The Stable Rank of Topological Algebras and a Problem of R. G. Swan

✍ Scribed by C. Badea

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 492 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3334

✦ Synopsis

Various notions of stable ranks are studied for topological algebras. Some partial answers to R. G. Swan's problem (Have two Banach or good Fre chet algebras as in the density theorem in K-theory the same stable rank?) are obtained. For example, a Fre chet dense V -subalgebra A of a C*-algebra B, closed under C -functional calculus of self-adjoint elements, has the same Bass stable rank as B. 1998 Academic Press Contents. 1. Introduction. 1.1. Preamble. 1.2. Stable ranks. 1.3. Swan's problem. 2. Characterizing Bsr and tsr. 2.1. The bass and the bilateral Bass stable ranks. 2.2. The topological stable rank. 2.3. Other stable ranks for normed algebras. 3. Stable ranks and connectedness properties. 3.1. Serre fibrations. 3.2. Characterizations for csr. 3.3. Characterizing acsr using tsr. 4. Subalgebras with the same stable rank. 4.1. Dense morphisms versus onto morphisms. 4.2. Reducible and bilateral reducible elements. 4.3. Swan's problem for subalgebras of C*-algebras. 4.4. Swan's problem for tsr and acsr. 5. The nonunital case.