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Discretization of C*-algebras (1607.03376v2)
Published 12 Jul 2016 in math.OA and math.FA
Abstract: We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra $A$ is a $$-homomorphism $A \to M$ that factors through the canonical inclusion $C(X) \subseteq \ell\infty(X)$ when restricted to a commutative C-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where $M$ is a C*-algebra. Any subhomogenous C*-algebra admits an injective functorial discretization, where $M$ is a W*-algebra. However, any functorial discretization, where $M$ is an AW*-algebra, must trivialize $A = B(H)$ for any infinite-dimensional Hilbert space $H$.