2010/10/04 by Marcelo Laca, Sergey Neshveyev, Laca, Marcelo +1
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.1010.0599
openalex publication_date 2010/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We complete the analysis of KMS-states of the Toeplitz algebra of the affine\nsemigroup over the natural numbers, recently studied by Raeburn and the first\nauthor, by showing that for every inverse temperature beta in the critical\ninterval [1,2], the unique KMSbeta-state is of type III1. We prove this by\nreducing the type classification from the Toeplitz algebra to that of the\nsymmetric part of the Bost-Connes system, with a shift in inverse temperature.\nTo carry out this reduction we first obtain a parametrization of the Nica\nspectrum of the Toeplitz algebra in terms of an adelic space. Combining a\ncharacterization of traces on crossed products due to the second author with an\nanalysis of the action of the affine semigroup on the Nica spectrum, we can\nalso recover all the KMS-states originally computed by Raeburn and the first\nauthor. Our computation sheds light on why there is a free transitive circle\naction on the extremal KMSbeta-states for beta>2 that does not ostensibly come\nfrom an action on the C*-algebra.\n