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An operator is compact relative to a semifinite von Neumann algebra, i.e., belongs to the two-sided closed ideal generated by the finite projections relative to the algebra, if and only if it maps vector sequences converging weakly relative to the algebra into strongly converging ones (generalized Hilbert condition). The generalized Wolf condition characterizes the class of almost Fredhoim operators. Introduction. The elements of the two-sided closed ideal i generated by the projections finite relative to a von Neumann algebra 6, are called compact operators of C, and it has been shown (see [7, 4]) that they satisfy many of the properties of the compact operators on a Hilbert space. The weak convergence of vectors plays an important role in classical operator theory. The aim of this paper is to study a relative weak (RW) convergence that could play an analogous role in the operator theory relative to a semifinite von Neumann algebra. A bounded sequence of vectors x,, is defined to converge RW to x if Px -> Px for every projection P finite relative to ,. This convergence is shown to be not equivalent (apart from trivial cases) to weak or strong convergence. The classical Hilbert characterization is extended to the compact operators of 6C and a generalized Wolf property (see [8]) is used to characterize a new class Ci+ called almost left-Fredholm [5]. We remark that the RW convergence can be used, in analogy with Calkin's construction (see [2]), to obtain a representation of the generalized Calkin algebra C, / i. Finally a RW topology is defined on the unit ball of the predual of 6, and both the generalized Hilbert and Wolf properties are reformulated in a space-free setting. The author wishes to thank M. Sonis, L. Brown and P. Fillmore for valuable suggestions. 1. The relative weak convergence. Let H be a Hilbert space, let L(H) be the algebra of all bounded linear operators on H, let 6i be a semifinite von Neumann algebra on H and P (C) be the set of the projections of (B. Let i be the ideal of compact operators (relative to ei), i.e., the norm closed two-sided ideal of a, generated by the finite projections of ,. It is known that i is proper iff C, is Received by the editors April 28, 1980. Some of the results in this paper were presented to the American Mathematical Society at the 85th Annual Meeting at Biloxi, Mississippi, January 1979. 1980 Mathematics Subject Classification. Primary 46L10, 47C15; Secondary 47A53, 47B05.

Relative weak convergence in semifinite von Neumann algebras