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A Non‐Separable C*‐Algebra with only Separable Abelian C*‐Subalgebras

A Non‐Separable C*‐Algebra with only Separable Abelian C*‐Subalgebras A NON-SEPARABLE C*-ALGEBRA WITH ONLY SEPARABLE ABELIAN C*-SUBALGEBRAS CHARLES A. AKEMANN f AND JOH N E. DONER In this paper we construct an example of a non-separable C*-algebra U such that every abelian C*-subalgebra of U is separable. This answers a question raised by J. Dixmier at the 1974 International Congress. Further, no abelian C*-subalgebra of U contains an approximate unit for U, so it serves as a correct example of a phenomenon mentioned in [1, Example 2.1]. The authors were assisted by helpful conversations with our colleagues Andy Bruckner, Phil Ostrand and Jim Robertson about Lemma 2. Let H be a two-dimensional complex Hilbert space with the elements sometimes written as ordered pairs (r,s) with r, seC and with scalar product written <e,/> for e,feH . Let B(H ) be the C*-algebra of all bounded linear operators on H 2 2 2 (i.e. 2x 2 matrices). Let M be the C*-algebra of all bounded infinite sequences from B(H ), and let M be the subalgebra of M consisting of all sequences converging to 0. 2 o An element deM will represent a sequence whose nth term is a(n). Let n : M -»M/M be the quotient map. Using http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Non‐Separable C*‐Algebra with only Separable Abelian C*‐Subalgebras

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/11.3.279
Publisher site
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Abstract

A NON-SEPARABLE C*-ALGEBRA WITH ONLY SEPARABLE ABELIAN C*-SUBALGEBRAS CHARLES A. AKEMANN f AND JOH N E. DONER In this paper we construct an example of a non-separable C*-algebra U such that every abelian C*-subalgebra of U is separable. This answers a question raised by J. Dixmier at the 1974 International Congress. Further, no abelian C*-subalgebra of U contains an approximate unit for U, so it serves as a correct example of a phenomenon mentioned in [1, Example 2.1]. The authors were assisted by helpful conversations with our colleagues Andy Bruckner, Phil Ostrand and Jim Robertson about Lemma 2. Let H be a two-dimensional complex Hilbert space with the elements sometimes written as ordered pairs (r,s) with r, seC and with scalar product written <e,/> for e,feH . Let B(H ) be the C*-algebra of all bounded linear operators on H 2 2 2 (i.e. 2x 2 matrices). Let M be the C*-algebra of all bounded infinite sequences from B(H ), and let M be the subalgebra of M consisting of all sequences converging to 0. 2 o An element deM will represent a sequence whose nth term is a(n). Let n : M -»M/M be the quotient map. Using

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Oct 1, 1979

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