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Let A be a unital Banach algebra and 8 the group of isometrics in A. The norm in A is uniquely maximal if S is not contained in any larger bounded group in A and there is no equivalent norm on A with the same group of isometrics. We use a group theory result of B. H. Neumann to prove that the discrete measure algebra l'(Ci) is uniquely maximal if and only if G has no finite conjugacy classes. 1. Uniquely maximal norms. Let A be a complex unital Banach algebra. Let ^i — {u G A; Hull = lit/-1 II = 1). We shall call § the group of isometries in A. We shall say A has a maximal norm if e is a maximal bounded group in A. This is equivalent to saying that there is no equivalent algebra norm on A with more isometries. When A is maximal, there may be many equivalent algebra norms on A with the same group of isometries. What shall concern us in this paper is when a maximal norm is unique. The unital Banach algebra A has a uniquely maximal norm if it is maximal and there is no equivalent algebra norm on A with the same group of isometries. The s*-algebras are uniquely maximal. For commutative and finite dimensional alge- bras, uniquely maximal is equivalent to B* (see (1)). For a general unital Banach algebra we shall say the norm is convex transitive if ~co~ {u; u G $} = (a G A; \\a\\ \ such that (a, (?) ( the semigroup generated by a and § ) is bounded. Proof. We shall first prove uniquely maximal implies convex transitive. We

An analytic characterization of groups with no finite conjugacy classes