We show that the dynamical system associated by Putnam to a pair of graph embeddings is identical to the shift map on the limit space of a self-similar groupoid action on a graph. Moreover, performing a certain out-split on said graph gives rise to a Katsura–Exel–Pardo groupoid action on the out-split graph whose associated limit space dynamical system is conjugate to the previous one. We characterise the self-similar properties of these groupoids in terms of properties of their defining data, two matrices A, B. We prove a large class of the associated limit spaces are bundles of circles and points that fibre over a totally disconnected space, and the dynamics restricted to each circle are of the form $z\to z^{n}$. Moreover, we find a planar embedding of these spaces, thereby answering a question Putnam posed in his paper.

We introduce positive correspondences as right $C^*$-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the $C^*$-correspondence (Enchilada) category as a ‘retract’. Kasparov’s KSGNS construction provides a semi-functor from this semi-category onto the $C^*$-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz–Pimsner algebras, and we describe classes of examples arising from projections on $C^*$-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index.

We establish some interactions between uniformly recurrent subgroups (URSs) of a group G and cosets topologies $\tau _{\mathcal {N}}$ on G associated to a family $\mathcal {N}$ of normal subgroups of G. We show that when $\mathcal {N}$ consists of finite index subgroups of G, there is a natural closure operation $\mathcal {H} \mapsto \mathrm {cl}_{\mathcal {N}}(\mathcal {H})$ that associates to a URS $\mathcal {H}$ another URS $\mathrm {cl}_{\mathcal {N}}(\mathcal {H})$, called the $\tau _{\mathcal {N}}$-closure of $\mathcal {H}$. We give a characterization of the URSs $\mathcal {H}$ that are $\tau _{\mathcal {N}}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when G belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal {A}_G$ and prove that for certain coset topologies on G, almost all subgroups $H \in \mathcal {A}_G$ have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal {A}_G$ to be a singleton based on residual properties of G.

Starting from a uniquely ergodic action of a locally compact group G on a compact space $X_0$, we consider non-commutative skew-product extensions of the dynamics, on the crossed product $C(X_0)\rtimes _\alpha {\mathbb Z}$, through a $1$-cocycle of G in ${\mathbb T}$, with $\alpha $ commuting with the given dynamics. We first prove that any two such skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity with respect to the fixed-point subalgebra by characterizing both in terms of the cocycle assigning the dynamics. The set of all invariant states is also determined: it is affinely homeomorphic with ${\mathcal P}({\mathbb T})$, the Borel probability measures on the one-dimensional torus ${\mathbb T}$, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity with respect to the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra.

Let G be a compact group, let $\mathcal {B}$ be a unital C$^*$-algebra, and let $(\mathcal {A},G,\alpha )$ be a free C$^*$-dynamical system, in the sense of Ellwood, with fixed point algebra $\mathcal {B}$. We prove that $(\mathcal {A},G,\alpha )$ can be realized as the G-continuous part of the invariants of an equivariant coaction of G on a corner of $\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$ for a certain Hilbert space ${\mathfrak {H}}$ that arises from the freeness of the action. This extends a result by Wassermann for free and ergodic C$^*$-dynamical systems. As an application, we show that any faithful $^*$-representation of $\mathcal {B}$ on a Hilbert space ${\mathfrak {H}}_{\mathcal {B}}$ gives rise to a faithful covariant representation of $(\mathcal {A},G,\alpha )$ on some truncation of ${\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}}$.

We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.

The action on the trace space induced by a generic automorphism of a suitable finite classifiable ${\mathrm {C}^*}$-algebra is shown to be chaotic and weakly mixing. Model ${\mathrm {C}^*}$-algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe $KK$-contractible stably projectionless ${\mathrm {C}^*}$-algebras as crossed products.

For every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz–Pimsner $C^*$-algebra $\mathcal {O}_X$ has nuclear dimension $1$ when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X, which also recovers an exact sequence, discovered before by Carlsen and Eilers.

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.

We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.

Given a self-similar set K defined from an iterated function system $\Gamma =(\gamma _{1},\ldots ,\gamma _{d})$ and a set of functions $H=\{h_{i}:K\to \mathbb {R}\}_{i=1}^{d}$ satisfying suitable conditions, we define a generalized gauge action on Kajiwara–Watatani algebras $\mathcal {O}_{\Gamma }$ and their Toeplitz extensions $\mathcal {T}_{\Gamma }$. We then characterize the KMS states for this action. For each $\beta \in (0,\infty )$, there is a Ruelle operator $\mathcal {L}_{H,\beta }$, and the existence of KMS states at inverse temperature $\beta $ is related to this operator. The critical inverse temperature $\beta _{c}$ is such that $\mathcal {L}_{H,\beta _{c}}$ has spectral radius 1. If $\beta <\beta _{c}$, there are no KMS states on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$; if $\beta =\beta _{c}$, there is a unique KMS state on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ which is given by the eigenmeasure of $\mathcal {L}_{H,\beta _{c}}$; and if $\beta>\beta _{c}$, including $\beta =\infty $, the extreme points of the set of KMS states on $\mathcal {T}_{\Gamma }$ are parametrized by the elements of K and on $\mathcal {O}_{\Gamma }$ by the set of branched points.

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

We characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$-algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$.

This paper is a continuation of the paper, Matsumoto [‘Subshifts, $\lambda $-graph bisystems and $C^*$-algebras’, J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $, there exists a $\lambda $-graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $-graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$, and define a $C^*$-algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $-graph bisystems come from two-sided subshifts, these $C^*$-algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$-algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$-algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts.

We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$ -torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ , we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$ , for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.