The study of the geometry of norm-unital complex Banach algebras at their units [5], [6] takes its first impetus from the celebrated Bohnenblust-Karlin theorem [3] asserting that the unit of such an algebra A is a vertex of the closed unit ball of A. As observed in [5, pp. 33 34], the Bohnenblust-Karlin paper contains a stronger result, namely that, for such an algebra A, the inequality n(A, 1) ≥ (1/e) holds. Here 1 denotes the unit of A, and n(A, 1) is a suitably defined nonnegative real number which depends only on the Banach space of A and the norm-one distinguished element 1. As the main result, we prove in this paper that the product of every nonzero C * -algebra A is a vertex of the closed unit ball of the Banach space Π(A) of all continuous bilinear mappings from A × A into A. As in the above mentioned case, the vertex property follows from stronger "numerical" conditions. Indeed, if A is a nonzero C * -algebra, and if p A denotes the product of A, then n(Π(A), p A ) is equal to 1 or 1/2 depending on whether or not A is commutative (Theorem 1.1). We note that our main result improves the recent one in [24, Corollary 2.7] asserting that the product of every nonzero C * -algebra A is an extreme point of the closed unit ball of Π(A).In Section 2 we show that the main result remains true for the socalled alternative C * -algebras (Theorem 2.5). Alternative C * -algebras are defined by means of the Gelfand-Naimark abstract system of axioms but relaxing the familiar requirement of associativity to that of alternativity. Alternative C * -algebras arise in a natural way in functional analysis. Indeed, Gelfand-Naimark axioms on a general nonassociative unital algebra imply the alternativity [22, Theorem 14] (see also [9]) and the existence of alternative C * -algebras failing to be associative is well known (see [17, Example 13] and [8, Theorem 3.7]). Alternative C * -algebras are studied in detail in [20] and [8] and have shown
We prove that if A is a prime non-commutative JB * -algebra which is neither quadratic nor commutative, then there exist a prime C * -algebra B and a real number λ with 1 2 < λ 6 1 such that A = B as involutive Banach spaces, and the product of A is related to that of B (denoted by •, say) by means of the equality xy = λx • y + (1 − λ)y • x.
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