Commutators modulo the center in a properly infinite von Neumann algebra

Abstract

1. Introduction. An element in a von Neumann algebra stf is said to be a commutator in stf if there are elements A and B in stf such that C=AB — BA. For finite homogeneous discrete algebras and for properly infinite factor algebras the set of commutators has been completely described [l]-[5], [10]. In each of these special cases any element is a commutator modulo a central element depending on C. In this paper we show that given any element in a properly infinite von Neumann algebra stf there is an element C0 in the center of stf depending on such that C— C0 is a commutator in stf. The element C0 is an arbitrary element in the intersection criTc of the center with the uniform closure of the convex hull of {U*CU | U unitary in ?tf} [6, III, §5]. We then present a few facts about those elements such that 0 e Xc or what is the same as far as determining commutators is concerned about those elements such that OeJf^-ics for some invertible S in stf. 2. Commutators. Let stf be a C*-algebra with identity and let / be a closed two-sided ideal in stf. The image of the element A e stf in the factor algebra stf(T) = stf/1 under the canonical homomorphism of stf onto stf/I will be denoted by A(T). If £ is a maximal ideal of the center of stf, the smallest closed two-sided ideal in stf containing £ is denoted by [{]. For simplicity we write A([Q) as A(t). The set of maximal (respectively, primitive) ideals of stf with the hull-kernel topology is called the strong structure space (respectively, structure space) of stf. If stf is a von Neumann algebra, then the strong structure space M(stf) of stf is homeomorphic with the spectrum of the center S of stf under the map M —> M C J? [13]. This means M(stf) is extremely disconnected. Proposition 1. Let stf be a properly infinite von Neumann algebra and let A be a fixed element of stf. The function M -> ||^(M)|| of the strong structure space M (stf) of stf into the real numbers is continuous. Proof. For every «?Owe know that the set X={M e M (stf) \\A(M) ^«} is closed. If /=p| X, then \\A(I)\\^a [8, Lemma 1.9] and so M(M)|| ^a for every MeM(stf) containing /. Thus X={M e M (stf) | I<=M}.