Topological invariant means on the von Neumann algebra $\textrm {VN}(G)$

Abstract:

Let $VN(G)$ be the von Neumann algebra generated by the left regular representation of a locally compact group $G$, $A(G)$ the Fourier algebra of $G$ and $TIM(\hat G)$ the set of topological invariant means on $VN(G)$. Let ${\mathcal {F}_1} = \{ \mathcal {O} \in {({l^\infty })^ \ast }\} :\mathcal {O} \geqslant 0,\;||\mathcal {O}|| = 1$ and $\mathcal {O}(f) = 0\;{\text {if}}\;f \in {l^\infty }$ and $f(n) \to 0\}$. We show that if $G$ is nondiscrete then there exists a linear isometry $\Lambda$ of ${({l^\infty })^ \ast }$ into $VN{(G)^ \ast }$ such that $\Lambda ({\mathcal {F}_1}) \subset TIM(\hat G)$. When $G$ is further assumed to be second countable then ${\mathcal {F}_1}$ can be embedded into some predescribed subsets of $TIM(\hat G)$. To prove these embedding theorems for second countable groups we need the existence of a sequence of means $\{ {u_n}\}$ in $A(G)$ such that their supports in $VN(G)$ are mutually orthogonal and $||u{u_n} - {u_n}|| \to 0\;{\text {if}}\;u$ is a mean in $A(G)$. Let $F(\hat G)$ be the space of all $T \in VN(G)$ such that $m(T)$ is a constant as $m$ runs through $TIM(\hat G)$ and let $W(\hat G)$ be the space of weakly almost periodic elements in $VN(G)$. We show that the following conditions are equivalent: (i) $G$ is discrete, (ii) $F(\hat G)$ is an algebra and (iii) $(A(G) \cdot VN(G)) \cap F(\hat G) \subset W(\hat G)$.