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A Dauns-Hofmann theorem for TAF-algebras

Author: D. W. B. Somerset
Journal: Proc. Amer. Math. Soc. 127 (1999), 1379-1385
MSC (1991): Primary 46K50, 47D25
Published electronically: January 28, 1999
MathSciNet review: 1616597
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Abstract: Let $A$ be a TAF-algebra, $Z(A)$ the centre of $A, Id(A)$ the ideal lattice of $A$, and $Mir(A)$ the space of meet-irreducible elements of $Id(A)$, equipped with the hull-kernel topology. It is shown that $Mir(A)$ is a compact, locally compact, second countable, $T_0$-space, that $Id(A)$ is an algebraic lattice isomorphic to the lattice of open subsets of $Mir(A)$, and that $Z(A)$ is isomorphic to the algebra of continuous, complex functions on $Mir(A)$. If $A$ is semisimple, then $Z(A)$ is isomorphic to the algebra of continuous, complex functions on $Prim(A)$, the primitive ideal space of $A$. If $A$ is strongly maximal, then the sum of two closed ideals of $A$ is closed.

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Additional Information

D. W. B. Somerset
Affiliation: Department of Mathematical Sciences, University of Aberdeen, AB24 UE United Kingdom

Received by editor(s): December 20, 1996
Received by editor(s) in revised form: May 13, 1997, and August 7, 1997
Published electronically: January 28, 1999
Communicated by: Dale Alspach
Article copyright: © Copyright 1999 American Mathematical Society