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# The von Neumann algebra $\VN(G)$ of a locally compact group and quotients of its subspaces

Published:1997-12-01
Printed: Dec 1997
• Zhiguo Hu
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## Abstract

Let $\VN(G)$ be the von Neumann algebra of a locally compact group $G$. We denote by $\mu$ the initial ordinal with $\abs{\mu}$ equal to the smallest cardinality of an open basis at the unit of $G$ and $X= \{\alpha; \alpha < \mu \}$. We show that if $G$ is nondiscrete then there exist an isometric $*$-isomorphism $\kappa$ of $l^{\infty}(X)$ into $\VN(G)$ and a positive linear mapping $\pi$ of $\VN(G)$ onto $l^{\infty}(X)$ such that $\pi\circ\kappa = \id_{l^{\infty}(X)}$ and $\kappa$ and $\pi$ have certain additional properties. Let $\UCB (\hat{G})$ be the $C^{*}$-algebra generated by operators in $\VN(G)$ with compact support and $F(\hat{G})$ the space of all $T \in \VN(G)$ such that all topologically invariant means on $\VN(G)$ attain the same value at $T$. The construction of the mapping $\pi$ leads to the conclusion that the quotient space $\UCB (\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$ has $l^{\infty}(X)$ as a continuous linear image if $G$ is nondiscrete. When $G$ is further assumed to be non-metrizable, it is shown that $\UCB(\hat{G})/F (\hat{G})\cap \UCB(\hat{G})$ contains a linear isomorphic copy of $l^{\infty}(X)$. Similar results are also obtained for other quotient spaces.
 MSC Classifications: 22D25 - $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 22D15 - Group algebras of locally compact groups 43A07 - Means on groups, semigroups, etc.; amenable groups 47D35 - unknown classification 47D35