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

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.