0$ such that \begin{equation}\label{34} \left\| \sum_{j=1}^n \epsilon_j d_{\alpha_j} *f \right\|_1 \leq a \|f\|_1. \end{equation} Furthermore, \begin{equation} \label{ucc of h1 equation} f = \mu_{\alpha_0}*f + \sum_\alpha d_\alpha*f, \end{equation} where the right hand side converges unconditionally in the norm topology on $H^1(G)$. \end{thm} % {\bf Proof.} The second part of Theorem~\ref{ucc of h1 fts} follows easily from the first part and Fourier inversion. Now let us show that if we have the result for compact $G$, then we have it for locally compact $G$. Let $\pi_{\alpha_0} :\ \Gamma\rightarrow \Gamma/C_{\alpha_0}$ denote the quotient homomorphism of $\Gamma$ onto the discrete group $\Gamma/C_{\alpha_0}$ (recall that $C_{\alpha_0}$ is open), and define a measurable order on $\Gamma/C_{\alpha_0}$ to be $\pi_{\alpha_0}(P)$. By Remarks \ref{remarkstructureorder} (c), the decomposition of the group $\Gamma/C_{\alpha_0}$ that we get by applying Theorem \ref{structureorder} to that group, is precisely the image by $\pi_{\alpha_0}$ of the decomposition of the group $\Gamma$. Let $G_0$ denote the compact dual group of $\Gamma/C_{\alpha_0}$. Thus if Theorem \ref{ucc of h1 fts} holds for $ H^1(G_0)$, then applying Theorem \ref{homth2}, we see that Theorem \ref{ucc of h1 fts} holds for $G$. Henceforth, let us suppose that $G$ is compact. We will suppose that the Haar measure on $G$ is normalized, so that $G$ with Haar measure is a probability space. Since each one of the subgroups $C_\alpha$, and $D_\alpha$ ($\alpha<\alpha_0$) is open, it follows that their annihilators in $G$, $G_\alpha=A(G,C_\alpha)$, and $A(G,D_\alpha)$, are compact. Let $\mu_\alpha$ and $\nu_\alpha$ denote the normalized Haar measures on $A(G,C_\alpha)$ and $A(G,D_\alpha)$, respectively. We have $\widehat{\mu}_\alpha=1_{C_\alpha}$ (for all $\alpha$), and $\widehat{\nu}_\alpha=1_{D_\alpha}$ (for all $\alpha\neq \alpha_0$), so that $d_\alpha=\mu_\alpha -\nu_\alpha$. For each $\alpha$, let ${\cal B}_\alpha$ denote the $\sigma$-algebra of subsets of $G$ of the form $A+G_\alpha$, where $A$ is a Borel subset of $G$. We have ${\cal B}_{\alpha_1}\subset {\cal B}_{\alpha_2}$, whenever $\alpha_1>\alpha_2$. It is a simple matter to see that for $f\in L^1(G)$, the conditional expectation of $f$ with respect to ${\cal B}_\alpha$ is equal to $\mu_\alpha*f$ (see \cite[Chapter 5, Section 2]{eg}). We may suppose without loss of generality that $\alpha_1>\alpha_2>\ldots>\alpha_n$. Thus the $\sigma$-algebras ${\cal B}_{\alpha_k}$ form a filtration, and the sequence $(d_{\alpha_1}*f, d_{\alpha_2}*f,\ldots,d_{\alpha_n}*f)$ is a martingale difference sequence with respect to this filtration. In that case, we have the following result of Burkholder \cite[Inequality (1.7)]{bur}, and \cite{bur1}. If $0

0$, for only finitely many $k$ do we have that $\| d_{\alpha_k} *_T \mu \| > \delta$. Since this is true for all such countable sets, we deduce that the set of $\alpha$ for which $ d_\alpha *_T \mu \ne 0$ is countable. Hence we have that $\sum_\alpha d_{\alpha} *_T \mu$ is unconditionally convergent to some measure, say $\nu$. Clearly $\nu$ is weakly measurable. To prove that $\mu=\nu$, it is enough by Proposition \ref{prop hypa} to show that for every $A\in\Sigma$, we have $T_t\mu(A)=T_t\nu(A)$ for almost all $t\in G$. We first note that since for every $f\in L^1(G)$ the series $\mu_{\alpha_0}*f+ \sum_\alpha d_\alpha *f$ converges to $f$ in $L^1(G)$, it follows that, for every $g\in L^\infty(G)$, the series $\mu_{\alpha_0}*g+ \sum_\alpha d_\alpha *g$ converges to $g$ in the weak-* topology of $L^\infty(G)$. Now on the one hand, for $t\in G$ and $A\in \Sigma$, we have $\mu_{\alpha_0}*_TT_t\mu(A)+ \sum_\alpha d_\alpha *_T T_t\mu(A)=T_t\nu(A)$, because of the (unconditional) convergence of the series $\mu_{\alpha_0}*_T\mu+ \sum_\alpha d_\alpha *_T\mu$ to $\nu$. On the other hand, by considering the $L^\infty(G)$ function $t\mapsto T_t(A)$, we have that $\mu_{\alpha_0}*_TT_t\mu(A)+ \sum_\alpha d_\alpha *_T T_t\mu(A)= \mu_{\alpha_0}*T_t\mu(A)+ \sum_\alpha d_\alpha * T_t\mu(A)=T_t\mu(A)$, weak *. Thus $T_t\mu(A)=T_t\nu(A)$ for almost all $t\in G$, and the proof is complete. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Generalized F. and M. Riesz Theorems} %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% Throughout this section, we adopt the notation of Section 5, specifically, the notation and assumptions of Theorem \ref{decomp of measures}. Suppose that $T$ is a sup path attaining representation of $\R$ by isomorphisms of $M(\Sigma)$. In \cite{amss}, we proved the following result concerning bounded operators $\cP$ from $M(\Sigma)$ into $M(\Sigma)$ that commute with the representation $T$ in the following sense: $$\cP\circ T_t=T_t\circ \cP$$ for all $t\in \R$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \begin{thm} Suppose that $T$ is a representation of $\R$ that is sup path attaining, and that $\cP$ commutes with $T$. Let $\mu\in M(\Sigma)$ be weakly analytic. Then $\cP \mu$ is also weakly analytic. \label{caseofR} \end{thm} %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% To describe an interesting application of this theorem from \cite{amss}, let us recall the following. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{defin} Let $T$ be a sup path attaining representation of $G$ in $M(\Sigma)$. A weakly measurable $\sigma$ in $M(\Sigma)$ is called quasi-invariant if $T_t\sigma$ and $\sigma$ are mutually absolutely continuous for all $t\in G$. Hence if $\sigma$ is quasi-invariant and $A\in \Sigma$, then $|\sigma|(A)=0$ if and only if $|T_t(\sigma)|(A)=0$ for all $t\in G$. \label{qi} \end{defin} %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Using Theorem \ref{caseofR} we obtained in \cite{amss} the following extension of results of de Leeuw-Glicksberg \cite{deleeuwglicksberg} and Forelli \cite{forelli}, concerning quasi-invariant measures. \begin{thm} Suppose that $T$ is a sup path attaining representation of $\R$ by isometries of $M(\Sigma)$. Suppose that $\mu\in M(\Sigma)$ is weakly analytic, and $\sigma$ is quasi-invariant. Write $\mu=\mu_a+\mu_s$ for the Lebesgue decomposition of $\mu$ with respect to $\sigma$. Then both $\mu_a$ and $\mu_s$ are weakly analytic. In particular, the spectra of $\mu_a$ and $\mu_s$ are contained in $[0,\infty)$. \label{lebesgue-decomp-forR} \end{thm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Our goal in this section is to extend Theorems \ref{caseofR} above to representations of a locally compact abelian group $G$ with ordered dual group $\Gamma$. More specifically, we want to prove the following theorems. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \begin{thm} \label{application1} Suppose that $T$ is a sup path attaining representation of $G$ by isomorphisms of $M(\Sigma)$ such that $T_{\phi_\alpha}$ is sup path attaining for each $\alpha$. Suppose that $\cP$ commutes with $T$ in the sense that $$\cP\circ T_t=T_t\circ \cP$$ for all $t\in G$. Let $\mu\in M(\Sigma)$ be weakly analytic. Then $\cP \mu$ is also weakly analytic. \end{thm} %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% As shown in \cite[Theorem (4.10)]{amss} for the case $G=\R$, an immediate corollary of Theorem \ref{application1} is the following result. \begin{thm} \label{application2} Suppose that $T$ is a sup path attaining representation of $G$ by isometries of $M(\Sigma)$, such that $T_{\phi_\alpha}$ is sup path attaining for each $\alpha$. Suppose that $\mu\in M(\Sigma)$ is weakly analytic with respect to $T$, and $\sigma$ is quasi-invariant with respect to $T$. Write $\mu=\mu_a+\mu_s$ for the Lebesgue decomposition of $\mu$ with respect to $\sigma$. Then both $\mu_a$ and $\mu_s$ are weakly analytic with respect to $T$. In particular, the $T$-spectra of $\mu_a$ and $\mu_s$ are contained in $\overline{P}$. \end{thm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% {\bf Proof of Theorem \ref{application1}.}\quad Write $$\mu=\mu_{\alpha_0}*_T\mu +\sum_\alpha d_\alpha *_T\mu,$$ as in (\ref{decomp of measures}), where the series converges unconditionally in $M(\Sigma)$. Then \begin{equation} \label{-3} \cP\mu=\cP(\mu_{\alpha_0}*_T\mu) +\sum_\alpha \cP(d_\alpha *_T\mu). \end{equation} It is enough to show that the $T$-spectrum of each term is contained in $\overline{P}$. Consider the measure $\mu_{\alpha_0}*_T\mu$. We have $\spec_T(\mu_{\alpha_0}*_T\mu)\subset S_{\alpha_0}$. Hence by Theorem \ref{equiv-def}, $\mu_{\alpha_0}*_T\mu$ is $T_{\phi_{\alpha_0}}$-analytic. Applying Theorem \ref{caseofR}, we see that \begin{equation} \label{-2} \spec_{T_{\phi_{\alpha_0}}} (\cP (\mu_{\alpha_0}*_T\mu))\subset [0,\infty[. \end{equation} Since $\cP$ commutes with $T$, it is easy to see from Proposition \ref{proposition5.9} and Corollary \ref{corollary5.10} that $$\spec_T (\cP (\mu_{\alpha_0}*_T\mu))\subset C_{\alpha_0}.$$ Hence by (\ref{-2}) and Theorem \ref{equiv-def}, $$\spec_T (\cP (\mu_{\alpha_0}*_T\mu))\subset S_{\alpha_0},$$ which shows the desired result for the first term of the series in (\ref{-3}). 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