A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras

Abstract

In [2]2 we classified the isometric mappings of one C*-algebra (uniformly closed, self-adjoint operator algebra) onto another. It was remarked in that paper that the results obtained were a non-commutative extension of results of Banach [1] and Stone [7]. While this was true in spirit, we were well aware that it was not accurate to the letter. Banach and Stone deal with the algebra of real continuous functions on a compact-Hausdorff space, and our results concerning C*-algebras are actually the non-commutative analogue of results concerning the complex function algebra. The strict non-commutative analogue of the real function algebra is the Jordan algebra of self-adjoint elements in a C*-algebra (Jordan C*-algebra). The complex and real theorems follow very easily from one another in the commutative case, so that one might justifiably consider the C*-algebra theorem an extension of both of the function algebra theorems. Despite such trifling considerations, two questions still remain: what are the isometries of one C*-algebra onto another, and what are the isometries of one Jordan C*-algebra onto another? At the time [21 was written, the C*-algebra seemed the more natural object to consider. In view of the results obtained, answering the Jordan C*-algebra questions appeared to be an unnecessary decoration to the theory. We felt that the Jordan C*-algebra results could be obtained from the C*-algebra results in the same way that the real function algebra theorem follows from the complex function algebra theorem (viz., by showing that the complexified linear map is everywhere isometric). Subsequent investigations have changed our attitude in this matter. An important application of these considerations requires a Jordan C*-algebra theorem for one thing, and our attempts to derive this theorem directly from the C*-algebra theorem failed for another. The result in question is contained in Theorem 2 of ?2 and states (in normalized form) that an isometry between two Jordan C*-algebras which carries the identity into the identity is a C* (Jordan) -isomorphism. This theorem was eventually proved with the aid of a Generalized Schwarz Inequality (cf. Theorem 1 of ?2). In effect, an alternative ending has been given to the proof of [Theorem 7; 2]. This ending is by no means simpler or shorter than the one given in [2] (though it is, perhaps, less contrived), but it is flexible enough to allow us to draw the desired Jordan C*-algebra conclusion. The critical application of these results is contained in Corollary 3. A discussion accompanies Corollaries 3 and 4, but a few additional remarks are in order.