%\input begplain \input amssym.def \input amssym.tex \magnification=1200 \overfullrule=0pt \def\m{{\cal M}} \def\r{{\Bbb R}^+} \def\mt{\widetilde{\cal M}} \def\p{\prec\prec} \def\ch{\raise 0.5ex \hbox{$\chi$}} \def\v{ \Vert_{_{E(0,\infty)}}} \def\vm{ \Vert_{_{E(\m,\tau)}}} \def\nmt{\widetilde {\cal M}} \def\nn{{\cal N}} \def\nm{{\cal M}} \def\nt{\ $\widetilde {\cal N}$} \def\nnt{\widetilde {\cal N}} \def\r{{\Bbb R}^+} \def\h{H({\cal M})} \def\g{G({\cal M})} \def\up{M^{(p)}(E)} \def\do{M_{(p)}(E)} \def\doq{M_{(q)}(E)} \def\t{\Bbb T} \def\f{{\cal F}} \def\em{E({\cal M},\tau )} \def\rem{{\rm Rad}\ \em} \def\vem{\Vert _{_{E({\cal M},\tau )}}} \def\lm{L^1({\cal M},\tau )\cap {\cal M}} \def\xa{x_{_{\alpha }}} \def\ya{y_{_{\alpha }}} \def\ea{e_{_{\alpha }}} \def\mpc{$(L^p(\nm),{\cal M})$} \def\mpcn{(L^p(\nm),{\cal M})} \def\mp+{$L^p(\nm)+{\cal M}$} \def\mp+n{L^p(\nm)+{\cal M}} \def\mtl{ L^\infty \overline \otimes \nm} \def\emtl{E(\mtl ,m \otimes \tau )} \def\d{{\Bbb D}} \def\z{{\Bbb Z}} \def\n{{\Bbb N}} \def\t{{\Bbb T}} \def\h{{\Bbb H}} \def\c{{\Bbb C}} \def\dis{\displaystyle} \centerline{ KHINTCHINE AND PALEY INEQUALITIES FOR ${\Bbb D}$-SYSTEMS} \centerline{ IN SYMMETRIC OPERATOR SPACES} \bigskip \bigskip \centerline{by} \bigskip \bigskip \centerline{ \bf P.G.Dodds, T.K.Dodds, S.V.Ferleger and F.A.Sukochev \footnote*{\rm Research supported by A.R.C.\hfil\break \indent 1980 {\sl Mathematics Subject Classification}. Primary 46E30; Secondary 46A40, 46B30.} } \bigskip \bigskip \indent {\bf Abstract}\quad {\it We consider Khintchine inequality in symmetric operator spaces in the situation when the sequence of Rademacher functions is replaced by sequence of eigenvectors of some representation of dyadic group ${\Bbb D}$ corresponding to lacunary sequence of characters from $\hat {\Bbb D}$. The same approach we apply to study of Paley inequality in non-commutative Hardy spaces and its dyadic generalizations.} \bigskip \bigskip \noindent {\bf 0. Introduction.\quad }Let $\t$ be the circle group with Lebesgue measure $dt$ and let $\z$ be its dual group. As usual we identify the character $n\in \z$ with the function $e^{int}, t\in \t$. Let $\z_2$ be the discrete cyclic group of order 2, that is $\z_2$ is the set $\{0,1\}$ with the discrete topology and addition modulo 2 and equipped with Haar measure. The {\it dyadic group} $\d$ is defined to be the compact abelian group formed by taking the Cartesian product of countably many copies of $\z_2$ so that $$\d:= \z_2\times \z_2\times \dots ,$$ equipped with the product topology and induced product (again Haar) measure $\mu$. We may identify the dual group $\hat \d$ with the set of all finitely non-zero $\{0,1\}$-valued sequence $\gamma =\{n_{_{k}}\}$, acting on $x=\{x_n\}_{n=0}^\infty \in \d$ as follows $$\gamma (x)=\prod _{k=0}^\infty (-1)^{n_k x_k}.$$ \noindent Let us set $$U_0=\{0\},\quad U_n=\{\{\gamma_k\}_{k=1}^\infty \in \hat \d:\gamma _n=1, \gamma _k=0\quad {\rm \ for\ all}\ k>n\}.$$ \noindent and for each $k\in \z^+$, and $x=\{x_n\}_{n=0}^\infty \in \d$, we define the character $\rho_k \in \hat \d$ by setting $$\rho_k(x)=(-1)^{x_k}=\cases {1&if x_k=0\cr-1&if x_k=1\cr}.$$ \noindent The sequence $\{\rho_k\}_{_{k=1}}^{\infty }$ generates $\hat \d$ i.e. for each $\gamma \in \hat \d$ there is unique sequence $(n_{_{k}})$ such that $$\gamma =\prod_{k=0}^\infty \rho _k^{n_k})$$ \noindent and is called Rademacher sequence, since its image under canonical isomorphism between $L_{_{p}}(\d,\mu)$ and $L_{_{p}}([0,1],m)$ is usual Rademacher sequence. Moreover, this isomorphism identifies the whole group $\hat \d$ with usual Walsh system $\{w_{_{n}}\}_{_{n=1}}^{\infty }$ taken in Walsh-Paley numeration, i.e. $\gamma \in \hat \d$ corresponds to $w_{_{n}}$ whenever $n=\sum _{_{k=0}}^{\infty }n_{_{k}}2^k$. The next two theorems are the classical ones and have been generalized in many ways. Both of them describe the behavior of certain sequences of characters in functional spaces. \bigskip \noindent {\bf Theorem 0.1 (Khintchine's Inequality)}\quad {\sl For all $1\le p<\infty$, there exist constant $A_{_{p}}, B_{_{p}}>0$ such that for all sequences $\{a_{_{k}}\}_{_{k=1}}^{\infty }$ and for all $n=1,2...$} $$A_{_{p}}{\biggl (} \sum _{_{k=1}}^{n}\mid a_{_{k}}\mid ^{2}{\biggr )}^{1\over 2}\le \Vert \sum _{_{k=1}}^{n}a_{_{k}}\rho _{_{k}}\Vert _{_{L^p(\d,\mu)}}\le B_{_{p}}{\biggl (} \sum _{_{k=1}}^{n}\mid a_{_{k}}\mid ^{2}{\biggr )}^{1\over 2}.$$ \bigskip \noindent {\bf Theorem 0.2\quad (Paley Gap Theorem)} {\sl There exists a constant $C>0$ such that for all functions $f=\sum _{_{n=0}}^{\infty }\hat f(n)e^{int} \in H^1 (\t)$ we have $$(\sum _{n=0}^\infty \vert \hat f(2^n)\vert ^2)^{1\over 2}\approx \Vert \sum _{_{n=0}}^{\infty } \hat f(2^n)e^{i2^n(\cdot )}\Vert _{_{H_{_{1}}(\t,dt)}}\leq \Vert f\Vert _{_{H_{_{1}}(\t,dt)}}.$$} \bigskip \noindent There is one very natural way to distinguish the characters of compact Abelian group $G$ with Haar measure $m$ among the functions from $L_{_{p}}(G,m)$. Namely, to consider them as eigenvectors of representation of $G$ at the $L_{_{p}}(G,m)$ as forward translation group (see [E 1]). This way can be extended to the case of vector-valued spaces $L_{_{p}}(G ,X )$ and $H_{_{p}}(G ,X )$ where again group $G$ is represented as forward translation group. Such generalization obtained in the recent years a great deal of attention. In other words it is just the replacement of the scalars $a_{_{k}}$ on the elements $x_{_{k}}$ of Banach space $X$ and studying the behavior of the norms $\Vert \sum _{_{k=1}}^{n}x_{_{k}}\rho _{_{k}} \Vert _{_{L^p(\d,X)}}$ and $\Vert \sum _{_{n=0}}^{\infty } \hat x_{_{k}}e^{i2^n(\cdot )}\Vert _{_{H_{_{1}}(\t,X)}}$ respectively for different classes of $X$. We mention here well-known Maurey's theorem (see [LT 2, 1.d.6]) for the case when $X$ is a Banach lattice and recent series of results [L-P 1,2], [L-PP], [BP], [HP] where many results were obtained for $X$ being either symmetrically normed ideal of compact operators with additional assumptions at its order-topological structure or predual to von Neumann algebra. Important information is also contained in [P 1,2]. We follow to the same idea of study eigenspaces of action groups $\d$ and $\t$ but represented on $X$ rather than $X$-valued space. This idea is well-known in the theory of non-commutative Hardy spaces, which has been defined to be span of \lq \lq positive" eigenspaces of non-commutative $L_{_{p}}$-spaces. Thus, we represent $G$ as group of authomorphisms of Banach space $X$ and study appearing systems of eigenspaces. We mostly concentrate at the representations of the groups $\d$ and $\t$ on the \lq \lq non-commutative" $L_{_{p}}(M,\tau )$-spaces, which usually appear as extentions to some authomorphic action of these groups on von Neumann algebra $M$. For instance, besides the algebra $L^\infty ({\Bbb D}, \mu)$ itself another imporatant example of von Neumann algebras which admit natural representation of ${\Bbb D}$ is approximatively finite-dimensional factor ${\cal R}$ of type $II_{_{1}}$. The systems of eigenvectors appearing there are called ${\Bbb D}$-systems have strong resemblance with usual Rademacher system and their behavior in symmetric operator spaces associated with underlying von Neumann algebra was recently examined in papers [SF 1,2] (see also [AFS], [FS]). Here we examine whether the \lq \lq Rademacher" subspaces generated by eigenvectors are isomorphic to $l_{_{2}}$ and complemented in enveloping symmetric operator space. \bigskip \noindent We will present here the direct analogues of Theorems 0.1 and 0.2 for actions of group $\d$ and $\t$ respectively and also introduce non-commutative $H_{_{1}}$-spaces connected with actions of the group $\d$ ( which are straight generalizations of dyadic $H_{_{1}}$-spaces), where some analogue of theorem 0.2 has been also held. \bigskip \noindent {\bf 1. Preliminaries.}\quad Throughout the article $\nm$ is a semifinite von Neumann algebra, $\tau$ is a semifinite normal faithful trace on $\nm$ with $\tau ({\bf 1})=a\le \infty$ (here {\bf 1} is the unit of $\nm$.) The $\ast$-algebra of all $\tau$ -measurable operators (see [FK]) affiliated with $\nm$ is denoted by $K(\nm,\tau )$. For symmetric function space $E[0,a )$ (see [KPS], [LT 2]) we define the symmetric space of measurable operators $E(\nm,\tau )$ associated with $E[0,a)$ and the semifinite von Neumann algebra $(\nm,\tau )$, as follows: $$E(\nm,\tau )=\{x\in K(\nm,\tau ):\mu (x)\in E[0,a )\},\ \Vert x \Vert _{_{E(\nm,\tau )}} =\Vert \mu (x) \Vert _{_{E[0,a)}},$$ \noindent where $\mu _{t}(x)$ for fixed $t\in (0,\infty )$ is defined by $$\mu _{t}(x)=\inf\{s\ge 0:\tau (\chi _{_{[s,\infty )}}(\mid x\mid )\le t\},$$ \noindent where $\chi _{_{[s,\infty )}}(\mid x\mid )$ is the spectral projection of $\mid x\mid =(x^{*}x)^{1/2}$ corresponding to the interval $(s,\infty )$. The function $\mu(x):[0,\infty)\to [0,\infty ]$ is called the generalized singular value function (or decreasing rearrangement) of $x$; note that $\mu_t(x)<\infty$ for all $t>0$. If $\nm=L_{_{\infty }}(0,a)$ the function $\mu (x)$ is the usual decreasing rearrangement ${\tilde x}(t)$ (see [KPS]). If $E[0,a )$ is separable, then $(E(\nm,\tau ),\Vert \cdot \Vert _{_{E(\nm,\tau )}})$ is a Banach space whose Banach dual $E(\nm,\tau )^{*}$ can be identified as follows $$E(\nm,\tau ) ^{*}= \{x\in K(\nm,\tau ): \Vert x\Vert _{_{E(M,\tau ) ^{*}}}=\sup \{\tau (\mid xy\mid ):y\in E(\nm,\tau ) , \Vert y \Vert _{_{E(\nm,\tau ) }}\le 1\}<\infty \}$$ \noindent and the action of $y\in E(\nm,\tau )^{*}$ on $x$ is $\tau (xy^{*})$. It is easily can be seen that if $M=L_{_{\infty }}(0,a),\ a\le \infty$, equipped with the trace $\mu$(=Lebesgue integral) generated by Lebesgue measure, then $E(\nm,\mu )$ coincides with $E(0,a)$. Proofs of assertions above and another information about these spaces can be found in [CS], [DDP 1,2,3], [SC].\bigskip \noindent Let us now recall some of the elements of the theory of representation of compact Abelian groups. Let $X$ be an arbitrary Banach space, let $G$ be a compact Abelian group with Haar measure $dt$ and dual group $\hat G$ and let $\{R_t\}_{t\in G}$ be a strongly continuous group of isometries of $X$ (or even group of invertible maps of $X$ onto itself with $c=\sup _{_{t\in G}}\Vert R_{t}\Vert _{_{X\rightarrow X}}<\infty$). For each $\gamma \in \hat G$, we define the projection $E_{\gamma }:X\to X$ by setting $$E_{\gamma }x=\int _G\gamma (u)R_{-u}xdu,$$ where the integral on the right is a Bochner integral. The range of $E_{\gamma }$ is the eigenspace $$X_\gamma :=\{x\in X: R_tx =\gamma (t)x \quad \forall t\in G\}.$$ \bigskip \noindent If $X=L^p(G,dt), 1\leq p<\infty$, and if $S=\{S_t\}_{t\in G}$ is given by forward translation $$(S_tf)(x):=f(x+t),\quad x\in G,\quad t\in G,$$then \eqalign {X_\gamma &=\{f\in L^p(G,dt):(S_tf)(x)=\gamma (t)f(x)\quad \forall t\in G\}\cr &=\{f\in L^p(G,dt):f(x+t)=\gamma (t)f(x)\quad \forall t\in G\}\cr &=\{f\in L^p(G,dt):f(t)=\gamma (t)f(0)\quad \forall t\in G\}\cr} and this is just the (one-dimensional) linear subspace spanned by the character $\gamma$. In particular, the usual trigonometric system $\{e^{2\pi in(\cdot )}\}_{n\in \z}$ appears as a system of eigenvectors arising from the representation of the group $\t$ by forward translation on $L^p(\t,m)$, while the Walsh system appears as a system of eigenvectors arising from the representation of the group $\d$ by forward translation on $L^p(\d,\mu)$.\hfill\break \indent Recall also that the representations of $G$ on von Neumann algebra $\nm$ as automorphisms group is called {\it action } of $G$ on $\nm$. In this case we will always assume that authomorphisms $R_{_{t}}$ preserves trace for any $t\in G$. \bigskip \noindent Let $(A,\Sigma _{A},\mu _{A})$ be an arbitrary probability space. One says that Banach space $X$ has the unconditional property for martingale differences (written \lq \lq $X$ is an $UMD$-space" or \lq \lq $X\in (UMD)$") if for $10$ such that for all finitely non-zero scalar sequences} $\{a_k\}_{k=0}^\infty$, $${C_{_{E}} }^{-1}\left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2} \leq \Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{_{\em}} \leq C_{_{E}} \left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}.$$ \bigskip \noindent {\bf Proof }\quad From the discussion preceding the theorem it follows that if $2\leq p<\infty$, then there exists a constant $C_{_{p}}>0$ such that for all finitely non-zero scalar sequences $\{a_k\}_{k=0}^\infty$, $$C_{_{p}}^{-1}\left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2} \leq \Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _p \leq C_{_{p}}\left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}.$$ Let now $q$ be the conjugate index to $p$. We have \eqalign {\Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{_{L_{_{q}}(\nm ,\tau )}} &\geq {\tau \left ((\sum _{k=1}^n a_k x_{\gamma _{[k]}} )(\sum _{k=1}^n a_k x_{\gamma _{[k]}})^*\right )\over \Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{_{L_{_{p}}(\nm ,\tau )}}} ={\tau \left (\sum _{k=0}^n \vert a_k\vert ^2{\bf 1}\right )\over \Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{_{L_{_{p}}(\nm ,\tau )}}}\cr &\approx {\sum _{k=0}^n \vert a_k\vert ^2\over \left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}} =\left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}.\cr } The remainig inequality follows from the continuity of the embedding of $L^2$ into $L_{_{q}}(\nm ,\tau )$ and the fact that $$\left \Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\right \Vert _2 = \left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}.$$ We suppose now that $E$ is an intermediate space for some couple $(L^{p_1},L^{p_2}), 1< p_1\leq p_2< \infty$. Thus $L^{p_2}(\nm ,\tau ) =L^{p_1}(\nm ,\tau )\cap L^{p_2}(\nm ,\tau )$ embeds continuously with constant $K_2$ into $\em$, which in turn embeds continuously into $L^{p_1}(\nm ,\tau )=L^{p_1}(\nm ,\tau )+L^{p_2}(\nm ,\tau )$ with constant $K_1$. We obtain \eqalign {C_{_{p_{_{1}}}}^{-1}K_1^{-1}\left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2} \leq K_1^{-1}\Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{p_1} &\leq \Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{_{\em}} \cr \leq K_2\Vert \sum _{k=1}^n a_k x_{\gamma _{[k]}}\Vert _{p_2} \leq K_2C_{_{p_{_{2}}}}\left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}.\quad \square \cr} \bigskip \noindent It is now of interest to exhibit some explicit lacunary systems and we turn to this question. We begin with a basic finite-dimensional example. Our exposition based on [SF 1,2], [FS], [AFS]. \bigskip \noindent {\bf Example 2.6}\quad We set $$U(0)=V(0)=1=\pmatrix {1&0\cr0&1\cr},\quad U(1)=\pmatrix {0&1\cr1&0\cr}, \quad V(1)=\pmatrix {1&0\cr0&-1\cr}.$$Let $$\h:=\z_2\times \z_2$$ and note that $\hat \h$ may also be identified with $\z_2\times \z_2$ via the duality given by setting $$\gamma (t)=(-1)^{\langle t ,\gamma \rangle},\quad t\in \h,\quad \gamma \in \hat \h.$$We define the representation $\{r_t\}_{_{t\in \h}}$ of the group $\h$ on $(M_2({\Bbb C}),tr )$, where $tr$ denotes normalised trace, by setting $$r_t(x)=U(t_1)V(t_2)xV(t_2)U(t_1),\quad t=(t_1,t_2)\in \h,\quad x\in =M_2({\Bbb C}).$$ It is clear that each $r_t$ is a trace preserving $*$-automorphism of $(M_2({\Bbb C}),tr )$. That $\{r_t\}_{_{t\in \h}}$ is a representation of $\h$ on $(M_2({\Bbb C}),tr )$ may be verified directly by tedious calculation. The eigenspaces corresponding to the representation $\{r_t\}_{_{t\in \h}}$ may be calculated by observing that $$E_\gamma x=\int _\h \gamma (t)r_{-t}xdt={1\over 4}\sum _{t\in \h}(-1)^{\langle t,\gamma \rangle}r_{t}x, \quad \gamma \in \hat \h, x\in M_2({\Bbb C}).$$ Direct computation using this formula now yields that the eigenspaces $(M_2({\Bbb C}))_\gamma ,\gamma \in \hat \h$ are the one-dimensional subspaces generated by the matrices $$e_{(0,0)}=\pmatrix {1&0\cr0&1\cr},\quad e_{(1,0)}=\pmatrix {1&0\cr0&-1\cr},\quad e_{(0,1)}=\pmatrix {0&1\cr1&0\cr},\quad e_{(1,1)}=\pmatrix {0&-1\cr1&0\cr}$$ \bigskip \noindent {\bf Example 2.7 \quad First (semi-commutative) Rademacher system}\quad We let \hfill\break \noindent ${\cal R}=\bigotimes _{k=1}^\infty (M_2({\Bbb C}),\tau )$ be the unique approximatevely finite-dimensional factor of type $II_1$ with faithful normal finite trace $\tau =\bigotimes _{k=1}^\infty tr$. The finite-dimensional factors $\bigotimes _{k=1}^n (M_2({\Bbb C}),tr)\otimes _{k=n+1}^\infty {\Bbb C}.1_{_{(M_2({\Bbb C}),tr}}$ will be denoted by ${\cal R}_n$. We let $$G=\prod _{n=1}^\infty(\z_2\times \z_2)$$ and define the representation $$R_t=\bigotimes _{k=1}^\infty r_{t_k},\quad t=(t_k)_{k=1}^\infty \in G$$ of $G$ on ${\cal R}$ by setting $$R_tx=\bigotimes _{k=1}^\infty r_{t_k}x_k, \quad t=(t_k)_{k=1}^\infty \in G,\quad x=\bigotimes _{k=1}^\infty x_k \in {\cal R}.$$ It is now easily checked that the restriction of each $R_t$ to ${\cal R}_n$ is a trace preserving $*$-automorphism, and we omit the further details needed to show that $R$ in fact defines a strongly continuous representation of $G$ on $({\cal R},\tau )$. Such an argument may be based on the density of $\bigcup _{k=1}^\infty {\cal R}_k$ in ${\cal R}$. \bigskip \noindent To compute the eigenspaces, let $\gamma =\{\gamma _k\}_{k=1}^\infty \in \hat G$ and observe that there exists a natural number $N$ such that $\gamma _k=0$ for all $k>N$. If $x=\bigotimes _{k=1}^\infty x_k \in {\cal R}$ then \eqalign {E_\gamma (x)&=\int _G\gamma (t)R_{-t}xdt\cr &=(\bigotimes _{k=1}^N\int _{\h}\gamma _k(t_k)r_{t_k}x_kdt_k)\otimes(\bigotimes _{k>N}x_k)\cr &=(\bigotimes _{k=1}^NE_{\gamma _k}(x_k))\otimes(\bigotimes _{k>N}x_k).\cr} Consequently, we may write formally $$E_\gamma =\bigotimes _{k=1}^\infty E_{\gamma _k},\quad \gamma \in \hat G,$$ and for each $\gamma =(\gamma _1,\gamma _2 \dots )\in \hat G$, it follows that the corresponding eigenspace $\left (({\cal R},\tau )\right )_\gamma$ is just the one-dimensional subspace spanned by $\displaystyle {\bigotimes _{k=1}^\infty }e_{\gamma _k}$,for each $\gamma \in \hat G$. \bigskip\noindent We now identify $$G=\prod_{k-1}^\infty (\z_2\times \z_2)=\d\times \d$$ with $D=\prod_{k-1}^\infty \z_2$ via the mapping $$\left ((t_1^{(1)},t_1^{(2)}),(t_2^{(1)},t_2^{(2)}),\dots ,(t_n^{(1)},t_n^{(2)}),\dots \right )\to (t_1^{(1)},t_1^{(2)},t_2^{(1)},t_2^{(2)},\dots ,t_n^{(1)},t_n^{(2)},\dots ),$$ with a similar identification of $\hat G=\hat \d \times \hat \d$ with $\hat \d$. We let $\{R_t\}_{t\in \d}$ be the induced representation on $({\cal R},\tau)$ given by $$R_tx=\bigotimes _{k=1}^\infty r_{(t_k,t_{k+1})}x_k, \quad t=(t_k)_{k=1}^\infty \in \d,\quad x=\bigotimes _{k=1}^\infty x_k \in {\cal R}.$$ The eigenspace $\left (({\cal R},\tau )\right )_\gamma$ is just the one-dimensional subspace spanned by $\displaystyle {\bigotimes _{k=0}^\infty }e_{(\gamma _{2k+1},\gamma _{2k})}$, for each $\gamma =(\gamma _1,\gamma _2,\dots )\in \hat D$. The Rademacher system $\{x_{\rho_n}\}_{n=1}^ \infty \subseteq ({\cal R},\tau )$ corresponding to the representation $R$ may now be taken to be given by the choice $$x_{\rho _n}=\cases {\dis{(\bigotimes _{k=1}^m}e_{(0,0)})\otimes e_{(1,0)}\otimes \bigotimes _{k=m+2}^\infty e_{(0,0)}& if n=2m+1,, m=0,1,2, \dots ;\cr \dis{(\bigotimes _{k=1}^m}e_{(0,0)})\otimes e_{(0,1)}\otimes\bigotimes _{k=m+2}^\infty e_{(0,0)}& if n=2m,, m=1,2, \dots .\cr}$$ \bigskip \noindent A further example of a lacunary sequence of eigenvectors, to which Theorem 2.5 applies, is given by setting, for example, $$y_{n}=\cases {(\bigotimes _{k=1}^m}e_{(0,0)})\otimes e_{(1,0)}\otimes\bigotimes _{k=m+2}^\infty e_{(0,0)}& if n=2m+1,, m=0,1,2, \dots ;\cr \dis{(\bigotimes _{k=1}^m}e_{(0,0)})\otimes e_{(1,1)}\otimes\bigotimes _{k=m+2}^\infty e_{(0,0)}& if n=2m,, m=1,2, \dots .\cr$$ \bigskip \noindent Let us return to the Rademacher system $\{x_{\rho_n}\}_{n=1}^\infty \subseteq ({\cal R},\tau )$. We observe first that the commutative subsystem $\{x_{\rho_{2m+1}}\}_{m=0}^\infty$ may be identified with the usual Rademacher sequence on the interval $[0,1]$. In fact, there is a trace preserving $*$-isomorphism of the algebra $\displaystyle {\bigotimes _{n=1}^\infty } {\rm diag} M_2({\Bbb C})$ onto $L^\infty [0,1]$ which maps $x_{\rho_{2m+1}}$ to $\rho _{m+1}, m=0,1,2,\dots$. A similar statement may be made concerning the commutative subsystem $\{x_{\rho_{2m}}\}_{m=1}^ \infty .$ To see this, let $$A=\pmatrix {0&1\cr 1&0\cr},$$ and observe that the $*$-automorphism of $M_2({\Bbb C})$ given by the map $X\to U^*XU,\ X\in M_2({\Bbb C})$, where $$U={1\over \sqrt 2}\pmatrix {1&1\cr 1&-1\cr},$$ maps $A$ onto the matrix $$\pmatrix {1&0\cr 0&-1\cr }.$$ This $*$-automorphism extends in a natural way to a $*$-automorphism of $\displaystyle {\bigotimes _{n=1}^\infty } {\rm diag} M_2({\Bbb C})$ which maps $x_{\rho_{2m}}$ to $x_{\rho_{2m-1}},\ m=1,2, \dots$. \bigskip \noindent We now recall the result of Rodin-Semyonov ([LT2] 2.b.4(i)) that if $E$ is any rearrangement invariant Banach function space on $[0,1]$, then the span of the Rademacher sequence $\{r_k\}_{k=0}^\infty$ in $E$ is equivalent to the the unit vector basis of $l^2$ if and only if there exists a constant $K>0$ such that $$\Vert f\Vert _{_{E}}\leq K\Vert f\Vert _{_{M}},\quad f\in L^\infty [0,1],$$ where $M$ denotes the Orlicz function $$M(t)=(e^{t^2}-1)/(e-1)$$ and $\Vert \cdot \Vert _{_{ M}}$ denotes the norm in the Orlicz space $L_{_{M}}[0,1]$. We shall refer to this condition as the condition of Rodin-Semyonov. We can now state the following form of the Khintchine inequalities for the first Rademacher system $\{x_{\rho_n}\}_{n=1}^\infty \subseteq ({\cal R},\tau )$. The result which follows shows that the Khintchine inequalities hold in the non-commutative space $E({\cal R},\tau)$ if and only if the classical form of the inequalities hold in the commutative space $E$. \bigskip \noindent {\bf Theorem 2.8}\quad {\sl Let $E$ be a rearrangement-invariant Banach function space on $[0,1]$, and let $\{x_{\rho_n}\}_{n=1}^\infty \subseteq ({\cal R},\tau )$ be the first Rademacher system. The following statements are equivalent.} \item {(i)}\quad {\sl There exists a constant $C>0$ such that for all finitely non-zero scalar sequences} $\{a_k\}_{k=0}^\infty$, $$C^{-1} \left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2} \leq \Vert \sum _{k=1}^n a_k x_{\rho _{k}}\Vert _{_{E({\cal R},\tau )}} \leq C \left (\sum _{k=1}^n\vert a_k\vert ^2\right )^{1/2}.$$ \item {(ii)}\quad {\sl $E$ satisfies the condition of Rodin-Semyonov.} \bigskip \noindent {\bf Proof }\quad The implication (i)$\Rightarrow$(ii) follows from the theorem of Rodin-Semyonov and the fact that $$\Vert \sum _{k=1}^n a_k x_{\rho _{2k}}\Vert _{_{E({\cal R},\tau )}} =\Vert \sum _{k=0}^n a_k x_{\rho _{2k+1}}\Vert _{_{E({\cal R},\tau )}} =\Vert \sum _{k=1}^n a_k r_k\Vert _{_{E}}.$$ \bigskip \noindent (ii)$\Rightarrow$ (i).\quad We assume that $E$ satisfies the condition of Rodin-Semyonov. Again using the equalities $$\Vert \sum _{k=1}^n a_k x_{\rho _{2k}}\Vert _{_{E({\cal R},\tau )}} =\Vert \sum _{k=0}^n a_k x_{\rho _{2k+1}}\Vert _{_{E({\cal R},\tau )}} =\Vert \sum _{k=1}^n a_k r_k\Vert _{_{E}},$$ and the theorem of Rodin-Semyonov, it follows that \eqalign {\Vert \sum _{k=1}^n a_k x_{\rho _{k}}\Vert _{_{E({\cal R},\tau )}} &\leq \Vert \sum _{1<2k\le n} a_{2k} x_{\rho _{2k}}\Vert _{_{E({\cal R},\tau )}} +\Vert \sum _{1\le 2k+1\le n} a_{2k+1} x_{\rho _{2k+1}}\Vert _{_{E({\cal R},\tau )}}\cr &\leq C \left (\sum _{1<2k\le n}\vert a_{2k}\vert ^2\right )^{1/2} + \left (\sum _{1\le 2k+1\le n}\vert a_{2k-1}\vert ^2\right )^{1/2} \cr &\leq 2C \left (\sum _{k=1}^{n}\vert a_k\vert ^2\right )^{1/2}.\cr } Further, \eqalign { \left (\sum _{1<2k\le n}\vert a_k\vert ^2\right )^{1/2} &\leq \left (\sum _{k=1}^n\vert a_{2k}\vert ^2\right )^{1/2} + \left (\sum _{1\le 2k+1\le n}\vert a_{2k-1}\vert ^2\right )^{1/2} \cr &\leq C^{-1}\left ( \Vert \sum _{1<2k\le n} a_{2k} x_{\rho _{2k}}\Vert _{_{E({\cal R},\tau )}} +\Vert \sum _{1\le 2k+1\le n} a_{2k+1} x_{\rho _{2k+1}}\Vert _{_{E({\cal R},\tau )}} \right )\cr.}To complete the proof, it suffices therefore to establish the inequalities $$\Vert \sum _{k=1}^n a_{2k} x_{\rho _{2k}}\Vert _{_{E({\cal R},\tau )}}, \Vert \sum _{k=0}^n a_{2k+1} x_{\rho _{2k+1}}\Vert _{_{E({\cal R},\tau )}} \leq \Vert \sum _{k=1}^n a_k x_{\rho _{k}}\Vert _{_{E({\cal R},\tau )}}.$$ To this end, it suffices to observe that $\sum _{k=0}^n a_{2k+1} x_{\rho _{2k+1}}$ (respectively, $\sum _{k=1}^n a_{2k} x_{\rho _{2k}}$) is the image of $\sum _{k=1}^n a_k x_{\rho _{k}}$ under the conditional expectation of ${\cal R}$ onto $\displaystyle {\bigotimes _{n=1}^\infty } {\rm diag} M_2({\Bbb C})$ (respectively, $\displaystyle {\bigotimes _{n=1}^\infty }N$, where $N$ is commutative subalgebra of $M_2({\Bbb C})$ generated by $1$ and $A=\pmatrix {0&1\cr1&0\cr}$). In fact, it is easily checked that if $T$ is conditional expectation of $M_2({\Bbb C})$ onto diag$M_2({\Bbb C})$, then $T(A)=0$. Similarly, it is not difficult to check that the image of $\pmatrix {1&0\cr 0&-1\cr}$ under conditional expectation of $M_2({\Bbb C})$ onto $N$ is also $0$. \bigskip \noindent {\bf Example 2.9\quad Second (anti-commutative) Rademacher system}\quad The following example is taken from [AFS]. Consider the Pauli matrices $$\sigma _1=\pmatrix {1&0\cr 0&1\cr },\quad \sigma _2=\pmatrix {0&1\cr1&0\cr }\quad \sigma _3=\pmatrix {0&i\cr-i&0},$$and define the sequence $\{r_k\}_{k=0}^\infty \subseteq {\cal R}$ by setting $$r_0=1,\quad r_1=\sigma _1\otimes \bigotimes _{k=2}^\infty 1_{M_2(\c)},\quad r_2=\sigma _2\otimes \bigotimes _{k=2}^\infty 1_{M_2(\c)},$$ and $$r_n=\cases {(\dis{\bigotimes _{k=1}^m}\sigma _3)\otimes\sigma _1\otimes (\dis {\bigotimes _{k=m+2}^\infty }1_{M_2(\c)})& if n=2m+1, m=1,2, \dots ;\cr (\dis{\bigotimes _{k=1}^{m-1}}\sigma _3)\otimes\sigma _2\otimes (\dis {\bigotimes _{k=m+1}^\infty }1_{M_2(\c)})& if n=2m, m=2,3, \dots .\cr}$$It is shown in [AFS] Lemma 1 that $r_n^*=r_n, r_n^2=1,$ and $r_nr_m=-r_mr_n$ for all $n,m=1,2, \dots ,n\neq m$. We will now show that this system arises as a Rademacher sequence corresponding to some representation of $\d$ on $L^p({\cal R},\tau )1\leq p<\infty .$ \bigskip \noindent Following [AFS], we set $$w_0=1,\quad w_\gamma =\prod _{n\in A_\gamma }r_n,\quad 0=\neq \gamma \in \hat \d,$$ where $$A_\gamma = \{n\in {\Bbb N}: \gamma _n=1\},\quad \gamma =(\gamma _1,\gamma _2 ,\dots )\in \hat \d .$$ For each finite subset ${\cal F}\subseteq \hat \d,t\in \d$ and scalars $\{\alpha _\gamma \}_{\gamma \in {\cal F}}$, we set $$R_t (\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma ) =\sum _{\gamma \in {\cal F}}\alpha _\gamma \gamma (t)w_\gamma .$$ We observe first that $R_0=Id$, and that for all $s,t\in \d$, \eqalign {R_tR_s (\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma ) &=\sum _{\gamma \in {\cal F}}\alpha _\gamma \gamma (t)\gamma (s)w_\gamma\cr &=\sum _{\gamma \in {\cal F}}\alpha _\gamma \gamma (t+s)w_\gamma \cr &=R_{t+s}(\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma ).\cr} Using now [AFS] Lemma 2(i), we obtain \eqalign {R_t \left (\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma \sum _{\mu \in {\cal F^\prime}}\beta _\mu w_\mu \right ) &=R_t(\sum_{\gamma ,\mu}\alpha _\gamma \beta _\mu w_{\gamma +\mu}(-1)^{I(\gamma +\mu)})\cr &=\sum_{\gamma ,\mu}\alpha _\gamma \beta _\mu (\gamma +\mu)(t)w_{\gamma +\mu}(-1)^{I(\gamma +\mu)}\cr &=\sum_{\gamma ,\mu}\alpha _\gamma \beta _\mu \gamma (t) \mu(t)w_{\gamma +\mu}(-1)^{I(\gamma +\mu)}\cr & =(\sum _{\gamma \in {\cal F}}\alpha _\gamma \gamma (t)w_\gamma ) (\sum _{\mu \in {\cal F^\prime}}\beta _\mu \mu (t)w_\mu )\cr &=R_t(\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma ) R_t(\sum _{\mu \in {\cal F^\prime}}\beta _\mu w_\mu ).\cr} Further, from [AFS] Lemma 2(ii), it follows immediately that $$\tau \left (R_t (\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma )\right ) =\tau (\sum _{\gamma \in {\cal F}}\alpha _\gamma w_\gamma ).$$ \bigskip \noindent The above calculations show that, for each $t\in \d$ ,the restriction of the mapping $R_t$ to each finite-dimensional factor ${\cal R}_n$ is a trace preserving $*$-automorphism, and consequently is an isometry in the $L^p$-norm for each $1\leq p<\infty$. Since the set $\cup _{n=1}^\infty {\cal R}_n$ is dense in $L^p({\cal R},\tau )$ for every $1\leq p<\infty$, it follows that $R=\{R_t\}_{t\in \d}$ extends to a group of isometries (which should also be checked to be strongly continuous) on $L^p({\cal R},\tau )$ for every $1\leq p<\infty$, whose restriction to $({\cal R},\tau )$ is a group of trace-preserving $*$-automorphisms of $({\cal R},\tau )$. It now remains to check that the system $\{w_\gamma\}_{\gamma \in \hat \d}$ is in fact a Walsh system corresponding to the representation $R$ on $\L^p({\cal R},\tau ),1\leq p<\infty$. For each $\gamma \in \hat \d$, it follows from the definition of $R_t$ that $$R_tw_\gamma =\gamma (t)w_\gamma,\quad \forall t\in \d,$$ and consequently, it follows that $w_\gamma \in \left (L^p ({\cal R},\tau )\right )_\gamma$, for all $\gamma \in \hat \d$. Denoting by $E_\gamma$ the projection onto the eigenspace $\left (L^p ({\cal R},\tau )\right )_\gamma$, observe that it follows [BGM] from the fact that $$\left (L^p ({\cal R},\tau )\right )_\gamma \cap \left (L^p ({\cal R},\tau )\right )_\mu=\{0\}, \quad \gamma \neq \mu .$$ On the other hand, for each $\gamma \in \hat \d$, \eqalign {E_\gamma (L^p ({\cal R},\tau ))&=E_\gamma (\overline {\{\sum _{\mu \in {\cal F}}\beta _\mu w_\mu )\}}\cr &=\overline {\{E_\gamma(\sum _{\mu \in {\cal F}}\beta _\mu w_\mu )\}}\cr &=\overline {\{\sum _{\mu \in {\cal F}}\beta _\mu E_\gamma w_\mu \}} =\overline {\{\beta _\gamma w_\gamma\}}.\cr } This shows that the eigenspaces correspnding to the representation ${\cal R}$ are just the one dimensional eigenspaces spanned by elements of the system $\{w_\gamma \}_{\gamma \in \hat \d}$. \bigskip \noindent Let us note that the real linear span of the anti-commutative Rademacher system $\{r_n\}$ in any $L^p({\cal R},\tau )$ is even isometric to $l^2$. This follows by observing that, if $x=\sum _{k=1}^N \alpha _nr_n$ is any finite linear combination with real coefficients, then $$\vert x\vert ^2=\sum _{n,m}\alpha _n\alpha _mr_n^*r_m =\sum _{n,m}\alpha _n\alpha _mr_nr_m =\sum _{n=1}^N\vert \alpha _n\vert ^21.$$ Suppose that $E$ is {\it any} rearrangement invariant space on the interval $[0,1]$ and suppose, for the sake of definiteness, that $\Vert \chi _{_{ [0,1]}}\Vert _{_{E({\cal R},\tau )}}=1$. It follows immediately that $$\Vert \sum _{k=1}^N \alpha _nr_n\Vert _{_{E({\cal R},\tau )}} =\left (\sum _{k=1}^N \vert \alpha _k\vert ^2 \right )^{1/2}=\Vert\{\alpha _k\}_{k=1}^N\Vert _2$$ for any real sequence $\{\alpha _k\}_{k=1}^N$. Now suppose that $\{\alpha _k\}_{k=1}^N\subseteq {\Bbb C}$ and observe that \eqalign {\Vert \sum _{k=1}^N \alpha _nr_n\Vert _{_{E({\cal R},\tau )}} &\leq \Vert \sum _{k=1}^N \Re (\alpha _n)r_n\Vert _{_{E({\cal R},\tau )}} + \Vert \sum _{k=1}^N \Im (\alpha _n)r_n\Vert _{_{E({\cal R},\tau )}}\cr &=\Vert \{\Re (\alpha _k)\}_{k=1}^N\Vert _2 +\Vert \{\Im (\alpha _k)\}_{k=1}^N\Vert _2\cr &\leq 2\Vert\{\alpha _k\}_{k=1}^N\Vert _2 .\cr } On the other hand, \eqalign {2\Vert \sum _{k=1}^N \alpha _nr_n\Vert _{_{E({\cal R},\tau )}} &\geq \Vert \sum _{k=1}^N \alpha _nr_n +\sum _{k=1}^N {\overline \alpha } _nr_n^*\Vert _{_{E({\cal R},\tau )}}\cr &= \Vert \sum _{k=1}^N \Re (\alpha _n)r_n\Vert _{_{E({\cal R},\tau )}} =\Vert \{\Re (\alpha _k)\}_{k=1}^N\Vert _2.\cr } Similarly, $$2\Vert \sum _{k=1}^N \alpha _nr_n\Vert _{_{E({\cal R},\tau )}} \geq \Vert \{\Im (\alpha _k)\}_{k=1}^N\Vert _2.$$ We obtain therefore that the Khintchine inequalities $${1\over 4} \Vert \{\alpha _k\}_{k=1}^N\Vert _2\leq \Vert \sum _{k=1}^N \alpha _nr_n\Vert _{_{E({\cal R},\tau )}}\leq 2 \Vert \{\alpha _k\}_{k=1}^N\Vert _2$$ are valid for any rearrangement invariant space $E$ on $[0,1]$.\bigskip \noindent The restriction of ergodicity of the action on the von Neumann algebra is natural. Indeed, the conclusion of Theorem 2.5 fails in the case of the Schatten ideals ${\cal C}_p$, even for natural (but non-ergodic !) representations of ${\Bbb D}$, as the following example shows. \bigskip \noindent {\bf Example 2.10}\quad Let $(\nm ,\tau )$ be ${\cal L}({\cal H})$, with ${\cal H}$ separable, equipped with the canonical trace $\tau$. We let $\{\varphi _n\}_{n=1}^\infty$ be some orthonormal basis in ${\cal H}$ and let $e_{nm}=\varphi _n\otimes \varphi _m,\ n,m=1,2, \dots$ be the corresponding system of matrix units. For each $x\in \nm$, we set $$R_tx=u_txu_t,\quad t\in \d$$ where $$u_t =\sum _{n=1}^\infty \rho _n(t)e_{nn},\quad t\in \d$$ with convergence in the strong operator topology. It is clear that each $u_t,t\in \d$ is a self-adjoint unitary operator and moreover $$u_su_t=u_{s+t},\quad s,t\in \d.$$ To calculate the eigenspaces, let $x\in \nm ,\ \gamma \in \d$ and observe that \eqalign {e_{mm}(E_\gamma x)e_{nn}&=\int _{\d}e_{mm}(\sum _{k=1}^\infty \rho _k(t)e_{kk} )x(\sum _{j=1}^\infty \rho _j(t)e_{jj})e_{nn}\gamma (t)dt\cr &=\int _{\d}\rho_m(t)e_{mm}xe_{nn}\rho _n(t)\gamma (t)dt\cr &=e_{mm}xe_{nn}\int _\d (\rho _n +\rho _m+\gamma )(t)dt\cr &=\cases {0, &if \rho _n+\rho _m+\gamma \neq 0;\cr e_{mm}xe_{nn}, &if \rho _n+\rho _m+\gamma = 0.\cr}\cr } We obtain that \eqalign { E_0({\cal L}({\cal H}))&=\left \{\sum _{n=0}^\infty \lambda _ne_{nn}:\{\lambda _n\}\in l^\infty \right \},\cr E_\gamma ({\cal L}({\cal H}))&=\{\lambda e_{mn}+\mu e_{nm}:\lambda ,\mu \in {\Bbb C}\}\cr } if $\gamma =\rho _n+\rho _m$, and $\{0\}$ otherwise. Thus, in contrast to the earlier examples for the finite case, the eigenspace corresponding to the $0$ character is far from one-dimensional; moreover, $E_{\rho _n}=0$ for all $n=1,2, \dots$, so that the only Rademacher system corresponding to the representation $R$ is trivial. Nonetheless, there is an abundance of lacunary systems. We consider, in particular, the lacunary system of characters $\{\gamma _{[n]}\}_{n=1} ^\infty$ defined by setting $$(\gamma _{[n]})_k=\cases {1,&if k=2n-1,2n;\cr 0,& otherwise .\cr}$$ We let $p\geq 2$ and let $$x_{\gamma _{[n]}} =\alpha _ne_{2n,2n-1}+\beta _ne_{2n-1,2n},\quad n\geq 1,$$ where $$\vert \alpha _n\vert ^p+\vert \beta \vert ^p=1, \quad n\geq 1,$$ be a corresponding lacunary system in the Schatten ideal ${\cal C}_p$. It is clear that $$x_{\gamma _{[n]}} x_{\gamma _{[m]}}=0,\quad n\neq m.$$ Let $\{a_n\}_{n=1}^N$ be any finitely non-zero sequence of scalars. Observe that $\left (\sum _{k=0}^N \vert a_k x_{\gamma _{[k]}}\vert ^2\right )^{1/2}$ is a diagonal matrix with entries $$\vert a_1\beta _1\vert , \vert a_1\alpha _1\vert,\vert a_2\beta _2\vert , \vert a_2\alpha _2\vert,\dots ,\vert a_N\beta _N\vert , \vert a_N\alpha _N\vert,$$ while $\left (\sum _{k=0}^N \vert {\overline a}_k x^*_{\gamma _{[k]}}\vert ^2\right )^{1/2}$ is a diagonal matrix with entries $$\vert a_1\alpha _1\vert ,\vert a_1\beta _1\vert ,\vert a_2\alpha _2\vert, \vert a_2\beta _2\vert\dots ,\vert a_N\alpha _N\vert,\vert a_N\beta _N\vert.$$ \bigskip \noindent {\bf 3. Paley Gap Theorem in non-commutative $H_{_{1}}$-spaces\quad } Our aim in this section is to establish analogue of the Theorem 0.2 for non-commutative $H_{_{1}}$-spaces associated with actions of group $\t$ and $\d$. The first ones are well-known (see , for example [Z], [S])the second ones are being defined here are the non-commutative counterparts of well-known dyadic $H_{_{1}}$-spaces (see, for example [SWS] and [G]). \bigskip \noindent {\bf Definition 3.1\quad }{\sl If $R=\{R_t\}_{t\in \t}$ is some $\sigma$-weakly continuous action of $\t$ on $(\nm ,\tau )$, then non-commutative $H_{_{1}}$-space (associated with $R$ and $(\nm ,\tau )$) is defined to be the closure in $L_{_{1}}(\nm ,\tau )$-norm of all finite linear sums $\sum x_{_{\gamma }}$ where $x_{_{\gamma }}\in (L_{_{1}} (\nm ,\tau))_\gamma ,\gamma \in \z , \gamma \ge 0$.} \bigskip \noindent We suppose that $R=\{R_t\}_{t\in \d}$ is some $\sigma$-weakly continuous ergodic action of $\d$ on $\nmt$. We will assume also that each eigenspace $\nm _\gamma ,\gamma \in \hat \d$ is one -dimensional (see discussion in Remark 2.3). It should be pointed out, that all results of the present section will hold also without the latter assumption, but our main examples of action of $\d$ on non-commutative algebras exposed in the Examples 2.7 and 2.9 obey this requirement. This assumption allows us to define generalized Walsh system (i.e. the system of eigenvectors) directly, exactly as classical functional system (see [SWS],p.1). We are going at first to discuss this matter in more details.\hfill\break indent Let us first choose our Rademacher system $\{r_n \}_{n=1}^\infty$ by the requirements that $r_n\in \nm _{\rho _n}$ is a self-adjoint unitary operator. We may now {\it define } the corresponding Walsh system by setting $$w_0=1,\quad w_\gamma := \prod_{n\in A_\gamma }r_n , 0\neq \gamma \in \hat \d,$$ where $A_\gamma =\{n\in {\Bbb N}:\gamma (n)=1\}$. Equivalently, $$w_\gamma =r_{n_1}r_{n_2}\cdots r_{n_k},\quad {\rm if }\quad \gamma =\rho _{n_1}\rho _{n_2}\cdots \rho _{n_k}.$$ From Proposition 2.2, it follows that $w_\gamma$ is a unitary operator in $\nm _\gamma$ for all $\gamma \in \hat \d$. The argument of [AFS], Lemma 2 (vi) shows that the system $\{w_\gamma \}_{\gamma \in \hat \d}$ is a minimal and total system in each $L^p (\nm ,\tau),1\leq p <\infty$ and [FS] Mat.Zam. shows that the system $\{w_\gamma \}_{\gamma \in \hat \d}$ is a Schauder basis in each $L^p (\nm ,\tau),1< p <\infty$. \bigskip \noindent For each $n=1,2, \dots$, we let $$G_n:=\{\gamma \in \hat \d: \gamma (k)=0, k>n\},\quad \nm _n={\rm clm} \{w_\gamma:\gamma \in G_n\}.$$ Since each $r_n$ is a self-adjoint unitary, it follows now that each $\nm _n$ is a finite-dimensional von Neumann subalgebra of $\nm$. It is clear that $\nm _n\uparrow _n$, and that $\nm =\overline {\cup _{n=1}^\infty \nm _n}^{w.o.}$ The argument of [AFS], Lemma 3, shows that the projection along $G_n$ exists in any $L^p (\nm ,\tau),1\leq p <\infty$, and coincides with the conditional expectation ${\cal E}(\cdot \mid \nm _n)$ for each $n\in {\Bbb N}$. Observe that $U_{n+1}=G_{n+1}\backslash G_n ,n\geq 1$. If $1n\},\quad \nm _n={\rm clm} \{w_\gamma:\gamma \in G_n\}.$$Since each$r_n$is a self-adjoint unitary, it follows now that each$\nm _n$is a finite-dimensional von Neumann subalgebra of$\nm $. It is clear that$\nm _n\uparrow _n$, and that$\nm =\overline {\cup _{n=1}^\infty \nm _n}^{w.o.}$The argument of [AFS], Lemma 3, shows that the projection along$G_n$exists in any$L^p (\nm ,\tau),1\leq p <\infty $, and coincides with the conditional expectation$ {\cal E}(\cdot \mid \nm _n)$for each$n\in {\Bbb N}$. Observe that$U_{n+1}=G_{n+1}\backslash G_n ,n\geq 1$. If$1