Banach algebras with unitary norms (Q1816527)

The metric nature of the unitary group of a (unital) \(C^*\)-algebra is studied in the Banach algebra framework. Using the Russo-Dye theorem and its recent refinements as a point of departure, a unitary norm on a (complex, unital) Banach algebra is defined to be a norm such that the unit ball is the norm closure of the convex hull of a group of invertible elements in the algebra. If the group is maximal as a bounded subgroup of the group of invertible elements of the algebra, the norm is said to be maximal unitary. It is shown that the unitary group of a \(C^*\)-algebra is a maximal bounded subgroup of the group of invertible elements, that a norm-closed algebra of bounded operators on a Hilbert space whose norm is unitary is a \(C^*\)-algebra, that each finite-dimensional Banach algebra with a maximal unitary norm is isometrically isomorphic to a \(C^*\)-algebra (this fails without the assumption of maximality). The unitary norms on the algebra of continuous complex-valued functions on a compact Hausdorff space are studied. The \(L^1\) norm is shown to be unitary on a Wiener algebra. Using Effros-Ruan and Blecher-Ruan-Sinclair, it is shown that a (unital) \(L^\infty\)-matricially normed Banach algebra with unitary norm is (completely) isometric and isomorphic to a \(C^*\)-algebra.