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Algebraic Inner Derivations on Operator Algebras

Published online by Cambridge University Press:  20 November 2018

C. Robert Miers  and
John Phillips
C. Robert Miers
Affiliation:
University of Victoria, Victoria, British Columbia
John Phillips
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Let A be a C*-algebra, let p be a polynomial over C, and let a in M(A) (the multiplier algebra of A) be such that p(ad a) = 0. In this paper we study the following problem: when does there exist λ in Z(M(A)) (the centre of M(A)) such that p(a – λ) = 0? The first result of this type known to us is due to I. N. Herstein [7], who showed that for a simple ring with identity, such a λ always exists when p is of the form p(x) = xk for some positive integer k. Later, in [8], C. R. Miers showed that the result is true for any primitive unital C*-algebra and any polynomial whatever. It was also shown in [8] that if A is a unital C*-algebra acting on H and p is any polynomial, then such a λ exists in the larger algebra Z(A″). In particular, the strict result holds for any von Neumann algebra, A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 4 , 01 August 1983 , pp. 710 - 723
Copyright
Copyright © Canadian Mathematical Society 1983

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