The structure of KMS weights on étale groupoid \(C^\ast\)-algebras (Q6111635)

The author studies KMS weights for \(C^\ast\)-dynamical systems constructed from continuous real-valued homomorphisms on second-countable locally compact Hausdorff étale groupoids. This work is motivated by the growing trend (ignited by the work of \textit{J.~B. Bost} and \textit{A.~Connes} [Sel. Math., New Ser. 1, No.~3, 411--457 (1995; Zbl 0842.46040)]) of giving concrete descriptions of KMS states for different examples of \(C^\ast\)-dynamical systems. In the context of unital \(C^\ast\)-algebras, KMS weights are just scaled KMS states, and so studying KMS weights is essentially the same as studying KMS states. However, in the context of non-unital \(C^\ast\)-algebras (such as those studied in this paper), the two notions do not coincide. Moreover, as indicated in \textit{K.~Thomsen}'s work [Adv. Math. 309, 334--391 (2017; Zbl 1358.81120)] on KMS weights on graph \(C^\ast\)-algebras, KMS weights are a more appropriate invariant for non-unital \(C^\ast\)-dynamical systems than KMS states. For this reason, the author extends several important results regarding KMS states to KMS weights in the setting of the aforementioned \(C^\ast\)-dynamical systems. In particular, the author finds an embedding of the set of \(\beta\)-KMS weights in a certain locally convex topological vector space, and proves that many important properties of \(\beta\)-KMS states also hold for \(\beta\)\nobreakdash-KMS weights. The author then proves that there is a bijective correspondence between \(\beta\)-KMS weights for \(C^\ast\)-dynamical systems associated to continuous real-valued Hausdorff étale groupoid homomorphisms and pairs consisting of a regular Borel measure \(\mu\) on the unit space of the groupoid and a \(\mu\)-measurable field of quasi-invariant states on the \(C^\ast\)-algebras of the isotropy groups of the groupoid satisfying certain properties. This correspondence restricts to a bijection between KMS states and the pairs for which \(\mu\) is a probability measure, and thus the result is an extension of Neshveyev's theorem [\textit{S.~Neshveyev}, J. Oper. Theory 70, No.~2, 513--530 (2013; Zbl 1299.46067), Theorem~1.3] for KMS states to the more general setting of KMS weights. The author uses this result to resolve an open question posed by Thomsen at the end of Section 2 of \textit{K.~Thomsen} [J. Funct. Anal. 266, No.~5, 2959--2988 (2014; Zbl 1308.46073)], and also to solve a problem left open in [\textit{J.~Christensen} and \textit{K.~Thomsen}, J. Oper. Theory 76, No.~2, 449--471 (2016; Zbl 1389.46084)]. The author establishes two key technical results in order to prove the main theorems of the paper. The first of these results is that all KMS weights for a \(C^\ast\)-dynamical system are finite on a certain subset of the Pedersen ideal. This result allows the author to extend Neshveyev's theorem to KMS weights. The second of these results is that the extremal measures in the convex set of quasi-invariant regular Borel measures on Hausdorff étale groupoids are exactly the ergodic measures. This observation allows the author to use ideas from ergodic theory to study KMS weights. The author uses this result to extend the refinement of Neshveyev's theorem given in [\textit{J.~Christensen}, Commun. Math. Phys. 364, No.~1, 357--383 (2018; Zbl 1408.46058), Theorem~5.2] from the setting of KMS states to the setting of KMS weights.