On the Jordan structure of 𝐶*-algebras

1. Introduction. The following problem presents itself in operator theory: how well does the Jordan structure of the self-adjoint operators determine the ring structure of a C*-algebra? We shall be concerned with two aspects of this; first we shall give an intrinsic characterization for a Jordan algebra of self-adjoint operators to be the self-adjoint part of a C*-algebra (Theorem 2.16), then we shall show that a C*-homomorphism from one C*-algebra into another is the sum of a *-homomorphism and a *-anti-homomorphism (Theorem 3.3). The first result roughly states that the Jordan algebras in question are those which are algebraically the same as the self-adjoint parts of C*-algebras while at the same time not too real (cf. the real symmetric matrices which satisfy the first property but not the second). In the finite-dimensional case this result is immediate from a paper by Jordan, von Neumann and Wigner C*-homomorphisms, or rather Jordan homomorphisms, have been studied by several authors; for an exposition see The key result for our applications in Jacobson and Rickart's

Перевод пока недоступен