Let A be a unital separable non-elementary amenable simple stably finite C^{*} -algebra such that its tracial state space has a \sigma -compact countable-dimensional extremal boundary. We show that A is {\mathcal{Z}} -stable if and only if it has strict comparison and stable rank one. We show that this result also holds for non-unital cases (which may not be Morita equivalent to unital ones).

We show that Villadsen algebras, which are not \mathcal{Z} -stable, are singly generated. More generally, we show that any simple unital AH algebra with diagonal maps is singly generated.

We study a class of Gorenstein isolated singularities which are the quotients of generic and unimodular representations of the one-dimensional torus, or of the product of the one-dimensional torus with a finite abelian group. Based on the works of Špenko and van den Bergh [Invent. Math. 210 (2017), 3–67] and Mori and Ueyama [Adv. Math. 297 (2016), 54–92], we show that the graded singularity categories of these varieties admit tilting objects, and hence are triangulated equivalent to the perfect categories of some finite-dimensional algebras.

Let \mathscr{A} be a unital C ^{*} -algebra and E_{n} be the Hilbert \mathscr{A} -module defined as the completion of the \mathscr{A} -valued Schwartz function space \mathcal{S}^{\mathscr{A}}(\mathbb{R}^{n}) with respect to the norm \|f\|_{2} := \|{ \int_{\mathbb{R}^n} f(x)^*f(x) \, dx}\|_\mathscr{A}^{1 / 2} . Also, let \operatorname{Ad}\mathcal{U} be the canonical action of the (2n + 1) -dimensional Heisenberg group by conjugation on the algebra of adjointable operators on E_{n} , and let J be a skew-symmetric linear transformation on \mathbb{R}^{n} . We characterize the smooth vectors under \operatorname{Ad}\mathcal{U} which commute with a certain algebra of right multiplication operators R_{h} , with h \in \mathcal{S}^{\mathscr{A}}(\mathbb{R}^{n}) , where the product is “twisted” with respect to J according to a deformation quantization procedure introduced by M.A. Rieffel. More precisely, we establish that they coincide with an algebra of left multiplication operators and show that this solves, in particular, a conjecture posed by Rieffel.

Given a connected semisimple Lie group G and an arithmetic subgroup \Gamma , it is well known that each irreducible representation \pi of G occurs in the discrete spectrum L^{2}_{\mathrm{disc}}(\Gamma\backslash G) of L^{2}(\Gamma\backslash G) with at most a finite multiplicity m_{\Gamma}(\pi) . While m_{\Gamma}(\pi) is unknown in general, we are interested in its limit as \Gamma is taken to be in a tower of lattices \Gamma_{1}\supset \Gamma_{2}\supset\cdots . For a bounded measurable subset X of the unitary dual \hat{G} , we let m_{\Gamma_{k}}(X) be the integration of the multiplicity m_{\Gamma_{k}}(\pi) over all \pi in X , which can be proved finite. Let H_{X} be the direct integral of the irreducible representations in X with respect to the Plancherel measure of \hat{G} , which is also a module over the group von Neumann algebra \mathcal{L}(\Gamma_{k}) . Based on the work of Sauvageot and Finis–Lapid–Müller, we prove \lim_{k\to \infty}\frac{m_{\Gamma_{k}}(X)}{\dim_{\mathcal{L}(\Gamma_{k})}H_{X}}=1 for any bounded subset X of \hat{G} when (i) \{\Gamma_{k}\}_{k\geq 1} are cocompact or (ii) G=\mathrm{SL}(n,\mathbb{R}) and \{\Gamma_{k}\} are principal congruence subgroups.

In this paper, we introduce a class of twisted matrix algebras of M_{2}(E) and twisted direct products of E\times E for an algebra E . Let A be a noetherian Koszul Artin–Schelter regular algebra, z\in A_{2} be a regular central element of A and B=A_{P}[y_{1},y_{2};\sigma] be a graded double Ore extension of A . We use the Clifford deformation C_{A^!}(z) of Koszul dual A^{!} to study the noncommutative quadric hypersurface B/(z+y_{1}^{2}+y_{2}^{2}) . We prove that the stable category of graded maximal Cohen–Macaulay modules over B/(z+y_{1}^{2}+y_{2}^{2}) is equivalent to certain bounded derived categories, which involve a twisted matrix algebra of M_{2}(C_{A^!}(z)) or a twisted direct product of C_{A^!}(z)\times C_{A^!}(z) depending on the values of P . These results are presented as skew versions of Knörrer’s periodicity theorem. Moreover, we show B/(z+y_{1}^{2}+y_{2}^{2}) may not be a noncommutative graded isolated singularity even if A/(z) is.

We present a classification framework for saturated Fell bundles over groups, utilizing data associated with their base group and unit fiber. This framework provides a unified perspective on the structure and properties of such bundles and yields key insights into their classification. In the case of discrete groups, we obtain a complete and transparent classification in terms of generalized factor systems, leading to a cohomological description of equivalence classes. For general locally compact groups, the situation is more delicate: While our construction extends to this setting, the classification depends on choices of topology on the underlying Banach bundle.

We provide a construction of Roe ( C^{*} -)algebras of general coarse spaces in terms of coarse geometric modules. This extends the classical theory of Roe algebras of metric spaces and gives a unified framework to deal with either uniform or non-uniform Roe algebras, algebras of operators of controlled propagation and algebras of quasi-local operators, both in the metric and general coarse geometric settings. The key new definitions are those of coarse geometric module and coarse support of operators between coarse geometric modules. These let us construct natural bridges between coarse geometry and operator algebras. We then study the general structure of Roe-like algebras and investigate several structural properties, such as admitting (non-commutative) Cartan subalgebras or computing their intersection with the compact operators. Lastly, we prove that assigning to a coarse space the K -theory groups of its Roe algebra(s) is a natural functorial operation.

In this paper, we present generalizations of some results on the asymptotic property C for wreath products. Specifically, we prove that certain wreath-like products admit asymptotic property C, thus providing some new examples for further study.

In this paper, we introduce a concept of A-by-FCE coarse fibration structure for metric spaces, which serves as a generalization of the A-by-CE structure for a sequence of group extensions proposed by Deng, Wang, and Yu. We prove that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry that admit an A-by-FCE coarse fibration structure. As an application, the relative expanders constructed by Arzhantseva and Tessera, as well as the box spaces derived from an “amenable-by-Haagerup” group extension, admit the A-by-FCE coarse fibration structure. Consequently, the maximal coarse Baum–Connes conjecture holds for these spaces, which may not admit an FCE structure, that is, fibered coarse embedding into Hilbert space.