JOURNAL OF FUNCTIONAL ANALYSIS 11, l-16 (1972) Banach Algebras with Involution and Mijbius Transformations LAWRENCE A. HARRIS Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by J. Dixmier Received January 18, 1971 This paper introduces the use of Potapov’s Miibius transformations in the study of Banach algebras with involution. 1. INTRODUCTION Our purpose is to present simplified proofs of some important results in the theory of Banach algebras with involution. The novelty of our approach consists in the use of a generalized kind of Mobius transformations together with some elementary facts in the theory of vector-valued holomorphic functions. These methods are particularly successful in obtaining extensions of the Russo-Dye theorem. The Mobius transformations we use are defined in Section 2, and two extended versions of the Russo-Dye theorem are deduced almost immediately in Section 3. A sharper form of the Russo-Dye theorem for Banach algebras with hermitian involution is obtained in Section 4 and then applied to give a simplified proof of a metric characterization of C*-algebras due to Palmer [II]. In Section 5 the symmetry of hermitian involutions is deduced from an extension of the fact that Mobius transformations of the form A -+ (A + Ml + fa), Irll cl, map the closed unit disc of the complex plane into itself. Section 6 contains two highly general maximum principles for holomorphic functions which are extensions of the Russo-Dye theorems given in Sections 3 and 4. Finally, in Section 7 results are given on the shape of the closed unit ball of B*-algebras and on the size of the spectrum of non-unitary partial isometries. 1 Copyright 0 1972 by Academic Press, Inc. All rights of reproduction in any form reserved.