Classification ofC*-Algebras of Real Rank Zero and UnsuspendedE-Equivalence Types

✍ Scribed by Guihua Gong

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 734 KB
• Volume: 152
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3165

✦ Synopsis

In this article, examples are given to prove that the graded scaled ordered K-group is not the complete invariant for a C*-algebra in the class of unital separable nuclear C*-algebras of real rank zero and stable rank one, even for a C*-algebra in the subclass which consists of those real rank zero, stable rank one C*-algebras being expressed as inductive limits of

, where X n, i are two-dimensional finite CW complexes and [n, i] are positive integers. (In the case of simple such C*-algebras, it has been proved that the above invariant is the complete invariant by George Elliott and the author.) These examples prove that the classification conjecture of Elliott for the case of non simple real rank zero C*-algebras should be revised one needs extra invariants. The obstruction preventing two such C*-algebras with the same graded scaled ordered K-group from being isomorphic is that they have different unsuspended E-equivalence types (a refinement of KK-equivalence type of C*-algebras due to Connes and Higson). In this article, it is proved that for the above class of inductive limit C*-algebras, the obstruction of unsuspended E-equivalence type is the only obstruction (i.e., if two C*-algebras in the class are unsuspended E-equivalence, then they are isomorphic). It is a surprise that in the case of simple such C*-algebras, or even the case of C*-algebras with finitely many ideals, the obstruction will disappear (see Section 4).