Convergence in measure and related results in finite rings of operators

Introduction. The foundations of a noncommutative integration theory were laid by Segal in [6], in connection with investigations in quantum mechanics, operator algebras and harmonic analysis on groups. In the aforesaid disciplines, there arise systems which are (noncommutative) analogues of measurable functions on a conventional measure space, the simplest instances of such systems being the so-called factors of type I1, e.g., the ring of all complex n x n matrices together with the trace. If the trace is normalized so as to assume the value unity, on the identity operator, then this system becomes the noncommutative analogue of complex random variables, on a probability space generated by n atoms. Let G be a locally compact unimodular group (such as, for some fixed n, the group of all real n x n nonsingular matrices), tu the Haar measure on G, and A the algebra of all bounded, integrable functions on G, with multiplication defined as convolution. The pair (A, ,uk) provides an example of the sort described in the beginning. Let H=L2(G, ,u) where G is the group of all transformations cx+ d, c, d rational, and JL is the counting measure on G. Forf, g in H, and a, b in G, let U0 be defined thus: Ujf=g, where g(b) =f(ba). Let M be the ring of all bounded operators which commute with Ua for all a. Every bounded operator A in H is representable in the form of a bounded numerical matrix A | , a, b in G. For any A in M, set (A)= e, (e being the identity of G). Let I denote the identity operator. For arbitrary C, D in M, r(CD) = r(DC), and r(I) = 1. M, a factor of type I1l, is the noncommutative analogue of bounded, complex random variables on a nonatomic probability space. A special case of a type I1 bearing the appellation approximately finite factor, arises in a natural way, in the theory of Fermi-Dirac quantization as described in the papers [7], [8], and [9]. The integration theory, developed by Segal in the most general setting, may be epitomized thus. A ring of operators, with a trace defined on some elements thereof is given. The trace is then extended to a wider class via suitable convergence concepts. And, for this enlarged ensemble (whose elements are called integrable) analogues of standard measure-theoretic results are obtained. In an arbitrary gage space, Segal introduced, and made a fairly extensive study of, the concepts of measurable operators, convergence nearly everywhere, and

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