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C*-algebras, approximately proper equivalence relations and thermodynamic formalism

Published online by Cambridge University Press:  09 August 2004

R. EXEL
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, Brazil (e-mail: exel@mtm.ufsc.br)
A. LOPES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil (e-mail: alopes@mat.ufrgs.br)

Abstract

We introduce a non-commutative generalization of the notion of (approximately proper) equivalence relations and propose the construction of a ‘quotient space’. We then consider certain one-parameter groups of automorphisms of the resulting C*-algebra and prove the existence of KMS states at every temperature. In a model originating from thermodynamics we prove that these states are unique as well. We also show a relationship between maximizing measures (the analogue of the Aubry–Mather measures for expanding maps) and ground states. In the last section we explore an interesting example of phase transitions.

Type
Research Article
Copyright
2004 Cambridge University Press

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