First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

We study homeomorphisms of a Cantor set with $k$ ( $k<+\infty$ ) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$ , the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

We prove that for any countable group $\unicode[STIX]{x1D6E4}$ , there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure.

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

We investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

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.

Given an infinite, compact, monothetic group $G$ we study decompositions and structure of unbounded derivations in a crossed product $\text{C}^{\ast }$-algebra $C(G)\rtimes \mathbb{Z}$ obtained from a translation on $G$ by a generator of a dense cyclic subgroup. We also study derivations in a Toeplitz extension of the crossed product and the question whether unbounded derivations can be lifted from one algebra to the other.

The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by

$$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

$\unicode[STIX]{x1D6EF}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$\unicode[STIX]{x1D6E5}_{s}$

$Q_{s}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$w_{\unicode[STIX]{x1D6FF}}$

${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x1D6FF}\in (0,1)$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$Q_{s}$

$Q_{s}$

${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$

$J_{s}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$, every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$-algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

We provide a class of separable II1 factors $M$ whose central sequence algebra is not the ‘tail’ algebra associated with any decreasing sequence of von Neumann subalgebras of $M$. This settles a question of McDuff [On residual sequences in a II1 factor, J. Lond. Math. Soc. (2) (1971), 273–280].

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$-endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$-semigroup over $P$ on $M$.

In this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decomposition and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities.

We consider fibrewise singly generated Fell bundles over étale 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 states 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 description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev’s main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs.

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$ -algebras, including: $D$ -absorbing $C^{\ast }$ -algebras, where $D$ is a strongly self-absorbing $C^{\ast }$ -algebra; stable $C^{\ast }$ -algebras; $C^{\ast }$ -algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$ -algebras with finite stable rank (or real rank); and $C^{\ast }$ -algebras whose $K$ -theory is either trivial, rational, or $n$ -divisible for $n\in \mathbb{N}$ . The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$ -algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$ -action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

We answer a question of Skalski and Sołan (2016) about inner faithfulness of the Curran’s map of extending a quantum increasing sequence to a quantum permutation. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran’s map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon (2018).

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

Kaplansky introduced the notions of CCR and GCR $C^{\ast }$-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.

This paper concerns HH-relations in the lattices P(M) of all projections of W*-algebras M. If M is a finite algebra, all these relations are generated by trails in P(M). If M is an infinite countably decomposable factor, they are either generated by trails or associated with them.