Non commutative Arens Algebras
by
Shavkat Abdullaevich Ayupov

Let U be a semifinite von Neumann algebra with a faithful normal semifinite trace \tau, L_p(U, \tau) the Banach space of all p-integrable measurable operators affiliated with U, equipped with L_p-norm ||  ||_p. Consider the set L^\omega(U, \tau) = \cap _{p >= 1} L_p(U, \tau). One can prove that L^\omega(U, \tau) is a complete metrizable locally convex *-algebra with respect to the family of norms {||  ||_p , p >= 1}.

In the particular case when U is commutative and U=L_\infty[0, 1] we obtain the algebra of functions L^\omega[0, 1] = \cap _{p >= 1} L_p[0, 1] firstly considered by Arens [1].

In the present talk we describe the conjugate space for the locally convex algebra L^\omega(U, \tau) and prove that L^\omega(U, \tau) is reflexive iff the trace \tau is finite. The main results are devoted to classification of Arens algebras up to isomorphism in terms of the underlying von Neumann algebras and traces. The most complete results are obtained in the case of commutative von Neumann algebras. In this case the set of projections P(U) forms a complete Boolean algebra, and criterions of isomorphism between Arens algebras L^\omega(M, \tau) and L^\omega(N, \nu) are given in terms of "passports" of corresponding Boolean algebras P(M) and P(N) in the sense of D.A.Vladimirov [2].

References. 1. Arens R. The space L^\omega and convex topological rings. Bull. AMS, 1946, V.52, 931-935. 2. Vladimirov D.A. Boolean Algebras. (Russian edition) Moscow: Nauka, 1969.

(T)

Date received: December 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cado-84.