We establish the following results on higher order ${\mathcal{S}}^{p}$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space:

1. (i) $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every bounded self-adjoint operator if and only if $f\in C^{n}(\mathbb{R})$;

2. (ii) if $f^{\prime },\ldots ,f^{(n-1)}\in C_{b}(\mathbb{R})$ and $f^{(n)}\in C_{0}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator;

3. (iii) if $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, then $f$ is $n-1$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable and $n$ times Gâteaux ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator.

We also prove that if $f\in B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{q}$-differentiable, $1\leqslant q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of Kissin et al. (Proc. Lond. Math. Soc. (3)108(3) (2014), 327–349) to arbitrary $n$ and unbounded operators as well as substantially extend the results of Azamov et al. (Canad. J. Math.61(2) (2009), 241–263); Coine et al. (J. Funct. Anal.; doi:10.1016/j.jfa.2018.09.005); Peller (J. Funct. Anal.233(2) (2006), 515–544) on higher order ${\mathcal{S}}^{p}$-differentiability of $f$ in a certain Wiener class, Gâteaux ${\mathcal{S}}^{2}$-differentiability of $f\in C^{n}(\mathbb{R})$ with $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, and Gâteaux ${\mathcal{S}}^{q}$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$. As an application, we extend ${\mathcal{S}}^{p}$-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives.

Let $n$ be a positive integer. A $C^{\ast }$-algebra is said to be $n$-subhomogeneous if all its irreducible representations have dimension at most $n$. We give various approximation properties characterising $n$-subhomogeneous $C^{\ast }$-algebras.

Let 𝔻n be the open unit polydisc in ℂn, $n \ges 1$, and let H2(𝔻n) be the Hardy space over 𝔻n. For $n\ges 3$, we show that if θ ∈ H∞(𝔻n) is an inner function, then the n-tuple of commuting operators $(C_{z_1}, \ldots , C_{z_n})$ on the Beurling type quotient module ${\cal Q}_{\theta }$ is not essentially normal, where

$${\rm {\cal Q}}_\theta = H^2({\rm {\open D}}^n)/\theta H^2({\rm {\open D}}^n)\quad {\rm and}\quad C_{z_j} = P_{{\rm {\cal Q}}_\theta }M_{z_j}\vert_{{\rm {\cal Q}}_\theta }\quad (j = 1, \ldots ,n).$$

In this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.

We characterize the class of RFD $C^{\ast }$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^{\ast }$-algebra is finite-dimensional, which is equivalent to the $C^{\ast }$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^{\ast }$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^{\ast }$-algebra that is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz–Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

Let a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

Let $A$ be the inductive limit of a sequence

$$\begin{eqnarray}A_{1}\xrightarrow[{}]{\unicode[STIX]{x1D719}_{1,2}}A_{2}\xrightarrow[{}]{\unicode[STIX]{x1D719}_{2,3}}A_{3}\longrightarrow \cdots\end{eqnarray}$$

$A_{n}=\bigoplus _{i=1}^{n_{i}}A_{[n,i]}$

$A_{[n,i]}$

$\unicode[STIX]{x1D719}_{n,n+1}$

$A$

$$\begin{eqnarray}B_{1}\xrightarrow[{}]{\unicode[STIX]{x1D713}_{1,2}}B_{2}\xrightarrow[{}]{\unicode[STIX]{x1D713}_{2,3}}B_{3}\longrightarrow \cdots\end{eqnarray}$$

$B_{n}=\bigoplus _{i=1}^{n_{i}^{\prime }}B_{[n,i]^{\prime }}$

$B_{[n,i]^{\prime }}$

$\unicode[STIX]{x1D713}_{n,n+1}$

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

In this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.

We give two characterisations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.

We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application, we give a dilation theoretic proof of Boca's Theorem.

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusions $L(H)\subset L(G)$.

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

We prove that any countable non-amenable group $\unicode[STIX]{x1D6E4}$ admits a free, minimal, amenable, purely infinite action on the non-compact Cantor set. This answers a question of Kellerhals, Monod and Rørdam [Non-supramenable groups acting on locally compact spaces. Doc. Math.18 (2013), 1597–1626].

We realize every small rigid C*-tensor category with simple unit object as Hilbert C*-bimodules.

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.