\cl H.\s\omega_0.$
and so the canonical extension of $\pi\s\omega_0.$ coincides on each
$\cg\alpha.$ with the canonical extension of $\pi\s\omega_0.
\,\big|\,\cg\alpha d.$ defined in Theorem 2.4(3\rn3).
Thus $\cg\alpha.\subset\ker\wt\pi\s\omega_0.$ for all $\alpha\in\Gamma$.\chop
{\bf (\rn3)} This follows from $(\rn2)$ since $\ker\wt\pi\s\omega_0.$
is an ideal.
\itemitem{{\bf Remark}} The ideal generated by $\cl A.$
in $\cl M.$ is $[\cl MAM.]$ and so since $\un\in\cl M.$,
3.2(\rn3) guarantees that $[\cl MAM.]$ is proper. In fact,
if $G=S$, $\rho=\sigma$ nondegenerate, we know that the CCR--algebra
$\ccr S,B.=C^*_\sigma(S_d)$ is simple,
hence $C^*_\sigma(S_d)\cap[\cl MAM.]=\{0\}$.\chop
\itemitem{{\bf Def. 3.3}}{\bf (\rn1)} A representation $\pi:C^*_\rho(G_d)\to
\cl B.(\cl H.)$ is called {\it regular}
if for each $\beta\in\Gamma$,
the restriction of
$\pi$ to $C^*_\rho(G_{\beta d})$ is regular
on its essential subspace.
A state $\omega$ is regular if its GNS--representation is regular.
(This is equivalent to $\lim\limits_\gamma\omega(\delta\s g_\gamma.)=1$
for any net $\{g_\gamma\}\subset G$ converging to $e$).
\noindent Recall the inductive limit topology on $G$, of which
the family of its open sets are all those sets $U\subseteq G$
such that $U\cap G_\beta$ is open in $G_\beta$ for all
$\beta\in\Gamma$. Thus for any topological space $V$, a map
$f:G\to V$ is continuous iff $f$ is continuous on each $G_\beta$.
Thus the regular representations of $G$ are precisely the strong operator
continuous representations of $G$ with the inductive limit topology.
On the CCR--algebra (with the inductive system of all
finite dimensional subspaces of $S$), this definition
of regularity agrees with the
usual one.
Clearly the definition depends on the choice of inductive system,
but as it is assumed to be fixed, usually this will not be
explicitly mentioned.
We also wish to examine those closed subgroups $H\subseteq G$
which can be written as an inductive limit of a subsystem of
the inductive system for $G$, so $H=\ilim G_\beta$, $\beta\in
\Gamma_H\subseteq\Gamma$, and each $\alpha\in\Gamma$ with
$\alpha<\beta\in\Gamma_H$ is in $\Gamma_H$.
These are called {\it inductive
subgroups} and include all $G_\beta$ and $G$, but also others.
For instance in the case $G=S$, the inductive subgroups can be
any closed linear subspace of $S$. (It was with this in mind
that we specified the finest available nondiscrete inductive
system for $G$).
\itemitem{{\bf Def. 3.3}}{\bf (\rn2)} Let $H\subset G$ be an inductive subgroup
of $G$, then we say that a representation $\pi$ or state
$\omega$ of $C^*_\rho(G_d)$ or $\cl M.$ is $H\hbox{--regular}$
if it is
regular on $C_\rho^*(H_d)\subset C^*_\rho(G_d)
\subset\cl M.$.
\noindent
It is clear that if a state or representation is $H\hbox{--regular}$,
then it is also $K\hbox{--regular}$ for all inductive subgroups
$K\subseteq H$. These are the ``partially regular'' representations
and states which we encountered in [10] for the CCRs.
\thrm Lemma 3.4."
$(1)$ A representation $\pi\in\rep\cg d.,$ is $G\hbox{--regular}$
iff its canonical extension $\wt\pi$ to $\cl M.$ is nondegenerate on each
$\cg\alpha.$. \chop
$(2)$ If $\pi\in\rep\cg d.,$ is a Borel representation on $G$,
and $H$ is an inductive subgroup of $G$,
then $\pi$ is $H\hbox{--regular}$ iff its canonical extension
$\wt\pi$ is nondegenerate on $\cg\alpha.$ for each $\alpha\in
\Gamma_H$."
{\bf (1)} If $\pi$ is $G\hbox{--regular}$ it is Borel, so
$\wt\pi\,\big|\,\cg\alpha.$ is the canonical extension of
$\pi\,\big|\,\cg\alpha d.$ as in Theorem 2.4(3).
Since $\un\in\cg\alpha d.$, $\pi\,\big|\,\cg\alpha d.$ is
nondegenerate, so by Theorem 2.4 since $\pi$ is
$G_\alpha\hbox{--regular}$, $\wt\pi\,\big|\,\cg\alpha.$
is nondegenerate, and this is true for all $\alpha
\in\Gamma$.\chop
Conversely, if $\wt\pi$ is nondegenerate on each
$\cg\alpha.$, then it is uniquely determined on
each $\cg\alpha d.$ and is regular on each $G_\alpha$,
using Theorem 2.4(2).
Now since $\cg d.=\ilim\cg\alpha d.$, this means that
$\pi$ is uniquely determined on $\cg d.$ by its
values on the collection of algebras $\cg\alpha.$,
$\alpha\in\Gamma$, and it is regular on all $G_\alpha$, hence on $G$.
\chop
{\bf (2)} This is obvious, since if $\pi$ is Borel, then
$\wt\pi\,\big|\,\cg\alpha.$ is the canonical extension of
$\pi\,\big|\,\cg\alpha d.$ as in Theorem 2.4(3).
\item{\bf Remark:} Any inductive subgroup $H\subset G$ can determine
its own ``canonical extension'' of a given non--Borel
representation as follows. For an $H\hbox{--canonical}$ extension, define
$$\cl H._B^H:=\set\psi\in\cl H._\pi,{ h\in H\longrightarrow
\pi(\delta_h)\psi\in\cl H._\pi\quad\hbox{is Borel}}.$$
and proceed as before for $\wt\pi$, extending from $\cg d.$ to all
$\cg\beta.$, $\beta\in\Gamma_H$. (Extension to $\cg\beta.$ for
$\beta\in\Gamma\backslash\Gamma_H$ may not be possible on
$\cl H._B^H$). Observe that for
$H\subset K\subset G$ we have $\cl H._B^K\subseteq\cl H._B^H$.
Now part (2) of lemma 3.4 is true for non--Borel representations
if we use $H\hbox{--canonical}$ extensions instead.\chop
We now come to the argument for the construction of $\cl L.$.
Let $\pi\in\rep\cg d.,$ and define its regular subspace by:
$$\cl H._R:=\set\psi\in\cl H._\pi,g\to\pi(\delta_g)\psi\quad\hbox{is
continuous from $G$ to $\cl H.$}.\subset\cl H._B\,.$$
That $\cl H._R$ is a linear space, is obvious, that it is closed
follows from the fact that
$$\eqalignno{\cl H._R&:=\bigcap_{\alpha\in\Gamma}\cl H._R^\alpha,\qquad\quad
\hbox{where:}\cr
\cl H._R^\alpha&:=\set\psi\in\cl H._\pi,g\in G_\alpha\to
\pi(\delta_g)\psi\in{\cl H._\pi}\quad\hbox{is continuous}.\cr}$$
is the representation space of the $G_\alpha\hbox{--regular}$
part of $\pi\,\big|\,\cg \alpha d.$ which we know to be closed.
Now all $\pi(\delta_h)$ will preserve $\cl H._R$ because
$$g\to\pi(\delta_g)\cdot\pi(\delta_h)\psi=\rho(g,\,h)\,\pi(\delta_{gh})\psi$$
is continuous if $\psi\in\cl H._R$, using the fact that $\rho(g,\,\cdot)$
is continuous and group multiplication is continuous.
Thus both $\pi(\cg d.)$ and $\pi(G)$ will preserve $\cl H._R$,
and so $\pi$ restricts to a regular representation $\pi_R$
of $\cg d.$ on $\cl H._R$. For the complementary representation,
write $\pi_N$, so $\pi=\pi_R\oplus\pi_N$.
Call $\pi_N$ the {\it irregular part} of $\pi$.
(Note that $\pi_N$ may still be regular on some proper subgroup of $G$).
Denote the projection onto $\cl H._R$ by $P^\pi_R$.
Now in $\cl M.''$ we define a projection $P_R$ as the projection
from the universal representation space $\cl H._u$ onto
$$P_R\cl H._u:=\oplus\set P_R^\pi\cl H._\pi,{\pi\in{\rm Rep}_c\cl M.}.$$
i.e. $P_R=\mathop{\oplus}\limits_{\pi\in{\rm Rep}_c\cl M.}P_R^\pi$.
We need to show that $\pi(\cl M.)$ preserves $P_R^\pi\cl H._\pi$
for all $\pi\in\rep\cl M.,$ (at this point we only know that
the subalgebra $\pi(\cg d.)\subset\cl M.$ preserves $\cl H._R$).
It is only necessary to prove that $\pi(L_\rho^1(G_\alpha))$
preserves $P_R^\pi\cl H._\pi$ for all $\alpha$ and $\pi$.
Clearly $\pi(\cg\alpha d.)\subset\pi(\cg d.)$ preserves
$P_R^\pi\cl H._\pi$. For the ideal decomposition
$\pi\,\big|\,\cl E._\alpha=\pi_1\oplus\pi_2$ (recall
$\pi_2(L_\rho^1(G_\alpha))=0$ and $\pi_1(\cg\alpha.)$
is weak operator dense in $\pi_1(\cl E._\alpha)$),
we have that
$$\pi_1(f)=\int_{G_\alpha}f(g)\,\pi_1(\delta_g)\, d\mu_\alpha(g)
\qquad\hbox{for}\;\; f\in L^1(G_\alpha)$$
from which it is clear that $\pi_1(\cg\alpha.)$ preserves
$\cl H._{\pi_1}\cap P_R^\pi\cl H._\pi$.
Since $\pi_2(\cg\alpha.)=0$, this algebra trivially preserves
$\cl H._{\pi_2}\cap P_R^\pi\cl H._\pi$, and thus
$\pi(\cg\alpha.)$ preserves $P_R^\pi\cl H._\pi$.
Hence $\pi(\cl M.)$ preserves $P_R^\pi\cl H._\pi$ for all
representations $\pi$. Thus $P_R^\pi\in\cl M.'\cap\cl M.''$,
and so $P_R\in\cl M.'\cap\cl M.''$. Define
$$\cl L.:=P_R[\cl MAM.]\;.\eqno{\bf -(3.5)}$$
\thrm Theorem 3.6."Given the notation and framework above,
$\cl L.$ is a closed two--sided ideal of the C*--algebra
$\cl E.:={C^*\big(\cl L.\cup\cg d.\big)}\subset\cl M.''$,
hence there is a homomorphism
$$\varphi:\cg d.\to M(\cl L.)$$
such that in $\cl M.''$, $\varphi(A)L=AL$ for all $A\in\cg d.$,
$L\in\cl L.$, and $\varphi$ is injective when $\cg d.$
has a faithful regular representation."
To see that $\cl L.$ is an ideal of $\cl E.$:
$$\cg d.\cl L.=P_R\cg d.[\cl MAM.]=P_R[\cl MAM.]=\cl L.$$
and similarly from the right. Then the homomorphism $\varphi$
exist by the general properties of multiplier algebras
(cf. Pedersen [11]). Let $\pi$ be a faithful regular representation
of $\cg d.$. Then $\wt\pi(P_R)=\un$ and $\wt\pi$ is nondegenerate on
$\cl L.$. Hence $\pi$ is uniquely determined by $\wt\pi\,\big|\,\cl L.$
using the unique extension of a representation from the ideal
$\cl L.$ to $\cl E.$ on the same space. So
$$\eqalignno{\pi(\delta_g)\psi&=\lim_\lambda\wt\pi(\delta_gu_\lambda)\psi
=\lim_\lambda\wt\pi(\varphi(\delta_g)u_\lambda)\psi\cr
&=\hat\pi(\varphi(\delta_g))\psi &\hbox{for all $\psi\in\cl H._\pi$}\cr}$$
where $\{u_\lambda\}$ is an approximate identity of $\cl L.$
and $\hat\pi$ is the unique extension of $\wt\pi\,\big|\,\cl L.$
to $M(\cl L.)$. Thus $\pi(A)=(\hat\pi\circ\varphi)(A)$ for all
$A\in\cg d.$. However $\pi$ is assumed to be faithful, hence
so is $\varphi$ and $\hat\pi$.
\item{\bf Remark. } Clearly when $G$ has no regular $\rho\hbox{
--representations}$, $\cl L.=\{0\}$.
As far as the author knows, the problem of whether
there exists a strong--operator continuous $\rho\hbox{--representation}$
$\pi:G\to U(\cl H.)$ for a general topological group is unsolved.
(In this direction there
is a result in a preprint by Pickrell on the nonexistence of
a continuous representation of a gauge--type group on a separable
Hilbert space). However in the case of the CCR's, where the symplectic
space $S$ is a pre--Hilbert space, the symplectic form being the
imaginary part of the inner product, Fock--representations exist,
are regular, faithful and well--studied, so for this situation $P_S\not=0$.
There are certainly known symplectic spaces $(S,\,B)$ for
which the CCR--algebra has neither Fock representations, nor
quasi--free representations [20].
\thrm Theorem 3.7."$(\rn1)$ Let $\pi\in\rep\cl L.$, then the unique
extension of $\pi$ on the same space to $\cg d.$ defines a
regular representation on $\cg d.$.\chop
$(\rn2)$ Let $\pi\in\cl P.$ ($\equiv$ regular representations
of $\cg d.$), then the canonical extension $\wt\pi$ of $\pi$ to
$\cl M.$ is nondegenerate on $\cl L.$.\chop
$(\rn3)$ There is a bijection $t:\cl P.\to\rep\cl L.,$
given by $t(\pi)=\wt\pi\,\big|\,\cl L.$, preserving
irreducibility."
{\bf (\rn1)} Let $\pi\in\rep\cl L.,$, then since $\cl L.$
is an ideal of $\cl M.''$, $\pi$ extends uniquely
on the same space to a representation $\hat\pi$ of $\cl M.''$.
Since $\hat\pi$ is nondegenerate on $\cl L.$, checking the
definition of $\cl L.$ shows that $\hat\pi(P_R)=\un$.
This means that $\hat\pi$ is regular on $\cg d.$.\chop
{\bf (\rn2)} Let $\pi\in\cl P.$ have canonical extension $\wt\pi$
to $\cl M.''$. Since $\pi$ is regular, $\wt\pi(P_R)=\un$
and so $\wt\pi$ is nondegenerate on each
$\cg\alpha.\subset\cl A.\subset[\cl MAM.]$ by lemma 3.4,
hence it is nondegenerate on $P_R[\cl MAM.]=\cl L.$.\chop
{\bf (\rn3)} We show that $t$ is invertible
with inverse $t^{-1}(\pi)=\hat\pi\,\big|\,\cg d.$,
$\pi\in\rep\cl L.,$
where $\hat\pi$ is the unique extension of $(\rn1)$.
Start with $t(\nu)=\wt\nu\,\big|\,\cl L.$,
$\nu\in\cl P.$, then since
$\nu$ is regular, $\wt\nu(P_R)=\un$, so
$\wt\nu(P_R)=\un$, so $\wt\nu(\cl L.)=
\wt\nu([\cl MAM.])\supset\wt\nu(\cg\alpha.)$ for all
$\alpha\in\Gamma$.
By definition of the canonical extension
$$\wt\nu(f)=\int_{G_\alpha}f(g)\,\nu(\delta_g)\, d\mu_\alpha(g)
\qquad\forall\; f\in L_\rho^1(G_\alpha)\;,$$
i.e. the same canonical extension than in the
locally compact case on each $\cg\alpha.$.
Now $(t^{-1}\circ t)(\nu)=(\wt\nu\,\big|\,\cl L.)\hat{\,}\,\big|\,\cg d.$.
Since $\nu$ is regular, by lemma 3.4 $\wt\nu\,\big|\,\cg\alpha.$
is nondegenerate for all $\alpha$. Hence for a fixed $\alpha$
by Theorem 2.4(2) it has a unique extension to $\cg\alpha d.$
on the same space, which therefore must agree with
$(\wt\nu\,\big|\,\cl L.)\hat{\,}\,\bigg|\,\cg\alpha d.$.
However by 2.4(4) we know that these maps are inverses
of each other, so
$(\wt\nu\,\big|\,\cl L.)\hat{\,}\,\bigg|\,\cg\alpha d.=
\nu\,\big|\,\cg\alpha d.$.
Since this is true for all $\alpha\in\Gamma$,
we find $(t^{-1}\circ t)(\nu)=\nu$. The statement of irreducibility is
obvious.
\thrm Theorem 3.8." With notation and assumptions as above,
when $G$ is locally compact
we have $\cl L.=\cg .$."
When the inductive system $\set G_\beta,\beta\in\Gamma.$ terminates
in $G$, define
$$\cl B.:=C^*\big(\cg .\cup\cg d.\cup\cl L.\cup\{P_R\}\big)\subset
\cl M.''\qquad\qquad-(*)$$
Otherwise, observe $\cl M.\cup\cg .\subset M(\cg .)$,
so define $\cl B.$ by the formula $(*)$, but as a
subalgebra of $C^*(\cl M.\cup\cg .)''\supset\cl M.''\ni P_R$.
Using 3.6 and 3.5 as well as $\cg .\subset\cl A.\subset
[\cl MAM.]$, we conclude $\cl L.$ is a two--sided closed ideal for
$\cl B.$. Now since $\cl M.$ is generated by
$\cg\alpha.$ and $\cg d.$, it is clear that $\cg .$ is an ideal
in $\cl M.$, so we only need to show that $P_R\cg .\subset\cg .$
to establish that $\cg .$ is also an ideal of $\cl B.$.
This is obvious since $P_R$ is exactly the projection
of $\cl H._u$ onto the direct sum of the cyclic representations
of $\cg .$. So with the imbedding $\cg .\subset\cl M.''$
given by $\pi_u(\cg .)\subset\pi_u(\cl M.)''$ we see
$P_R\pi_u(\cg .)=\pi_u(\cg .)$.\chop
So since both $\cl L.$ and $\cg .$ are ideals of $\cl B.$,
we use the result in Pedersen 3.13.8 [11] that any closed ideal
of a C*--algebra is the intersection of the primitive ideals
containing it. We prove that $\cl L.=\cg .$ by showing
$\cl L.\subset\ker\pi$ iff $\cg .\subset\ker\pi$
for all $\pi\in{\rm Irrd}\,\cl B.$.
Fix $\pi\in{\rm Irrd}\,\cl B.$ and let $\cl L.\subset\ker\pi$.
Consider the ideal decomposition of $\pi$ w.r.t. $\cg .$.
Since $\pi$ is irreducible, either $\cg .\subset\ker\pi$
or $\pi$ is nondegenerate on $\cg .$ (and $\pi(\cg .)$
is strong operator dense in $\pi(\cl B.)$).
In the latter case $\pi$ has unique extension to $\cl B.$,
so by theorem 2.4 must be regular on $\cg d.$, so $\pi(P_R)
=\un$ and
$$\pi(\cl L.)=\pi(P_R[\cl MAM.])=\pi([\cl MAM.])\supset\pi(\cl A.)
\supset\pi(\cg .)$$
so if $\pi$ is nondegenerate on $\cg .$, this contradicts $\cl L.
\subset\ker\pi$. Thus $\cg .\subset\ker\pi$.\chop
Conversely, assume $\cg .\subset\ker\pi$ and consider
the ideal decomposition of $\pi$ w.r.t. $\cl L.$.
If $\pi$ is nondegenerate on $\cl L.$, then by definition
of $\cl L.$ we have $\pi(P_R)=\un$, and $\pi\,\big|\,\cl L.$
is the canonical extension of $\pi\,\big|\,\cg d.$
by 2.4(3). So since $\pi(\cg .)\subset\pi(\cl L.)$,
$\pi$ is also the canonical extension of $\pi\,\big|\,\cg d.$
to $\cg .$. Since it is regular, $\pi$ is nondegenerate
on $\cg .$, and this contradicts $\cg .\subset\ker\pi$.
Thus $\cl L.\subset\ker\pi$, and so we have shown
that $\cl L.\subset\ker\pi$ iff $\cg .\subset\ker\pi$
for all $\pi\in{\rm Irrd}\,\cl B.$.
\noindent Thus $\cl L.$ is an adequate infinite dimensional
analogue of the twisted group algebra $\cg .$.
So in the infinite
dimensional case of $S$, the regular representations form the representation
space
of a C*--algebra, and this may be useful to obtain
direct integral decompositions
of regular representations
into irreducible regular representations.
More general decompositions of regular representations are dealt with in
Schaflitzel [2].
\item{\bf Remarks:}$(\rn1)$ Clearly for an $H\hbox{--regular}$
representation we can construct the same structure for
$C^*_\rho(H_d)\subset\cg d.$, thus obtaining an algebra
$\cl L._H$ which has as representation space exactly
the continuous representations of $H$. More specifically, let
$$\leqalignno{\cl A._H&:=\lim_{\beta\in\Gamma_H}\cl A._\beta
=C^*(\set\cl N._\beta,\beta\in\Gamma_H.)\subset\cl A.\cr
\cl M._H&=\lim_{\beta\in\Gamma_H}\cl M._\beta=C^*(
\set\cl N._\beta\cup\cl N._\beta^d,\beta\in\Gamma_H.)\subset\cl M.\cr
&[\cl M._H\cl A._H\cl M._H]\subset[\cl MAM.]\;.&\hbox{so}\cr}$$
Now let $P_R^H\in\cl M._H''\cap\cl M._H'$ be the projection
which maps every representation space $\cl H._\pi\subset\cl H._u$
(the universal representation of $\cl M._H$ used here) onto
the space
$$P_R^H\cl H._\pi:=\set\psi\in\cl H._\pi,{h\in H\longrightarrow
\pi(\delta_h)\psi\in\cl H._\pi\;\;\hbox{is continuous}}.$$
(note that $P_R^H\not\in\cl M.''\cap\cl M.'$). Now define
$$\cl L._H:=P_R^H[\cl M._H\cl A._H\cl M._H]\;.$$
Note that $M(\cl L._H)\supset C^*_\rho(H_d)$ but it does not
contain $\cg d.$, and furthermore $\cl L._H\not\subset\cl L.$.
Since $P_R^HP_R=P_R$ (using $\cl M._H''\subset\cl M.''$ to make
sense of this), clearly $\cl L._H\subset M_{\cl M.''}(\cl L.)$.
Thus we have, just as in the locally compact case that
$\cl L.$, $\cl L._H$ and $\cg d.$ are all imbedded in a
larger C*--algebra $\cl M.''$ such that $\cl L.$ is an
ideal for $\cl L._H$ and $\cg d.$, i.e.
$\cl L._H\subset M_{\cl M.''}(\cl L.)\supset\cg d.$
for all inductive subgroups $H$, and regular representations
are obtained from the unique extensions of representations
from $\cl L.$ to $\cg d.$. Moreover, a representation
$\pi\in\cg d.$ is $H\hbox{--regular}$ iff its
$H\hbox{--canonical}$ extension to $\cl M.$ is nondegenerate
on $\cl L._H$. So the structure theory for partially regular
representations looks almost like what it is in the locally
compact case.
\item{$(\rn2)$} For the moment we leave open the questions
of when $\cl L.$ is simple or separable, and to what
extent it depends on the initial inductive system for
$G$.
\item{$(\rn3)$} At this point we would like to argue that
if one requires all physical representations of the CCRs
to be regular (this will be when no linear constraints are
present, or after the constraints have been imposed),
then $\cl L.$ should be the appropriate field algebra
for the CCRs rather than $\ccr S,B.$.
\item{$(\rn4)$} The irregular part $\pi_N$ of a representation
$\pi\in\rep\cg d.$ must have canonical extension vanishing
on $\cl L.$ (clearly since $\cl L.$ consists of products
with $P_R$, and $\pi_N$ is orthogonal to $P^\pi_R=\pi(P_R)$).
This is consistent with the definition of irregularity
in the locally compact case, by 3.8.
Note that the irregular part $\pi_N$ may
be regular on some inductive subgroup $H$, in which case
$\pi_N$ can of course be further decomposed into
$H\hbox{--regular}$ and irregular parts, but only as a
representation of $C^*_\rho(H_d)$, {\bf not}
of $\cg d.$, because $\pi(\delta_g)$ for some $g\in
G\backslash H$ may not preserve the $H\hbox{--regular}$
subspace:
$$\cl H._R^H:=\set\psi\in\cl H._\pi,{h\in H\longrightarrow\pi(\delta_h)\psi
\in\cl H._\pi\;\;\hbox{is continuous}}.\;.$$
Say an $H\hbox{--regular}$ representation $\pi$ is {\it simply}
$H\hbox{--regular}$ if it is $K\hbox{--irregular}$ for all inductive
subgroups $K$ properly containing $H$.
By the preceding discussion, we cannot in general expect to find a
decomposition of a nonregular representation into a direct integral
of partially simply regular subrepresentations of $\cg d.$.
However, if one allows more general direct integrals as in
Schaflitzel [2], where the subrepresentations are on different
subalgebras, such a decomposition may be possible.
\def\rep{{\rm Rep}\,} \def\cg#1.{{C_\rho^*(G_{#1})}}
\def\ilim{\displaystyle{\mathop{{\rm lim}}_{\mathord{\longrightarrow}}}\,}
\def\wt#1,{\widetilde{#1}}
\beginsection 4. Automorphisms.
In physics it is frequently the case that actions of the physical
transformation groups on the CCR--algebra
are weak operator continuous
in all regular representations but not pointwise norm
continuous on the CCR--algebra. Recall that
from the continuous group homomorphism
$\gamma:\cl G.\to\Sp(S,\,B)$ an action $\alpha:\cl G.\to
\aut\ccr S,B.$ is defined by $\alpha_g(\delta_x)=\delta\s
\gamma_g(x).$ for all $f\in\cl G.$, $x\in S$.
For the finite dimensional case we already have from theorem
2.6 that such a group action defines a pointwise
norm continuous action on $C_\sigma^*(S)=\cl L.$
and we aim to prove this for the algebra $\cl L.$
of the infinite dimensional
case.
{}From a continuous group action on $G$, it is possible
to prove that it defines a pointwise norm continuous
action on $\cl L.$ via the Ernest--Takesaki--Bichteler [14, 15]
realisation of a C*--algebra as admissable operator fields
on its representation space, but for
concreteness we do a more direct argument below.
Assume the structures of the last section,
$$\set G_\beta,\beta\in\Gamma.\,,\qquad G=\ilim G_\beta\,,\qquad
\rho\in Z^2(G,\,{\bf T})\quad\hbox{is continuous}$$
and define
\def\autr{\mathop{{\rm Aut}_\rho^0}}
$$\theta\in\autr G\quad\hbox{iff}\quad
\cases{\vphantom{\bigg|}\;\bullet\;
\hbox{$\theta\in\aut G\;$ is Borel on each $\; G_\beta$}& \cr
\vphantom{\bigg|}\;\bullet\;
\hbox{for each $\;\beta\in\Gamma\;$ there are $\;\delta\,,\;\gamma\in\Gamma$
with}& \cr \qquad\qquad\qquad \hbox{
$\theta(G_\beta)=G_\delta\;$ and $\;\theta^{-1}(G_\beta)=G_\gamma$}& \cr
\vphantom{\bigg|}\;\bullet\;
\hbox{$\rho\big(\theta(g),\,\theta(h)\big)=\rho(g,\,h)\;$ for all
$\; g,\, h\in G$.}& \cr}$$
so $\autr G$ are those $\rho\hbox{--preserving}$
automorphisms mapping members of the
inductive system to other members of the inductive system,
so for each $\theta\in\autr G$ we obtain a map $\theta:\Gamma\to\Gamma$
by $\theta(G_\beta)=:G\s\theta(\beta).$ and an abuse of notation.
\thrm Theorem 4.1." {\bf (1)} Each $\alpha\in\autr G$ defines an automorphism
of the measure algebra $\cl M._\rho(G)=\ilim\cl M.(G_\beta)$ by
\hfill$(\alpha\cdot\mu)(D):=\mu(\alpha^{-1}(D))$\hfill\break
for all $\mu\in\cl M.(G_\beta)$ and Borel sets $D\subset G\s\alpha(\beta).$.
\chop
{\bf (2)} The automorphism $\alpha\in\aut\cl M._\rho(G)$ of (1)
preserves $L_\rho^1(G_d)$ and maps $L_\rho^1(G_\beta)$
to $L_\rho^1(G\s\alpha(\beta).)$. Its restriction to $L_\rho^1(G_d)$
is given by $\alpha(\delta_g)=\delta\s\alpha(g).$, which extends to
$\cg d.$. Explicitly on $L_\rho^1(G_\beta)$ we have
$(\alpha\cdot f)(g)=C_\alpha^\beta\, f(\alpha^{-1}(g))$ where on the
Haar measures we have $\alpha\cdot\mu_\beta=C_\alpha^\beta\,\mu\s\alpha(
\beta).$, $C_\alpha^\beta\in\r_+$.\chop
{\bf (3)} Each $\alpha\in\autr G$ defines an automorphism on
$\cl M.$ which preserves $\cl A.$. There is an injective
homomorphism \chop $\Phi:\autr G\to\aut\cl M.$ such that
any $\alpha\in\Phi(\autr G)$ preserves $\cl A.$.\chop
{\bf (4)} Let $\pi\in\cl P.\equiv$regular representations of $\cg d.$.
Then $\pi\circ\alpha\in\cl P.$ for all $\alpha\in\autr G$. "
{\bf (1)} Clearly the defined $\alpha\cdot\mu$ is a measure on
$G\s\alpha(\beta).$, and $\alpha:\cl M.(G_\beta)\to\cl M.(G\s\alpha(\beta).)$
is an invertible map. Surjectivity is clear, we show that it is a
*--isomorphism. Let $\mu$ and $\nu\in\cl M.(G_\beta)$, $f\in C_0(G\s
\alpha(\beta).)$, so $\alpha(\mu *\nu)$ is defined by
$$\eqalignno{\int_{G_{\alpha(\beta)}}f(t)\, d(\alpha(\mu *\nu))(t)&=
\int_{G_{\alpha(\beta)}}f(t)\, d(\mu *\nu)(\alpha^{-1}(t))\cr
=\int_{G_\beta}f(\alpha(t))&\, d(\mu *\nu)(t)\qquad\quad
\hbox{using isomorphism $\alpha:G_\beta\to G\s\alpha(\beta).$}\cr
&=\int\int_{G_\beta}f(\alpha(st))\,\rho(s,\,t)\, d\mu(s)\, d\nu(t)\cr
=\int\int_{G_{\alpha(\beta)}}f(st)&\,\rho(s,\, t)\,
d\mu(\alpha^{-1}(s))\, d\nu(\alpha^{-1}(t))\qquad\quad
\hbox{using $\rho\circ\alpha=\rho$}\cr
&=\int_{G_{\alpha(\beta)}}f(t)\, d((\alpha\cdot\mu)*(\alpha\cdot\nu))(t)\cr}$$
and thus $\alpha(\mu *\nu)=(\alpha\cdot\mu)*(\alpha\cdot\nu)$, so
$\alpha $ is a homomorphism $\cl M.(G_\beta)\to\cl M.(G\s\alpha(\beta).)$.
To see that it
preserves the involution:
$$\eqalignno{\int_{G_{\alpha(\beta)}}f(t)\, d(\alpha\cdot\mu)^*(t)&=
\overline{\int_{G_{\alpha(\beta)}}\overline{f}(t^{-1})\,
\rho(t^{-1},\, t)\, d(\alpha\cdot\mu)(t)}\cr
&=\overline{\int_{G_\beta}\overline{f}(\alpha(t)^{-1})\,
\rho(\alpha(t)^{-1},\alpha(t))\, d\mu(t)}\cr
=\overline{\int_{G_\beta}\overline{f}(\alpha(t)^{-1})\,
\rho(t^{-1},\, t)\, d\mu(t)}&=\int_{G_\beta}f(\alpha(t))\, d\mu^*(t)\cr
&=\int_{G_{\alpha(\beta)}}f(t)\, d(\alpha(\mu^*))(t)\cr}$$
hence $\alpha(\mu^*)=(\alpha\cdot\mu)^*$, and so $\alpha$
is a *--isomorphism. \chop
Moreover if $\beta\leq\gamma$ then $G_\beta\subseteq G_\gamma$, and
so since $\alpha(G_\beta)\subseteq\alpha(G_\gamma)$,
we see that $\cl M.(\alpha(G_\beta))\subseteq
\cl M.(\alpha(G_\gamma))$, hence $\alpha$ preserves the partial
ordering of the inductive system of $\cl M._\rho(G)$.
Thus, by the universal property of inductive limit algebras
(cf. L.1.1 and L.2.1 in [26]) there is a homomorphism from
$\cl M._\rho(G)=\ilim\cl M.(G_\beta)$ to
$\wt\cl M.,_\rho(G):=\ilim\cl M.(\alpha(G_\beta))$,
which is an isomorphism since each $\alpha:\cl M.(G_\beta)
\to\cl M.(G\s\alpha(\beta).)$ is an isomorphism.
Thus $\cl M._\rho(G)\cong\wt\cl M.,_\rho(G)$ via $\alpha$.
Now there is also an identity isomorphism between these
algebras since we can identify each $\cl M.(G_\beta)$
with its copy in both $\cl M._\rho(G)$ and in $\wt\cl M.,_\rho(G)$.
Then with respect to this identification, the isomorphism
obtained from $\alpha$ becomes an automorphism, and on the
measures of $G_\beta$, agrees with the original definition
$(\alpha\cdot\mu)(D)=\mu(\alpha^{-1}(D))$ for Borel sets $D$.\chop
{\bf (2)} First, the restriction of $\alpha$ to $L^1_\rho(G_d)$ is:
$$\eqalignno{(\alpha\cdot\delta_g)(D)=\delta\s g.\big(\alpha^{-1}(
D)\big)&=\cases{1&if $\alpha(g)\in D$,\cr
0 & otherwise\cr}\cr
&=\vphantom{\bigg|}
\delta\s\alpha(g).(D)&\hbox{for all Borel sets $D$.}\cr}$$
Thus $\alpha\cdot\delta_g=\delta\s\alpha(g).$, and so $\alpha$
preserves $L_\rho^1(G_d)$.\chop
Next, we show that if $\alpha(G_\beta)=G\s\alpha(\beta).$,
then $\alpha(L_\rho^1(G_\beta))=L_\rho^1(G\s\alpha(\beta).)$.
Since for any Borel set $D\subset G\s\alpha(\beta).\ni g$ and the
Haar measure $\mu_\beta$ of $G_\beta$ we have
$$(\alpha\cdot\mu_\beta)(gD)=\mu_\beta\big(\alpha^{-1}(g)\cdot
\alpha^{-1}(D)\big)=\mu_\beta(\alpha^{-1}(D))=(\alpha\cdot\mu_\beta)
(D)$$
it is clear that $\alpha$ maps Haar measures to Haar measures.
Using the uniqueness of Haar measures up to positive multiples, write
$\alpha\cdot\mu_\beta=C_\alpha^\beta\mu\s\alpha(\beta).$
where $C_\alpha^\beta\in\r_+$. Now realise $f\in L_\rho^1(G_\beta)$
as a measure $\nu=f\,\mu_\beta$, so
$d(\alpha\cdot\nu)(g)=d\nu(\alpha^{-1}g)=f(\alpha^{-1}(g))\,
d\mu_\beta(\alpha^{-1}(g))=C_\alpha^\beta\,
f(\alpha^{-1}(g))\, d\mu\s\alpha(\beta).(g)$
which is in $L_\rho^1(G\s\alpha(\beta).)$, and so
$$\big(\alpha(f)\big)(g)=C_\alpha^\beta\, f(\alpha^{-1}(g))\qquad\quad
\forall\; g\in G\s\alpha(\beta).,\; f\in L_\rho^1(G_\beta)\;.$$
{\bf (3)} Since $\alpha\in\aut\cl M._\rho(G)$ is norm preserving,
it extends uniquely to the C*--enveloping algebras of the
$L_\rho^1\hbox{--algebras}$ above, and also to their multiplier
algebras. So we obtain $\alpha\in\aut\cg d.$ and homomorphisms
$\alpha:\cg\beta.\to\cg\alpha(\beta).$ which preserve
the inductive limit of these (see argument in proof of (1)),
so $\alpha$ defines an automorphism of $\cl M.$
preserving the subalgebras $\cg d.$ and $\cl A.$ (which is
generated by $\set L_\rho^1(G_\beta),\beta\in\Gamma.$).\chop
{\bf (4)} Recall that $\pi\in\cl P.$ iff the canonical
extension $\wt\pi,$ to $\cl M.$ is nondegenerate on each
$\cg\beta.$. We first show that $(\wt{\pi\circ\alpha},)=
\wt\pi,\circ\alpha$ for $\pi\in\cl P.$. Since
$\alpha$ is Borel on all $G_\beta$ and $\pi\in\cl P.$
(henceforth fixed) is Borel
on $\cl H._\pi$, clearly $\pi\circ\alpha$ is Borel on
$\cl H._\pi$, and so $\cl H._B=\cl H._\pi$.
Now $\wt{\pi\circ\alpha},$ is defined on each $L_\rho^1(G_\beta)$ by
$$\leqalignno{(\wt{\pi\circ\alpha},)(f)=\int_{G_\beta}f(g)&\,
(\pi\circ\alpha)(\delta_g)\, d\mu_\beta(g)=
\int_{G_\beta}f(g)\,\pi(\delta\s\alpha(g).)\, d\mu_\beta(g)\cr
(\wt\pi,\circ\alpha)(f)&=\wt\pi,(\alpha(f))=\int_{G_{\alpha(\beta)}}
\alpha(f)(t)\,\pi(\delta_t)\, d\mu\s\alpha(\beta).(t)&\hbox{whereas}\cr
&=\int_{G_{\alpha(\beta)}}C_\alpha^\beta\, f(\alpha^{-1}(t))\,\pi(\delta_t)\,
d\mu\s\alpha(\beta).(t)\cr
&=\int_{G_{\alpha(\beta)}}f(\alpha^{-1}(t))\,\pi(\delta_t)\,
d\mu_\beta(\alpha^{-1}(t))\cr
&=\int_{G_\beta}f(g)\,\pi(\delta\s\alpha(g).)\,
d\mu_\beta(g)=(\wt{\pi\circ\alpha},)(f)\cr}$$
for all $f\in L_\rho^1(G_\beta)$. Now since $\pi$ is regular,
$\wt\pi,$ is nondegenerate on all $L_\rho^1(G_\beta)$, so using
$\alpha(L_\rho^1(G_\beta))=L_\rho^1(G\s\alpha(\beta).)$
and the invertibility of $\alpha$ we see that
$\wt\pi,\circ\alpha=\wt{\pi\circ\alpha},$ is nondegenerate on
all $L_\rho^1(G_\beta)$. Hence $\wt{\pi\circ\alpha},\,\big|\,\cg d.
=\pi\circ\alpha$ is regular.
\thrm Corollary 4.2." Every automorphism $\alpha\in\autr G$
defines an automorphism of $\cl L.=P_R[\cl MAM.]$."
Since $\alpha$ preserves $\cl M.$ and $\cl A.$, it
preserves $[\cl MAM.]$. Now $\alpha\in\aut\cl M.$ extends
uniquely to $\cl M.''$, hence to $P_R$.
Since $\wt\pi,(P_R)=\un$ for all $\pi\in\cl P.$
and $\pi\circ\alpha\in\cl P.$ we have
$\wt\pi,(\alpha(P_R))=\un$. Hence, using
invertibility of $\alpha$,
$\wt\pi,(P_R)=\un$ iff $\wt\pi,(\alpha(P_R))=\un$.
Thus $\alpha(P_R)=P_R$, recalling that $P_R$
is the projection onto the regular part of $\cl H._u$.
Thus $\alpha$ preserves $\cl L.$.
Next we show that a pointwise continuous action
$\theta:\cl G.\to\autr G$ defines a pointwise continuous
action on $\cl L.$.
\thrm Theorem 4.3." Let $\{\alpha_\nu\}\subset\autr G$
be a net converging pointwise to $\alpha\in\autr G$
(i.e. $\alpha_\nu(x)\to\alpha(x)$ as $\nu\to\infty$
for each $x\in G$).\chop
Then $\|\alpha_\nu(A)-\alpha(A)\|\to 0$ as
$\nu\to\infty$ for each $A\in\cl L.$."
Let $\pi\in\cl P.$, so $\|\big[\pi(\delta\s\alpha_\nu(x).)
-\pi(\delta\s\alpha(x).)\big]\psi\|\to 0$ as $\nu\to\infty$,
hence $\big\|\left[\pi(\alpha_\nu(A))-\pi(\alpha(A))\right]
\psi\big\|\to\infty$ for all $A\in\cg d.$ and $\psi\in\cl H._\pi$.
Now recall that $\wt\pi,$ is defined on $\cl M._\rho(G)$ by
$$\leqalignno{\pi(\mu)&=\int_{G_\beta}\pi(\delta_g)\, d\mu(g)
\qquad\quad\forall\;\mu\in\cl M.(G_\beta)\;,\qquad\hbox{so}\cr
\pi(\alpha\cdot\mu)&=\int_{G_{\alpha(\beta)}}\pi(\delta_g)\,
d\mu(\alpha^{-1}(g))=\int_{G_\beta}\pi(\alpha(\delta_g))\,
d\mu(g)\;.\cr
\hbox{Then}\qquad\quad &\left\|\big[\pi(\alpha_\nu\cdot\mu)
-\pi(\alpha\cdot\mu)\big]\,\psi\right\|\cr
&\quad=\big\|\int_{G_\beta}\pi(\alpha_\nu(\delta_g)
-\alpha(\delta_g))\;\psi\, d\mu(g)\big\|\cr
&\leq\|\mu\|_1\cdot\sup\set\left\|\pi(\alpha_\nu(\delta_g)
-\alpha(\delta_g))\psi\right\|,g\in G.\longrightarrow 0\cr}$$
as $\nu\to\infty$ for all $\psi\in\cl H._\pi$.
We also obtain for the corresponding automorphisms $\alpha_\nu$
and $\alpha$ on $\cl M.''$ and on $\cl L.$ that for
$\pi\in\cl P.$ and $\psi\in\cl H._\pi$:
$$\left\|\wt\pi,(\alpha_\nu(A)-\alpha(A))\psi\right\|\longrightarrow 0
\qquad\hbox{as}\quad\nu\to\infty$$
for all $A\in\cl M.''\supset\cl L.$. Now recall that all the representations of
$\cl L.$ are obtained from restrictions to $\cl L.$ of canonical extensions
of regular representations $\pi\in\cl P.$. Thus
$$\left\|\pi(\alpha_\nu(A)-\alpha(A))\psi\right\|\to 0\qquad\hbox{as}
\quad\nu\to\infty$$
for all $A\in\cl L.$, $\psi\in\cl H._\pi$, $\pi\in{\rm Rep}\,\cl L.$.
In particular if $(\pi,\,\psi)$ is the GNS--representation
and cyclic vector of a state $\omega\in\wp(\cl L.)$, we have
$$\omega\Big(\big(\alpha_\nu(A)-\alpha(A)\big)^*\big(\alpha_\nu(A)-
\alpha(A)\big)\Big)\to 0\qquad\hbox{as}\quad\nu\to\infty$$
and since this is true for all states of $\cl L.$, we find by
Dixmier 2.7.1 [8] that $\|\alpha_\nu(A)-\alpha(A)\|\to 0$
in the C*--norm of $\cl L.$.
\thrm Corollary 4.4." Let $G=S$ and $\rho=\sigma$ with inductive system
consisting of all finite dimensional subspaces of $S$. Then
${\rm Sp}(S,\,B)={\rm Aut}_\sigma^0S$, ${\rm Sp}(S,\,B)$ is closed under the
pointwise convergent topology, and any continuous homomorphism
$\gamma:\cl G.\to{\rm Sp}(S,\,B)$ defines a pointwise norm continuous
action $\alpha:\cl G.\to\aut\ccr S,B.$."
Let $\{\theta^\nu\}\subset{\rm Sp}(S,\,B)$ be a sequence such that
$\{\theta^\nu(x)\}$ converges for each $x\in S$, and denote the
limit by $\theta(x)\in S$. Then we wish to show that this defines
an element $\theta\in{\rm Sp}(S,\,B)$. Linearity is obvious, and we
only need to show that $B(\theta(x),\,\theta(y))=B(x,\,y)$
(which will also establish invertiblity) and this follows from
the fact that $B$ is (jointly) continuous on $S\times S$ and that
$\theta^\nu\in{\rm Sp}(S,\, B)$. The rest of
the corollary follows directly from 4.3.
\itemitem{{\bf Remarks.}}$(\rn1)$ The requirement that $\alpha$ is in
${\rm Aut}_\rho^0G$ should be expected, because $\cl A.$
depends on the inductive system. (This restriction was not
visible for the finite dimensional case).
\itemitem{$(\rn2)$} The possibility now arises of considering actions
$\alpha:\cl G.\to{\rm Aut}_\rho^0G$ which are only partially
continuous, i.e. only continuous on the orbit of a subgroup
$H\subset G$. This will be appropriate for Bosonic systems with linear hermitian
constraints, where we only need continuity on the physical subalgebra.
\beginsection Acknowledgements.
I am deeply indebted to Colin Sutherland, in that the germinal idea
for this paper surfaced during a discussion we had, and
he was always an interested and enlightening presence.
The problem of nonregular representations is an old
favourite of Prof. C.A. Hurst, which is how I came to
know about it [10].
Special thanks are due to Dr. R. Schaflitzel whose critical
reading of several previous versions uncovered serious
mistakes in each.
I am also grateful to Norman Wildberger for his questions, which
made me return in the first place to a problem as old and venerable
as this one. Further discussions I have benefitted from were
with
Prof. John Roberts, Prof. W. Thirring, Prof. H. Narnhofer,
Prof. A. Verbeure and Prof. M. Rieffel.
This work was done with the sponsorship of an ARC grant.
\vfill\eject
\noindent {\bf Bibliography.}\chop
\item{1.} Amann, A.: Invariant states of C*--systems without norm
continuity properties. J. Math. Phys. {\bf 32}, 739--743 (1991)
\item{2.} Schaflitzel, R.: Decompositions of regular representations
of the canonical commutation relations. Publ. RIMS, Kyoto Univ.
{\bf 26}, 1019--1047 (1990)
\item{3.} Araki, H., Woods, E.J.: Topologies induced by representations
of the canonical commutation relations. Rep. Math. Phys. {\bf 4},
227--254 (1973)
\item{4.} Bratteli, O., Robinson, D.W.: Operator algebras and
quantum statistical mechanics II. New York Heidelberg Berlin,
Springer--Verlag 1979.
\item{5.} Slawny, J.: On factor representations and the C*--algebra
of the canonical commutation relations. Commun. Math. Phys.
{\bf 24}, 151--170 (1972)
\item{6.} Packer, J.A., Raeburn, I.: Twisted crossed products of
C*--algebras. Math. Proc. Camb. Phil. Soc. {\bf 106}, 293--311 (1989)
\item{7.} Manuceau, J.: C*--algebre de relations de commutation.
Ann. Inst. Henri Poincar\'e {\bf 8}, 139--161 (1968)
\item{8.} Dixmier, J.: C*--algebras. Amsterdam: North Holland 1977.
\item{9.} Bratteli, O., Robinson, D.W.: Operator algebras and
quantum statistical mechanics I. New York Heidelberg Berlin,
Springer--Verlag 1979.
\item{10.} Grundling, H., Hurst, C.A.: A note on regular states and
supplementary conditions. Lett. Math. Phys. {\bf 15}, 205 (1988)
Erratum: Lett. Math. Phys. {\bf 17}, 173 (1989).
\item{11.} Pedersen, G.K.: C*--algebras and their automorphism groups.
London: Academic Press 1979.
\item{12.} Takeda, Z.: Inductive limit and infinite direct product
of operator algebras. Tohoku Math. J. {\bf 7}, 67--86 (1955).
\item{13.} Blackadar, B.: K--theory for operator algebras.
New York Berlin Heidelberg, Springer--Verlag 1986.
\item{14.} Takesaki, M.: A duality in the representation theory of
C*--algebras. Ann. Math. {\bf 85}, 370--382 (1967)
\item{15.} Bichteler, K.: A generalisation to the non--separable case of
Takesaki's duality theorem for C*--algebras. Inventiones Math. {\bf 9},
89--98 (1969).
\item{16.} Segal, I.E.: Representations of the canonical commutation
relations, Carg\'ese lectures in theoretical physics p107--170,
Gordon and Breach, 1967.
\item{17.} Folland, G.: Harmonic analysis in phase space.
Annals of Mathematics Studies 122, Princeton University Press, 1989.
\item{18.} Segal, I.E.: Foundations of the theory of dynamical
systems of infinitely many degrees of freedom, II,
Can. J. Math. {\bf 13}, 1--18 (1962)
\item{19.} Kastler, D.: The C*--algebras of a free boson field.
Commun. Math. Phys. {\bf 1}, 14 (1965).
\item{20.} Robinson, P.L.: Symplectic Pathology, Quart. J. Math. Oxford
{\bf 44}, 101--107 (1993).
\item{21.} Reed, M., Simon, B.: Methods of Modern Mathematical Physics,
Vol. 1. Academic Press, New York 1980.
\item{22.} Acerbi, F.: Ph.D. thesis; and with Morchio G. and Strocchi F.:
Algebraic Fermion Bosonization, Lett. Math. Phys. {\bf 26}, 13--22 (1993).
\item{23.} Narnhofer, H., Thirring, W.: Covariant QED without indefinite metric,
Rev. Math. Phys., special issue pp 197--211 (1992).
\item{24.} Fannes, M, Verbeure, A.: On momentum states in quantum
mechanics, Ann. Inst. H. Poincar\'e Phys. Theor.
{\bf 20}, 291--296 (1974).
\item{25.} Rieffel, M.: Quantization and C*--algebras, Preprint March 1993,
Dept. Maths., University of California, Berkeley.
(To appear in Proceedings of the Conference
``C*--algebras 1943--1993 (a fifty year celebration)'' San Antonio,
Texas 1993.)
\item{26.} Wegge--Olsen, N.E.: K--theory and C*--algebras, Oxford University
Press, Oxford, 1993.
\item{27.} Manuceau, J., Sirugue, M., Testard, D, Verbeure, A.:
The Smallest C*--algebra for the Canonical Commutation Relations,
Commun. Math. Phys. {\bf 32}, 231--243 (1973).
\item{28.} Goderis, D., Verbeure, A., Vets, P.: Non--commutative Central
Limits, {\bf 82}, 527--544 (1989).
\bye