Classification of Non-Commutative Arens Algebras Associated with Semi-Finite Traces

Abstract

Let (Ω, Σ, μ) be a measurable space with a finite measure, L ^p (μ) = L ^p (Ω, Σ, μ) be the Banach space of all μ - measurable complex functions on Ω, the p ^th degrees of which are integrable p ∈ [1, ∞). P.Arens had introduced and studied the set ${{L}^{w}}\left( \mu \right) = \bigcap\limits_{{1 \leqslant p < \infty }} {{{L}^{p}}} \left( \mu \right)$ It was demonstrated, in particular, that L ^w(μ) is a metrizable locally convex ⋆ - algebra with respect to the topology t generated by the system of norms ${{\left\| f \right\|}_{p}} = {{\left( {{{{\int\limits_{\Omega } {\left| f \right|} }}^{p}}d\mu } \right)}^{{1/p}}},p1$ The additional study of the Arens algebras L^w(μ) was made by S.J.Bhaft [2,3] who has described the algebras L ^W(μ) and examined some classes of homomorphism of these algebras. B.Z. Zakirov [4] showed that L ^W (μ) is a EW* algebra and gave an example of two measures, μ, and v, on an atomic Boolean algebra for which the algebras L ^w(μ) and L ^w (v) are not isomorphic. The Arens algebras associated with the non-commutative von Neumann algebras were examined for the first time in the works [4,5].