Pure states and p-commutative Banach *-algebras
Banach *-algebras with bounded approximate identities are considered. If every element x of a Banach *-algebra A from this class satisfies the condition \(\sigma_ A(x)=\{f(x):\) \(f\in P(A)\}\), where \(\sigma_ A(x)\) is the spectrum of x in A and P(A) is the set of all pure states of A, then A is P-commutative (Theorem 1). An isomorphism theorem is proved, which involves P-commutative Banach *-algebras with bounded approximate identities, and pure states. It is shown that the norm of every positive functional f on a symmetric, P-commutative Banach *-algebra with an approximate identity \(\{e_{\alpha}\}\) bounded by one, is computed by the rule \(\| f\| =\lim_{\alpha}f(e_{\alpha})\) (Theorem 3).
Banach *-algebras with bounded approximate identities
scientific article; zbMATH DE number 4071674