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A joint spectral characterization of primeness for C-algebras
Journal article   Open access   Peer reviewed

A joint spectral characterization of primeness for C-algebras

Raül E Curto and G Carlos Hernändez
Proceedings of the American Mathematical Society, Vol.125(11), pp.3299-3301
1997
DOI: 10.1090/S0002-9939-97-03948-8
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https://doi.org/10.1090/S0002-9939-97-03948-8View
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Abstract

We prove that a C*-algebra A is prime iff YT((La, Rb), A) = a-(a) x a(b) for every a, b E A, where UT denotes Taylor spectrum and La, Rb are the left and right multiplication operators acting on A. Let A be a unital C*-algebra, let La, Rb denote the left and right multiplication operators induced by a,b E A (i.e., La( := ax, Rb(X) := xb, x E A), and set Ma,b := LaRb. We obtain a characterization of primeness for C*-algebras in terms of the spectral theory of (La, Rb). Recall that an ideal I in A is said to be prime if, whenever I1, 12 are ideals in A such that 1112 C 1, it follows that I1 C I or 12 C I; A is said to be prime if (0) is a prime ideal. In [Mal], M. Mathieu obtained the following result. Theorem 1 ([Mal]). Let A be a unital C*-algebra. The following statements are equivalent. (i) A is prime. (ii) HlMa,bl = 1lall 1IbI1 for all a, b E A. (iii) J(Ma,b) = a(a)a(b) for all a, b E A. In this note we prove a joint spectral analogue of Theorem 1. First, we recall the definition of the (joint) Taylor spectrum of (La, Rb). The Koszul complex associated with (La) Rb) is Ca~b 0 -? AXL4A0A f4Aa -0a , where Dab (x) := ax @ xb (x E A) and Dab(x ($ y) :=-xb + ay (x, y E A). We say that (La, Rb) is Taylor invertible if Cab is exact, i.e., if the following three implications hold: KerDab= : xEA, ax=O=xb =\u003ex=O; Received by the editors December 6, 1995 and, in revised form, June 12, 1996. 1991 Mathematics Subject Classification. Primary 46L05, 47A10, 47A13, 47C15, 47D25; Secondary 47A62, 18G35.

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