Almost periodic operators in $\textrm {VN}(G)$

Abstract:

Let $G$ be a locally compact group, $A(G)$ the Fourier algebra of $G$, $B(G)$ the Fourier-Stieltjes algebra of $G$ and ${\text {VN}}(G)$ the von Neumann algebra generated by the left regular representation $\lambda$ of $G$. Then $A(G)$ is the predual of ${\text {VN}}(G)$; ${\text {VN}}(G)$ is a $B(G)$-module and $A(G)$ is a closed ideal of $B(G)$. Let ${\text {AP}}(\hat G) = \{ T \in {\text {VN}}(G):u \mapsto u \cdot T$ is a compact operator from $A(G)$ into ${\text {VN}}(G)\}$, the space of almost periodic operators in ${\text {VN}}(G)$. Let $C_\delta ^*(G)$ be the ${C^*}$-algebra generated by $\{ \lambda (x):x \in G\}$. Then $C_\delta ^*(G) \subset {\text {AP}}(\hat G)$. For a compact $G$, let $E$ be the rank one operator on ${L^2}(G)$ that sends $h \in {L^2}(G)$ to the constant function $\int {h(x)dx}$. We have the following results: (1) There exists a compact group $G$ such that $E \in \text {AP}(\hat G)\backslash C_\delta ^*(G)$. (2) For a compact Lie group $G$, $E \in {\text {AP(}}\hat G{\text {)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G)$ has a unique left invariant mean $\Leftrightarrow G$ is semisimple. (3) If $G$ is an extension of a locally compact abelian group by an amenable discrete group then ${\text {AP}}(\hat G) = C_\delta ^*(G)$. (4) Let $G = {{\mathbf {F}}_r}$, the free group with $r$ generators, $1 < r < \infty$. If $T \in {\text {VN}}(G)$ and $u \mapsto u \cdot T$ is a compact operator from $B(G)$ into ${\text {VN}}(G)$ then $T \in C_\delta ^*(G)$.