Johannes Christensen
We generalise a number of classical results from the theory of KMS states to KMS weights in the setting of $C^{*}$-dynamical systems arising from a continuous groupoid homomorphism $c:\mathcal{G} \to \mathbb{R}$ on a locally compact second countable Hausdorff étale groupoid $\mathcal{G}$. In particular, we generalise Neshveyev's Theorem to KMS weights.
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The paper contains a description of the KMS weights for the one-parameter action on the reduced C*-algebra of a second countable locally compact Hausdorff etale groupoid, arising from a continuous real valued homomorphism satisfying two conditions. The results are subsequently applied to identify the KMS weights for the gauge action on a simple graph algebra. The von Neumann algebra generated by the GNS-representation of an extremal KMS weight is a factor, and tools are developed to determine its type. The paper concludes with three examples to illustrate the results.
KMS states on the C*-algebras of non-principal groupoids
本研究借助迷向群C*-代数上迹的可测场刻画平展广群C*-代数的KMS态，将其用于分析变换广群C*-代数的迹态，并为数域ax+b半群Toeplitz代数的已有结果给出简短证明。
We consider fiberwise singly generated Fell-bundles over etale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo-Martin-Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of traces on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a des…
The paper contains a description of a connection between diagonal actions and certain KMS weights on groupoid $C^{*}$-algebras. It furthermore contains the realization of a graph $C^{*}$-algebra of a countable graph as the groupoid $C^{*}$-algebra of a local homeomorphism and applies the theory obtained on these algebras.
A going-down principle for {é}tale groupoids and the Baum-Connes conjecture
本文扩展了étale群胚的下行原理，通过群胚单纯复形研究证明了一般étale群胚的Baum-Connes装配映射分裂单射，并应用于拓扑K-理论连续性和Künneth公式。
KMS states on the $\mathrm{C}^*$-algebras of Fell bundles over {é}tale groupoids
本文证明了étale群胚上饱和Fell丛的全C*-代数上KMS态与迷向群上Fell丛C*-代数的可测态场之间的一一对应，建立了积分-分解定理，并通过群胚交叉积、矩阵代数等例子加以验证。
A continuous groupoid homomorphism $c$ on a locally compact second countable Hausdorff étale groupoid $\mathcal{G}$ gives rise to a $C^{*}$-dynamical system in which every $β$-KMS state can be associated to a $e^{-βc}$-quasi-invariant measure $μ$ on $\mathcal{G}^{(0)}$. Letting $Δ_μ$ denote the set of KMS states associated to such a $μ$, we will prove that $Δ_μ$ is a simplex for a large class of groupoids, and we will show that there is an abelian group that acts transitively and freely on the extremal points of $Δ_μ$. This group can be described using the support of $μ$, so our theory of symm…
We describe the KMS-states and the ground states for the gauge action on the C*-algebra of the oriented transformation groupoid of a continuous piecewise monotone and exact map of the circle.
Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. I
本文为étale群胚真作用空间建立等变Roe代数与K-理论框架，构造粗指标映射的接收函子，并证明满足条件的群胚单纯复形存在通用模。
C*-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states
本研究引入群胚在有限高阶图上的自相似作用概念，构造对应乘积系得到泛Nica–Toeplitz与Cuntz–Pimsner代数，刻画了大逆温下前者的KMS态单纯形，并证明满足条件时后者存在唯一KMS态。
We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).
The groupoid approach to equilibrium states on right LCM semigroup C*-algebras
本文用群胚方法研究右LCM半群C*-代数上由同态N诱导的KMSβ-态，给出存在唯一性的充要条件，并在ζ函数有限时建立与核子代数迹态的显式双射。
We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph automaton. Self-similar groupoid actions have a Cuntz-Pimsner algebra and a Toeplitz algebra, both of which carry natural dynamics lifted from the gauge actions. We study the equilibrium states (the KMS states) on the resulting dynamical systems. Above a critical inverse temperature, the KMS states on the Toeplitz algebra are parametrised by the traces on the full $C^*$-algebra of the groupoid, and we describe a program…
From a non-constant holomorphic map on a connected Riemann surface we construct an 'etale second countable locally compact Hausdorff groupoid whose associated groupoid C*-algebra admits a one-parameter group of automorphisms with the property that its KMS states corresponds to conformal measures in the sense of Sullivan. In this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking.
We analyse certain Haar systems associated to groupoids obtained by certain natural equivalence relations of dynamical nature on sets like $\{1,2,...,d\}^\mathbb{Z}$, $\{1,2,...,d\}^\mathbb{N}$, $S^1\times S^1$, or $(S^1)^\mathbb{N}$, where $S^1$ is the unitary circle. We also describe properties of transverse functions, quasi-invariant probabilities and KMS states for some examples of von Neumann algebras (and also $C^*$-Algebras) associated to these groupoids. We relate some of these KMS states with Gibbs states of Thermodynamic Formalism. While presenting new results, we will also describe …
The Full Set of KMS-States for Abelian Kitaev Models
本文研究了阿贝尔基塔耶夫模型的KMS态，通过证明生成子代数与UHF代数的包含关系，构建了群oid C*代数，并证明了在β≥0时KMS态的唯一性及β→∞时极限态与无 frustrations 地面态一致。
A disintegration theorem for non-second-countable etale groupoids
研究非第二可数étale群胚的C* - 代数，给出雷诺分解定理自包含版本，可覆盖可数开双截面，还应用于讨论I - 范数、KMS态和权重等。
Recent results of Laca, Raeburn, Ramagge and Whittaker show that any self-similar action of a groupoid on a graph determines a 1-parameter family of self-mappings of the trace space of the groupoid C*-algebra. We investigate the fixed points for these self-mappings, under the same hypotheses that Laca et al. used to prove that the C*-algebra of the self-similar action admits a unique KMS state. We prove that for any value of the parameter, the associated self-mapping admits a unique fixed point, which is in fact a universal attractor. This fixed point is precisely the trace that extends to a K…
(Non)exotic completions of the group algebras of isotropy groups
本文研究étale群胚约化C$^*$-代数上的范数，证明其诱导迷向群群代数上的C$^*$-范数，并指出该范数在变换群胚中为约化范数，但在某些芽群胚中可为奇异范数，同时给出保持约化范数的分次群胚类。
We use a groupoid model for the spin algebra to introduce boundary conditions on quantum spin systems via a Poisson point process representation. We can describe KMS states of quantum systems by means of a set of equations resembling the standard DLR equations of classical statistical mechanics. We introduce a notion of quantum specification which recovers the classical DLR measures in the particular case of classical interactions. Our results are in the same direction as those obtained recently by Cha, Naaijkens, and Nachtergaele, differently somehow from the predicted by Fannes and Werner.
Ground states on the \(\mathrm{C}^*\)-algebras of Fell bundles over étale groupoids
该研究针对平展群胚上的 Fell 丛的 C*代数，建立其基态空间与对应边界群胚 C*代数态空间的仿射同胚，探讨基态与 KMS_∞ 态的关系，并将结果应用于两类扭曲 C*代数以得到基态的显式刻画。
We consider the Hecke pair consisting of the group $P^+_K$ of affine transformations of a number field $K$ that preserve the orientation in every real embedding and the subgroup $P^+_O$ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra $C^*(P^+_K,P^+_O)$ has a natural time evolution $σ$, and we describe the corresponding phase transition for KMS$_β$-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated to $K$ has an essentially unique arithmetic subalgebra. When we import this suba…
D. Ruelle considered a general setting where he is able to characterize equilibrium states for Hölder potentials based on properties of conjugating homeomorphism in the so called Smale spaces. On this setting he also shows a relation of KMS states of $C^*$-algebras and equilibrium probabilities of Thermodynamic Formalism. A later paper by N. Haydn and D. Ruelle presents a shorter proof of this equivalence. Here we consider similar problems but now on the symbolic space $Ω= \{1,2,...,d\}^{\mathbb{Z} - \{ 0 \} }$ and the dynamics will be given by the shift $τ$. In the case of potentials dependin…
We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the gro…
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