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


Published:19971201
Printed: Dec 1997
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
nonmetrizable, 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, 43A22, 43A30, 22D15, 43A07, 47D35 show english descriptions
$C^*$algebras and $W^*$algebras in relation to group representations [See also 46Lxx] Homomorphisms and multipliers of function spaces on groups, semigroups, etc. Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. Group algebras of locally compact groups Means on groups, semigroups, etc.; amenable groups unknown classification 47D35
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 FourierStieltjes 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
