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首页> 外文期刊>Canadian Journal of Mathematics >The von Neumann algebra $VN(G)$ of a locally compact group and quotients of its subspaces
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The von Neumann algebra $VN(G)$ of a locally compact group and quotients of its subspaces

机译：局部紧群的冯·诺依曼代数$ VN（G）$及其子空间的商

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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 $picirckappa = id_{l^{infty}(X)}$ 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.
机译：令$ VN（G）$为局部紧致群$ G $的冯·诺依曼代数。我们用$ mu $表示初始序数，其中$ abs {mu} $等于开放基数的最小基数，单位为$ G $和$ X = {alpha; alpha

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