Concrete barriers to quantifier elimination in finite dimensional C*‐algebras

Abstract

AbstractWork of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*‐algebras that admit quantifier elimination in continuous logic are , , , and the continuous functions on the Cantor set. We show that, among finite dimensional C*‐algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections, and one for pairs of unitarily conjugate elements. Both of these predicates are definable, but not quantifier‐free definable, in the usual language of C*‐algebras. We also show that adding just the predicate for minimal projections is sufficient in the case of full matrix algebras, but that in general both new predicate symbols are required.