2019/05/29 by Eagle, Christopher J., Schmid, Todd
#03C10 #03C65 #46L05 #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1905.12153
Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*-algebras that admit quantifier elimination in continuous logic are ℂ, ℂ2, M2(ℂ), 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 projections. 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.