Hermitian functionals on $B$-algebras and duality characterizations of $C^{\ast }$-algebras

Abstract:

The hermitian functionals on a unital complex Banach algebra are defined here to be those in the real span of the normalized states (tangent functionals to the unit ball at the identity). It is shown that every functional f in the dual A’ of A can be decomposed as $f = h + ik$, where h and k are hermitian functionals. Moreover, this decomposition is unique for every $f \in A’$ iff A admits an involution making it a ${C^\ast }$-algebra, and then the hermitian functionals reduce to the usual real or symmetric functionals. A second characterization of ${C^\ast }$-algebras is given in terms of the separation properties of the hermitian elements of A (real numerical range) as functionals on A’. The possibility of analogous theorems for vector states and matrix element functionals on operator algebras is discussed, and potential applications to the representation theory of locally compact groups are illustrated.