Abstract
This approach was abandoned by Kadison because “the sparseness of knowledge concerning the pure states of an operator algebra makes this procedure seem difficult” [14, p. 326]. Instead he gives an intrinsic proof, depending mainly on spectral theory and the geometry of the underlying Hilbert spaces on which A and B act. Kadison points out that ρ preserves the quantum mechanical structure of the C∗-algebras, i.e., the linear structure and the power structure of self-adjoint elements. It follows, and this is significant for the viewpoint expressed in this paper, that T , given by (1.1), preserves powers of the form aa∗a, aa∗aa∗a, . . . , and hence, by polarization, that T preserves the triple product ab∗c+ cb∗a.
Highlights
Introduction and preliminariesIn 1951, Kadison [14] proved the following non-commutative extension of the Banach Stone Theorem, thereby showing that the geometry of a C∗-algebra determines some aspects of its algebraic structure.THEOREM A
Recall that the proof of the Banach Stone theorem, i.e., the special case of Theorem A in which A and B are abelian, say A = C(X) and B = C(Y ), uses duality and the intimate relation between the topological space X and the algebra C(X) (respectively C(Y )). This approach was abandoned by Kadison because “the sparseness of knowledge concerning the pure states of an operator algebra makes this procedure seem difficult” [14, p. 326]
Kadison points out that ρ preserves the quantum mechanical structure of the C∗-algebras, i.e., the linear structure and the power structure of self-adjoint elements. This is significant for the viewpoint expressed in this paper, that T, given by (1.1), preserves powers of the form aa∗a, aa∗aa∗a, . . . , and by polarization, that T preserves the triple product ab∗c + cb∗a
Summary
Introduction and preliminariesIn 1951, Kadison [14] proved the following non-commutative extension of the Banach Stone Theorem, thereby showing that the geometry of a C∗-algebra determines some aspects of its algebraic structure.THEOREM A. The purpose of this note is to give a proof of Theorem D which is elementary in the sense that it uses only simple affine geometric properties of the dual unit ball of a JB∗-triple, together with analogs of standard operator algebraic theoretical tools (spectral, polar, and Jordan decompositions; biduals, and a Theorem of Effros). If A is a von Neumann algebra, by Lemma 1(c), the normal state space of A2(v) is affinely isometric to the norm exposed face Fv defined by Fv = {f ∈ A∗ : f (v) = ||f || = 1}.
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