Journal of Operator Theory

Volume 43, Issue 2, Spring 2000 pp. 389-407.

On the AH algebras with the ideal property

Authors: Cornel Pasnicu
Author institution: Department of Mathematics and Computer Science, University of Puerto Rico, Box 23355, San Juan, PR 00931--3355, USA

Summary: A $C^{*}$-algebra has the {\it{ideal property}} if any ideal (closed, two-sided) is generated (as an ideal) by its projections. We prove a theorem which implies, in particular, that an {\rm AH} algebra ({\rm AH} stands for ``approximately homogeneous") stably isomorphic to a $C^{*}$-algebra with the ideal property has the ideal property. It is shown that, for any {\rm AH} algebra $A$ with the ideal property and slow dimension growth, the projections in $M_\infty(A)$ satisfy the Riesz decomposition and interpolation properties and ${\rm K}_0(A)$ is a Riesz group. We prove a theorem which describes the partially ordered set of all the ideals generated by projections of an {\rm AH} algebra $A$; the special case when the projections in $M_\infty(A)$ satisfy the Riesz decomposition property is also considered. This theorem generalizes a result of G.A. Elliott which gives the ideal structure of an {\rm AF} algebra. We answer --- jointly with M. Dadarlat --- a question of G.K. Pedersen, constructing extensions of $C^{*}$-algebras with the ideal property which do not have the ideal property.

Keywords: $C^*$-algebra, {\rm AH} algebra, ideal property, stable isomorphism, Riesz decomposition and interpolation property, Riesz group, ideal generated by projections, extension of two $C^*$-algebras

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