A Dauns-Hofmann theorem for TAF-algebras

Somerset, D.

doi:10.1090/S0002-9939-99-04606-7

A Dauns-Hofmann theorem for TAF-algebras
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by D. W. B. Somerset PDF

Proc. Amer. Math. Soc. 127 (1999), 1379-1385 Request permission

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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