Non-commutative Gelfand theorem

Chi-Keung Ng  (Nankai University)

9:00-10:00,April 13,2023   A503

Abstract:

Let A be a unital C*-algebra with no quotient C*-algebra of the form M2(C). One can introduce a “generalized topology” on the set ΣA of pure states of A, via the usual metric on ΣA and a kind of Jacobson topology construction, resulting in what is called the “Gelfand spectrum” of A (that generalizes the Gelfand spectrum for a commutative C*-algebra). In this article, we define a notion of “continuity” for a kind of self-adjoint operator-valued functions on ΣA involving only its Gelfand spectrum structure. We will show that the set Cq b,her(ΣA) of such continuous functions forms a JB-algebra, and there is a canonical Jordan isomorphism ΘA : Asa → Cq b,her(ΣA), which, in the commutative case, is the restriction of the Gelfand transform. Furthermore, the C*-algebra structure on A induced a “canonical signature” σA on ΣA(which is trivial in the commutative case). It will be shown that the “σA-trimming” of Cq b,her(ΣA) + iCq b,her(ΣA) is a C*-algebra, and ΘA induces *-isomorphism from A onto this C*-algebra. These extend the usual Gelfand theorem (for commutative unital C*-algebras) to unital C*-algebras with no 2-dimensional irreducible*-representations. In particular, we recover such a C*-algebra in a constructive way from its signed Gelfand spectrum.

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