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 ¦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.

__About the speaker:__

__Attachments:__