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

Published online by Cambridge University Press:  20 November 2018

Steve Wright
Steve Wright*
Affiliation:
Oakland University, Rochester, Michigan
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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 has

Since 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 3 , 01 June 1984 , pp. 436 - 457
Copyright
Copyright © Canadian Mathematical Society 1984

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