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Central Double Centralizers on Quasi-Central Banach Algebras with Bounded Approximate Identity

Published online by Cambridge University Press:  20 November 2018

Sin-Ei Takahasi
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We assume throughout this paper that A is a semi-simple, quasi-central, complex Banach algebra with a bounded approximate identity {eα}. The author [6] has shown that every central double centralizer T on A can be, under suitable conditions, represented as a bounded continuous complex-valued function ΦT on Prim A, the structure space of A with the hull-kernel topology, such that

Here x + P for P ∊ Prim A denotes the canonical image of x in A/P. This map Φ is called Dixmier's representation of Z(M(A)), the central double centralizer algebra of A. We denote by τ the canonical isomorphism of A into the Banach algebra D(A) with the restricted Arens product as defined in [6]. Also denote by μ Davenport's representation of Z(M(A)). In fact, this map μ is given by

for each T ∊ Z(M(A)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 2 , 01 April 1983 , pp. 373 - 384
Copyright
Copyright © Canadian Mathematical Society 1983

References

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