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Concrete barriers to quantifier elimination in finite dimensional C*‐algebras
Author(s) -
Eagle Christopher J.,
Schmid Todd
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201900041
Subject(s) - quantifier elimination , mathematics , predicate (mathematical logic) , separable space , quantifier (linguistics) , discrete mathematics , algebra over a field , pure mathematics , computer science , programming language , artificial intelligence , mathematical analysis
Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*‐algebras that admit quantifier elimination in continuous logic are C , C 2 ,M 2 ( 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 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.