Isometries of Operator Algebras

Abstract

Well known results of Banach [1]2 and M. H. Stone [8] determine all linear isometric maps of one C(X) onto another (where 'C(X)' denotes, throughout this paper, the set of all real-valued, continuous functions on the compact Hausdorff space X). Such isometries are the maps induced by homeomorphisms of the spaces involved followed by possible changes of sign in the function values on the various closed and open sets. An internal characterization of these isometries would classify them as an algebra isomorphism of the C(X)'s followed by a real unitary multiplication, i.e., multiplication by a real continuous function whose absolute value is 1. The situation in the case of the ring of complex continuous functions (which we denote by 'C'(X)' throughout) is exactly the same; the real unitary multiplication being replaced, of course, by a complex unitary multiplication. It is the purpose of this paper to present the non-commutative extension of the results stated above. A comment as to why this noncommutative extension takes form in a statement about algebras of operators on a Hilbert space seems to be in order. The work of Gelfand-Neumark [2T has as a very particular consequence the fact that each C'(X) is faithfully representable as a self-adjoint, uniformly closed algebra of operators (C*algebra) on a Hilbert space. The representing algebra of operators is, of course, commutative. A statement about the norm and algebraic structure of C' (X) finds then its natural non-commutative extension in the corresponding statement about not necessarily commutative C*algebras. A cursory examination shows that one cannot hope for a word for word transference of the C'(X) result to the non-commutative situation. An isometry between operator algebras is as likely to be an anti-isomorphism as an isomorphism. The direct sum of two C* algebras, which is again a C* algebra, by [2], with an automorphism in one component and an anti-automorphism in the other shows that isomorphisms and anti-isomorphisms together do not encompass all isometries. It is slightly surprising, in view of these facts, that any orderly classification of the isometries of a C* algebra is at all possible. It turns out, in fact, that all isometric maps are composites of a unitary multiplication and a map preserving the C*or quantum mechanical structure (see Segal [7])of the operator algebra in question. More specifically, such maps are linear isomorphisms which commute with the * operation and are multiplicative on powers, composed with a multiplication by a unitary operator in the algebra.