Maximally Decomposable von Neumann Algebras on Locally Compact Groups and Duality

• Autores: Zhiguo Hu
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 31, Nº 3, 2005, págs. 857-881
• Idioma: inglés
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• Resumen

  • We present a decomposition of the abelian von Neumann algebra L8(G) of a locally compact group as an inductive union of certain maximally decomposable translation invariant sub von Neumann algebras. As an application of the decomposition, for all locally compact groups G, we precisely express the weight and the cardinality of the spectrum of L8(G) in terms of the character and the compact covering number of G. Using decomposability numbers of von Neumann algebras, we provide a unified formulation of the decomposition of L8(G) and its dual version on VN(G) in the setting of Kac algebras. A concept of Kakutani-Kodaira numbers for locally compact groups and general Kac algebras is introduced. It is used to reveal some quantitative intrinsic relations between L8(G), VN(G) and the underlying group G. A Kac algebraic Kakutani-Kodaira theorem on the dual pair L8(G) and VN(G) is obtained.