## $P$-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.## References

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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