Approximation properties for group $C^ *$-algebras and group von Neumann algebras

Abstract:

Let G be a locally compact group, let $C_r^\ast (G)$ (resp. ${\text {VN}}(G)$) be the ${C^\ast }$-algebra (resp. the von Neumann algebra) associated with the left regular representation l of G, let $A(G)$ be the Fourier algebra of G, and let ${M_0}A(G)$ be the set of completely bounded multipliers of $A(G)$. With the completely bounded norm, ${M_0}A(G)$ is a dual space, and we say that G has the approximation property (AP) if there is a net $\{ {u_\alpha }\}$ of functions in $A(G)$ (with compact support) such that ${u_\alpha } \to 1$ in the associated weak $^\ast$-topology. In particular, G has the AP if G is weakly amenable ($\Leftrightarrow A(G)$ has an approximate identity that is bounded in the completely bounded norm). For a discrete group $\Gamma$, we show that $\Gamma$ has the ${\text {AP}} \Leftrightarrow C_r^\ast (\Gamma )$ has the slice map property for subspaces of any ${C^\ast }$-algebra $\Leftrightarrow {\text {VN}}(\Gamma )$ has the slice map property for $\sigma$-weakly closed subspaces of any von Neumann algebra (Property ${S_\sigma }$). The semidirect product of weakly amenable groups need not be weakly amenable. We show that the larger class of groups with the AP is stable with respect to semidirect products, and more generally, this class is stable with respect to group extensions. We also obtain some results concerning crossed products. For example, we show that the crossed product $M{ \otimes _\alpha }G$ of a von Neumann algebra M with Property ${S_\sigma }$ by a group G with the AP also has Property ${S_\sigma }$.