Weakly Compact, Operator-Valued Derivations of Type I Von Neumann Algebras

Abstract

In [18], the author initiated an investigation of compact, Banach-module-valued derivations of C*-algebras. In collaboration with C. A. Akemann [3] and S.-K. Tsui [16], he determined the structure of all compact and weakly compact, A-valued derivations of a C*-algebra A, and of all compact, B(H)-valued derivations of a C*-subalgebra of B(H), the algebra of bounded linear operators on a Hilbert space H. In this paper, we begin the study of weakly compact, B(H)-valued derivations of C*-subalgebras of B(H).Let R be a C*-subalgebra of B(H), δ:R → B(H) a weakly compact derivation, i.e., a weakly compact linear map which hasSince δ has a unique weakly compact extension to a derivation of the closure of R in the weak operator topology (WOT) on B(H) (consult the proof of Theorem 3.1 of [16]), we may assume with no loss of generality that R is a von Neumann subalgebra of B(H). In this paper, we determine in Lemma 4.1 and Theorems 4.3 and 4.10 the structure of δ when R is type I, using I. E. Segal's multiplicity theory [14] for type I von Neumann algebras and results of E. Christensen [6], [7] on B(H)-valued derivations of von Neumann algebras.