A Classification of Simple Limits of Splitting Interval Algebras

✍ Scribed by Xinhui Jiang; Hongbing Su

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 477 KB
• Volume: 151
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3120

✦ Synopsis

Let A be a unital simple limit of finite direct sums of sub-homogeneous interval algebras of a certain type (cf. Definition 1.1). It is proved that A can be classified by the scaled ordered group K 0 (A), the simplex T(A), and the canonical pairing between them. It is also shown that K 0 (A) might fail to have the Riesz decomposition property. 1997 Academic Press 0. INTRODUCTION This paper is a contribution to the recent C*-algebra classification program initiated by George A. Elliott. Until now the K 0 groups of classified C*-algebras had all enjoyed the so-called Riesz decomposition property (see [E3] for a survey). In this paper we classify a class of simple C*-algebras whose K 0 -groups might fail to have this property.

Main Theorem. Let A, B be two simple unital inductive limits of finite direct sums of splitting interval algebras. If there is a homomorphism

of scaled ordered groups and a continuous affine map % : T(B) Ä T(A) of the tracial state spaces that are compatible with respect to the pairing between K 0 -groups and traces, then there exists a unital V-homomorphism : A Ä B which induces } and %.

Moreover, if } and % are isomorphisms, then \ can be chosen to be an isomorphism.

Besides providing new range for the invariants, the inductive limits of subhomogenous algebras are interesting for many other reasons (See [Su] for a good indication). This paper is a continuation of [EGJS] which classified simple C*-algebras that can be expressed as inductive limits of article no.