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The spectral theorem in Banach algebras

Published online by Cambridge University Press:  18 May 2009

Stephen Plafker
Stephen Plafker
Affiliation:
Tulane University, New Orleans, Louisiana 70118
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The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 1 , March 1972 , pp. 49 - 55
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

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