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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$P$-commutative Banach $^{\ast }$-algebras
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by Wayne Tiller PDF
Trans. Amer. Math. Soc. 180 (1973), 327-336 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 327-336
  • MSC: Primary 46K05
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0322515-5
  • MathSciNet review: 0322515