Non-commutative Gelfand theorem

Chi-Keung Ng  (Nankai University)

9:00-10:00,April 13,2023   A503


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 ¦²A of pure states of A, via the usual metric on ¦²A 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 ¦²A involving only its Gelfand spectrum structure. We will show that the set Cq b,her(¦²A) of such continuous functions forms a JB-algebra, and there is a canonical Jordan isomorphism ¦¨A : Asa ¡ú Cq b,her(¦²A), which, in the commutative case, is the restriction of the Gelfand transform. Furthermore, the C*-algebra structure on A induced a ¡°canonical signature¡± ¦ÒA on ¦²A(which is trivial in the commutative case). It will be shown that the ¡°¦ÒA-trimming¡± of Cq b,her(¦²A) + iCq b,her(¦²A) is a C*-algebra, and ¦¨A 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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