Algebraic equivalence of locally normal representations

Abstract

It will be shown that (i) the absolute value of every locally normal linear functional is again locally normal; (ii) two locally normal representations πi and π2 of S/ generate isomorphic von Neumann algebras ^(πϊ) and ^^(π 2) if and only if there exists an automorphism a of S>/ such that πx o β and π2 are quasi-equivalent, provided that either ^(πϊ) or ^^(τr 2) is afinite. This paper is motivated by a recent work [6] of R. Haag. R. V. Kadison and D. Kastler. As they mentioned, the recent progress in mathematical physics has made a precise analysis of representatio ns of a C*-algebra furnished with a net of von Neumann algebras a growing necessity. In the first half of this paper, we shall show that the space of all locally normal linear functionals of a C*-algebra with a net of von Neumann algebras is a closed invariant subspace of the conjugate space in the sense of [14], which will imply that the absolute value of a locally normal linear functional is locally normal too. The last half of this paper will be devoted to extending a result of Powers [11] for UHF algebra to a C*-algebra sf with a proper sequential type 1^ funnel. Namely it will be shown that two locally normal representations πλ and π2 of the C*-algebra s%? generate isomorphic von Neumann algebras if and only if they are connected by an automorphism of s^. This is proven under the assumption that one of the generated von Neumann algebras is σ-finite. 2* The locally normal conjugate space of a C*-algebra with a net of von Neumann algebras* Let Szf be a C*-algebra. Suppose a system % = (jQ (ϋ) \}a^fa is dense in s^ with respect to the norm topology. The system % = {s/a} is called a net (in Ssf) of von Neumann algebras and each j^ς is called local subalgebra of DEFINITION 1. A continuous linear functional φ (resp. representa