Characterizations of 𝐶*-algebras

Abstract

This note presents two related characterizations of those Banach algebras which are isometrically isomorphic to C*-algebras, i.e., to operator-norm closed, self-adjoint algebras of operators on a Hilbert space. The first characterization has evolved from a theorem of I. Vidav [7] and its extension by E. Berkson [l] and B. W. Glickfeld [2] (cf. [5]). The proof given below is considerably simpler than the proofs given in [l] and [2] for closely related, but weaker, results. It is based on Lemma 1 which refines a result of B. Russo and H. A. Dye [6]. All algebras considered here have complex scalars and an identity element I of norm one. In [6] it is shown that the closed unit ball 2ti of a C*-algebra SI is the norm closed convex hull clco £7(21) of the set U{%) of all unitary elements in SI. The set »« = {e: RE%,R = R*}> which can be defined in any Banach algebra with an involution, is a subset of U(%) in any C*-algebra. In a von Neumann algebra 2I« = Z7(2t), but in certain C*algebras 2Ie is a proper subset of £7(21). For instance if 21 is the usual Banach algebra of continuous functions on the unit circle in the complex plane, then multiplication by the complex variable belongs to £/(2I) but not to 2l«. Thus the following lemma strengthens Theorem 1 of [6].