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Ground States for Generalized Gauge Actions on UHF Algebras

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Abstract

We describe the structure of ground states and ceiling states for generalized gauge actions on a UHF algebra. It is shown that both sets are affinely homeomorphic to the state space of a unital AF algebra, and that any pair of unital AF algebras can occur in this way, independently of the field of KMS states. In addition we study the subset of the ground states called \({\text {KMS}}_\infty \)-states.

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Notes

  1. This can also be deduced from the identification \(\varprojlim ({\text {Br}}'^{(j)})^T = \varprojlim {\text {Br}}'(0)^{(j)}\) and Lemma 3.1 since the 0-KMS states are the trace states.

  2. There is an alternative definition which may be closer to what Connes and Marcolli had in mind. See [CMN]. However, the definition we use here has been adopted in much of the subsequent work, including [LR] and [LLN]. It is not clear if the various definitions agree or not.

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Acknowledgements

I am grateful to Johannes Christensen for discussions, and for reading and commenting on earlier versions of the paper. The work was supported by the DFF-Research Project 2 ‘Automorphisms and Invariants of Operator Algebras’, No. 7014-00145B.

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  1. Department of Mathematics, Aarhus University, Ny Munkegade, 8000, Aarhus C, Denmark

    Klaus Thomsen

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  1. Klaus Thomsen
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Correspondence to Klaus Thomsen.

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Communicated by H. Yau

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Thomsen, K. Ground States for Generalized Gauge Actions on UHF Algebras. Commun. Math. Phys. 386, 57–85 (2021). https://doi.org/10.1007/s00220-021-04075-1

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