We show that the method to construct C^*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from old ones, and study the relation of C^*-algebras constructed from them. We also prove that our C^*-algebras have a certain characterization. This gives us another definition of our C^*-algebras.
We introduce a subclass of recursive subhomogeneous algebras, in which each of the pullback maps is diagonal in a suitable sense. We define the notion of a diagonal map between two such algebras and show that every simple inductive limit of these algebras with diagonal bonding maps has stable rank one. As an application, we prove that for any infinite compact metric space $T$ and minimal homeomorphism $h\colon T\to T$, the associated dynamical crossed product $\mathrm{C^*}(\mathbb{Z},T,h)$ has stable rank one. This affirms a conjecture of Archey, Niu, and Phillips. We also show that the Toms-W…
Let $φ:X\to X$ be a homeomorphism of a compact metric space $X$. For any continuous function $F:X\to \mathbb{R}$ there is a one-parameter group $α^{F}$ of automorphisms on the crossed product $C^*$-algebra $C(X)\rtimes_φ\mathbb{Z}$ defined such that $α^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and $U$ is the canonical unitary in the construction of the crossed product. In this paper we study the 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 resul…
In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-correspondences over commutative $\mathrm{C}^*$-algebras from the point of view of the $\mathrm{C}^*$-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite-dimensional infinite compact metric space $X$ twisted by a vector bundle, the resulting Cuntz--Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these $\mathrm{C}^*$-algebras by the Elliott invariant. This establishes a dichotomy: when t…
We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.
This paper investigates the structure of C*-algebras built from one-sided Sturmian subshifts. They are parametrised by irrationals in the unit interval and built from a local homeomorphism associated to the subshift. We provide an explicit construction and description of this local homeomorphism. The C*-algebras are *-isomorphic exactly when the systems are conjugate, and they are Morita equivalent exactly when the defining irrationals are equivalent (this happens precisely when the systems are flow equivalent). Using only elementary dynamical tools, we compute the dynamic asymptotic dimension…
The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. More precisely, minimal homeomorphisms are constructed on space with prescribed $K$-theory or cohomology. We also allow for some control of the map on $K$-theory and cohomology induced from these minimal homeomorphisms. This allows for the construction of many minimal homeomorphisms that are not homotopic to the identity. Applications to $C^*$-algebras will be discuss…
A $λ$-graph bisystem $\mathcal{L}$ consists of two labeled Bratteli diagrams $(\mathcal{L}^-,\mathcal{L}^+)$, that presents a two-sided subshift $Λ_\mathcal{L}$. We will construct a compact totally disconnected metric space with a shift homeomorphism consisting of two-dimensional configurations from a $λ$-graph bisystem. The configuration space has a certain étale AF-equivalence relation written $R_{\mathcal{L}}$ with a natural shift homeomorphism $σ_{\mathcal{L}}$ coming from the shift homeomorphism $σ_{Λ_{\mathcal{L}}}$ on the subshift $Λ_{\mathcal{L}}$. The equivalence relation $R_{\mathcal…
We prove several rigidity results for corona $C^*$-algebras and Čech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todorčević's $\OCA$ and Martin's Axiom at level $\aleph_1$ imply: (i) that if $X$ and $Y$ are locally compact second countable topological spaces, then all homeomorphisms between $βX\setminus X$ and $βY\setminus Y$ are induced by homeomorphisms between cocompact subspaces of $X$ and $Y$; (ii) that all automorphisms of the corona algebra of a separable $C^*$-algebra are trivial in a topological sense; (iii) that if $A$ is …
