transactions of the american mathematical society Volume 148, April 1970 CHARACTERIZATIONS OF C-ALGEBRAS. II BY T. W. PALMERO) 1. Introduction. The present paper is a continuation of [8], in which two necessary and sufficient conditions were given for a complex Banach algebra with a multiplicative identity element of norm one to be isometrically isomorphic to a C*-algebra (a norm closed self-adjoint algebra of operators on Hubert space). In §2 of this paper the principal result of [8] is put into its proper context—the theory of numerical range of operators on a normed linear space. In §3 this theorem is given a simplified proof and several consequences are derived. In particular it is shown that a complex Banach algebra with an identity element of norm one is isometrically isomorphic to a C*-algebra iff it is linearly isometric. Thus C*- algebras are a type of Banach algebra which can be described in purely geometric terms without any reference to the multiplication. A proof is also given that the convex hull of the set of exponentials of skew-adjoint elements in a complex C*- algebra contains the open unit ball. This answers a question raised in [12] where it was shown that the closed convex hull of the set of unitary elements in a C*- algebra is the closed unti ball. In §4 characterizations are given of those complex Banach spaces which are linearly isometric to a commutative C*-algebra with identity, or respectively, to an arbitrary C*-algebra with identity. These results depend on the concept of a vertex of the unit ball in a normed space. The commuta- tive case is related to well-known characterizations of the Banach space of real valued continuous functions on a compact space. 2. Numerical range. We begin with a review of the basic facts about numerical range on a Banach space and the application of this concept to Banach algebras. This puts the statement of the main theorem from [8] into its proper context. Let X be a complex Banach space and let X* and [X] be, respectively, the Banach space of bounded linear functionals on X, and the Banach algebra of bounded linear operators from X into X. For x e X we define the conjugate set of x to be C(x) = {x* e X* : ||x*|| = ||x|| and x*(x)= ||x||2}. The numerical range of Te [X] is WiT) = {x*Tx : x e X, ||x|| = 1 ; x* e C(x)). This generalizes the classical concept of numerical range on a Hubert space. We denote the numerical radius of Te [X] by r)(£) = sup {| A| : Ag WiT)} and the spectral radius and spectrum by v(£) and &(£), respectively. These concepts have the following basic properties. Received by the editors May 30, 1969. (j) The author was supported during the preparation of this article by a grant from the General Research Fund of the University of Kansas. Copyright © 1970, American Mathematical Society 577 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use