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Operator algebras with contractive approximate identities: A large operator algebra in $ c_0$


Authors: David P. Blecher and Charles John Read
Journal: Trans. Amer. Math. Soc. 368 (2016), 3243-3270
MSC (2010): Primary 46B15, 47L30, 47L55; Secondary 43A45, 46B28, 46J10, 46J40
DOI: https://doi.org/10.1090/tran/6590
Published electronically: July 29, 2015
MathSciNet review: 3451876
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Abstract: We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of $ c_0$.


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Additional Information

David P. Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: dblecher@math.uh.edu

Charles John Read
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
Email: read@maths.leeds.ac.uk

DOI: https://doi.org/10.1090/tran/6590
Keywords: Singly generated operator algebra, algebras generated by orthogonal idempotents, spectral idempotent, approximate identity, semisimple, set of synthesis, Banach sequence algebra, Banach function algebra, Tauberian, socle.
Received by editor(s): February 27, 2014
Published electronically: July 29, 2015
Additional Notes: The first author was partially supported by a grant from the National Science Foundation
The second author is grateful for support from UK Research Council grant EP/K019546/1
Article copyright: © Copyright 2015 American Mathematical Society