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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ P$-commutative Banach $ \sp{\ast} $-algebras

Author: Wayne Tiller
Journal: Trans. Amer. Math. Soc. 180 (1973), 327-336
MSC: Primary 46K05
MathSciNet review: 0322515
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Abstract: Let $ A$ be a complex $ ^ \ast $-algebra. If $ f$ is a positive functional on $ A$, let $ {I_f} = \{ x \in A:f(x^ \ast x) = 0\} $ be the corresponding left ideal of $ A$. Set $ P = \cap {I_f}$, where the intersection is over all positive functionals on $ A$. Then $ A$ is called $ P$-commutative if $ xy - yx \in P$ for all $ x,y \in A$. Every commutative $ ^ \ast $-algebra is $ P$-commutative and examples are given of noncommutative $ ^ \ast $-algebras which are $ P$-commutative. Many results are obtained for $ P$-commutative Banach $ ^ \ast $-algebras which extend results known for commutative Banach $ ^ \ast $-algebras. Among them are the following: If $ {A^2} = A$, then every positive functional on $ A$ is continuous. If $ A$ has an approximate identity, then a nonzero positive functional on $ A$ is a pure state if and only if it is multiplicative. If $ A$ is symmetric, then the spectral radius in $ A$ is a continuous algebra seminorm.

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Keywords: Banach $ ^ \ast $-algebta, positive functional, $ ^ \ast $-representation, multiplicative linear functional, symmetric $ ^ \ast $-algebra
Article copyright: © Copyright 1973 American Mathematical Society