The Von Neumann Algebra VN(G) of a Locally Compact Group and Quotients of Its Subspaces

Hu, Zhiguo

Canadian journal of mathematics, Tome 49 (1997) no. 6, pp. 1117-1138

Résumé

Let VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = {α ; α < μ}.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l ∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l ∞(X) such that π o = idl ∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l ∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l ∞(X). Similar results are also obtained for other quotient spaces.

Détail

DOI : 10.4153/CJM-1997-055-5

Mots-clés : 22D25, 43A22, 43A30, 22D15, 43A07, 47D35

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