$. \item{}} Because $\norm\psi=1$, $V$ is an isometry. We omit the straightforward verification that $\rho$ is invariant under $\Eh\1V$. % % It is easy to check that $\rho$ is indeed an invariant state of % $\Eh\1V$: % $$\eqalign{ % \rho(\Eh\1V(\bigoplus_{p=0}\1{\ell-1}B_p)) % &=\rho(\bra\psi\mid\idty\otimes B_{\ell-1}\psi> % \oplus \bigoplus_{p=0}\1{\ell-2}\idty\otimes B_p)\cr % &={1\over \ell}\set{\bra\psi\mid\idty\otimes B_{\ell-1}\psi> % \bra\psi\mid\psi> % +\sum_{p=0}\1{\ell-2}\bra\psi\mid\idty\otimes B_p\psi>}\cr % &=\rho(\bigoplus_{p=0}\1{\ell-1}B_p)\cr % }$$ % For any $A\in\M_d$, the operator $\E\1V_A$ then leaves the block-diagonal algebra $\bigoplus_{p=0}\1{\ell-1}(\M_d)\1{\otimes p}$ invariant and it is given by: $$\eqalignno{ \E\1V_A(\bigoplus_{p=0}\1{\ell-1}B_p) &=V\1*A\otimes\bigoplus_{p=0}\1{\ell-1}B_p V\cr &=\bra\psi\mid A\otimes B_{\ell-1}\psi> \oplus \bigoplus_{p=0}\1{\ell-2}A\otimes B_p \quad.\cr}$$ Iterating this equation $\ell$ times we find for $A_1,\ldots A_\ell\in\M_d$: $$ (\E\1V_{A_1}\cdots\E\1V_{A_\ell}) \bigl(\bigoplus_{p=0}\1{\ell-1}B_p\bigr) =\bigoplus_{p=0}\1{\ell-1} A_1\otimes\cdots A_p \bra\psi, A_{p+1}\otimes\cdots A_\ell\otimes B_p \psi> \quad.$$ >From this equation it can be checked easily that $\om\1V=\omps$. \step4 We now show that, provided $\psi$ is not factorizable as $\psi_p\otimes\psi_{\ell-p}$, the $V$ constructed in the previous step satisfies condition (a) of the Lemma. Any operator $B\in\M_k$ can be written in block matrix form with respect to the decomposition $\Cx\1k =\bigoplus_{p=0}\1{\ell-1}\Cd p$. Thus the entry $B_{ij}$ of $B$ is an operator from $\Cd i$ to $\Cd j$, or a $d\1i\times d\1j$-matrix. $\Eh\1V$ then acts like $$\eqalign{ \Eh\1V(B) &=\Eh\1V\pmatrix{B_{00}&B_{01}&\cdots&B_{0,\lm}\cr \vdots& & & \vdots\cr B_{\lm,0}&B_{\lm,1}&\cdots&B_{\lm,\lm}\cr} \cr \hbox{\vbox to 5pt{}}\cr &=\pmatrix{ \bra\psi\mid\idty\otimes B_{\lm,\lm}\psi> &\langle\psi\mid\idty\otimes B_{\lm,0}&\cdots &\langle\psi\mid\idty\otimes B_{\lm,\ell-2}\cr \idty\otimes B_{0,\lm}\mid\psi\rangle &\idty\otimes B_{00}&\idty\otimes B_{01}&\cdots\cr \vdots& & & \vdots\cr \idty\otimes B_{\ell-2,\lm}\mid\psi\rangle &\idty\otimes B_{\ell-2,0}&\cdots& \idty\otimes B_{\ell-2,\ell-2}\cr} \quad.}$$ Thus the $(i,j)$-entry of $(\Eh\1V)\1n(B)$ depends only on $B_{i'j'}$ with $i-i'\equiv j-j'\equiv n\ {\rm mod} \ell$. A fixed point $B$ is uniquely determined by the entries $B_{m0}, m=0,\ldots\ell-1$, and the fixed point condition can be evaluated separately for each $m$. For $m=0$, $B_{00}$ is just a scalar. Thus the fixed point $B=\idty$ is the unique one for $m=0$, up to a factor. Now suppose $B_{m0}\neq0$ for some fixed point $B$. Since $B_{m0}:\Cx\to\Cd m$, this entry is given uniquely in terms of the vector $\phi_m=B_{m0}1$. Iterating the condition $\Eh\1V(B)=B$ we obtain $$\eqalign{ B_{\lm,\lm-m}\xi_{\lm-m} &=\xi_{\lm-m}\otimes\phi_m\cr B_{0,\ell-m} \xi_{\ell-m} &=\bra\psi,\xi_{\ell-m}\otimes\phi_m>\cr \bra\xi_{m-1},B_{m-1,\lm}\xi_{\lm}> &=\bra\xi_{m-1}\otimes\psi,\xi_{\lm} \otimes\phi_m>\cr \bra\xi_{m},B_{m0}\xi_0> &=\bra\xi_m\otimes\psi,\psi\otimes\phi_m>\xi_0 \cr &=\bra\xi_m,\phi_m> \xi_0 \quad, \cr}$$ where in every equation $\xi_j$ denotes an arbitrary vector in $\Cd j$. Hence with $\norm{\psi}=1$ we get the condition $\bra\phi_m\otimes\psi,\psi\otimes\phi_m> =\norm{\phi_m}\12\norm{\psi}\12$, which by the Cauchy-Schwartz inequality implies $$ \phi_m\otimes\psi=\psi\otimes\phi_m \quad.$$ Unless $\phi_m=B_{m0}1=0$, this implies a factorization of $\psi$ \tref\STEXb\ of the type we have excluded by assumption. Hence the fixed point $\idty$ is unique up to a factor. \step5 Combining the steps so far, we have shown that the set of states $\om\1V$ with $V$ satisfying the hypothesis (a) of the Lemma (\ie $V\in\Nu_a$) is w*-dense in $\Ti$. It is evident that $\Eh\1V$ depends continuously on $V$. By analytic functional calculus the fixed point $\rho$ depends continuously on $\Eh\1V$, where it is unique. Hence $\rho=\rho\1V$ is a continuous function of $V$ for $V\in\Nu_a$. For any finite $n$, and any local observable $A=A_1\otimes\cdots A_m\in\Aint m$ the function $V\mapsto \om\1V(A)=\rho\1V\circ\E\1V_{A_1}\cdots\E\1V_{A_m}(\idty)$ is continuous. By taking linear combinations and norm limits, this result carries over to arbitrary $A\in\chain\A$. Thus $V\mapsto\om\1V$ is w*-continuous on $\Nu_a$. By the Lemma we may approximate any $V\in\Nu_a$ by elements $V'\in\Nu_{abc}$, and by the Proposition the states $\om\1{V'}$ are pure. Hence the pure states of this form are w*-dense in $\Ti$. \QED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % second run of FCDR.tex : \let\REF\doref %%%%%%%%%%% \input fcdr \ACKNOW B.N. is partially supported by NSF Grant \# PHY-8912069. R.F.W is supported by a fellowship from the DFG in Bonn, and also acknowledges a travel grant to visit Princeton, where a part of this work was carried out. \REF AF \AFri \Jref L. Accardi, and A. Frigerio "Markovian Cocycles" Proc.R.Ir.Acad. @83A{(2)}(1983) 251--263 \REF AKLT \AKLT \Jref I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki "Valence bond ground states in isotropic quantum antiferromagnets" Commun.Math.Phys. @115(1988) 477--528 \REF BR \BraRo \Bref O. 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