Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T04:51:45.158Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Akemann, C. A., Elliott, G. A., Pedersen, G. K. and Tomiyama, J., Derivations and multipliers of C*-algebras, Amer. J. Math. 98 (1976), 679–708.Google Scholar
2. Dixmier, J., Ideal center of a C*-algebra, Duke Math. J. 35 (1968), 375–382.Google Scholar
3. Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1969).Google Scholar
4. Ernest, J., On the topology of the spectrum of a C*-algebra, Math. Ann. 216 (1975), 149–153.Google Scholar
5. Fell, J. M. G., The structure of algebras of operator fields, Acta. Math. 106 (1961), 233–280.Google Scholar
6. Halmos, P. R., A Hilbert space problem book (van Nostrand, Princeton, N. J., 1967).Google Scholar
7. Herstein, I. N., Sui commutator degli anelli semplici, Rendiconti del Seminaris Matematico e Fisico di Milano 33 (1963).Google Scholar
8. Miers, C. R., Centralizing maps of operator algebras, J. Algebra 59 (1979), 56–64.Google Scholar
9. Phillips, J. and Raeburn, I., Perturbations of C*-algebras, II, Proc. Lond. Math. Soc. 43 (1981), 46–72.Google Scholar
10. Rosenblum, M., On the operator equation BX - XA = Q, Duke Math. J. 23 (1956), 263–270.Google Scholar
11. Vaisman, I., Cohomology and differential forms (Marcel Dekker, New York, 1973).Google Scholar