ON THE AH ALGEBRAS WITH THE IDEAL PROPERTY

• Published in: Journal of operator theory Vol. 43; no. 2; pp. 389 - 407
• Main Author: PASNICU, CORNEL
• Format: Journal Article
• Language: English
• Published: Theta Foundation 01-04-2000
• Subjects: Algebra; Equivalence relation; Gongs; Interpolation; Mathematical theorems; Mathematics; Operator theory; Partially ordered sets; Property titles
• ISSN: 0379-4024; 1841-7744

A C*-algebra has the 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 AH algebra (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 AH algebra A with the ideal property and slow dimension growth, the projections in M∞(A) satisfy the Riesz decomposition and interpolation properties and K0(A) is a Riesz group. We prove a theorem which describes the partially ordered set of all the ideals generated by projections of an AH algebra A; the special case when the projections in M∞(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 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.