Operator algebras with contractive approximate identities: A large operator algebra in 𝑐₀

Blecher, David; Read, Charles

doi:10.1090/tran/6590

Operator algebras with contractive approximate identities: A large operator algebra in $c_0$
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Trans. Amer. Math. Soc. 368 (2016), 3243-3270 Request permission

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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