The Effros-Shen algebra corresponding to an irrational number $\theta$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $\theta$ encodes the dimensions of the summands, and how the matrix algebras at the $n$th level fit into the summands at the $(n+1)$th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the $C^*$-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of $\theta$. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In the present work, we define a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This article describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction.

We study modular theory in hyperfinite von Neumann algebras, i.e. in those of type II or type III, from the viewpoint of a subregion charge sector decomposition. We address this symmetry resolution by considering infinite tensor products of finite-dimensional algebras with fixed subregion charge values. An important ingredient is the combination of these algebras using direct integrals. This allows us to obtain the symmetry-resolved modular operator, modular flow, and modular correlation functions for hyperfinite algebras. Our approach establishes a mathematical foundation for recent results on symmetry resolution and modular theory in conformal field theory. Our analysis applies both to charges defined on a continuous range, or on a discrete set. The latter is of interest for condensed matter theory. Moreover, within the AdS/CFT correspondence we expect our findings to be relevant as a new ingredient for bulk spacetime reconstruction, including information from different boundary charge sectors.

For an étale groupoid, we define a pairing between the Crainic-Moerdijk groupoid homology and the simplex of invariant Borel probability measures on the base space. The main novelty here is that the groupoid need not have totally disconnected base space, and thus the pairing can give more refined information than the measures of clopen subsets of the base space. Our principal motivation is $C^*$-algebra theory. The Elliott invariant of a $C^*$-algebra is defined in terms of $K$-theory and traces; it is fundamental in the long-running program to classify simple $C^*$-algebras (satisfying additional necessary conditions). We use our pairing to define a groupoid Elliott invariant, and show that for many interesting groupoids it agrees with the $C^*$-algebraic Elliott invariant of the groupoid $C^*$-algebra: this includes irrational rotation algebras and the $C^*$-algebras arising from orbit breaking constructions studied by the first listed author, Putnam, and Strung. These results can be thought of as establishing a refinement of Matui's HK conjecture for the relevant groupoids.

We extend Lück's determinant conjecture from groups to invariant random subgroups (IRS) of free groups, a framework generalizing groups where a non-sofic object is known to exist. For every free group, we prove the existence of an IRS satisfying the determinant conjecture that is not co-hyperlinear, and hence not co-sofic. This provides evidence that satisfying the determinant conjecture might be a weaker property than soficity for groups, and consequently the conjecture possibly holds for all groups. We use techniques from non-local games and $\mathsf{MIP}^* = \mathsf{RE}$, showing more generally when the latter can be used to narrow down when a von Neumann algebra (or IRS) contains a non-Connes embeddable object.

We take a first step towards developing a new language to describe causal structure, event horizons, and quantum extremal surfaces (QES) for the bulk description of holographic systems beyond the standard Einstein gravity regime. By considering the structure of boundary operator algebras, we introduce a stringy ``causal depth parameter'', which quantifies the depth of the emergent radial direction in the bulk, and a certain notion of ergodicity on the boundary. We define stringy event horizons in terms of the half-sided inclusion property, which is related to a stronger notion of boundary ergodic or quantum chaotic behavior. Using our definition, we argue that above the Hawking--Page temperature, there is an emergent sharp horizon structure in the large $N$ limit of $\mathcal{N}=4$ Super-Yang--Mills at finite nonzero 't Hooft coupling. In contrast, some previously considered toy models of black hole information loss do not have a stringy horizon. Our methods can also be used to probe violations of the equivalence principle for the bulk gravitational system, and to explore aspects of stringy nonlocality.

We introduce the notion of biexactness for general von Neumann algebras, naturally extending the notion from group theory. We show that biexactness implies solidity for von Neumann algebras, and that many of the examples of solid von Neumann algebras contained in the literature are, in fact, biexact. We also give examples of certain crossed products arising from Gaussian actions that are solid but not biexact, and we give examples of certain $q$-Gaussian von Neumann algebras that are strongly solid but not biexact. The techniques developed involve studying a certain weak form of nuclear embeddings, and we use this setting to give a new description of weak exactness for von Neumann algebras, which allows us to answer several open problems in the literature about weakly exact von Neumann algebras.

Given a codimension one Riemannian embedding of Riemannian spin$^c$-manifolds $\imath:X \to Y$ we construct a family $\{\imath_!^ \epsilon\}_{0< \epsilon< \epsilon_0}$of unbounded$KK$-cycles from$C(X)$to$C_0(Y)$, each equipped with a connection $\nabla^\epsilon$ and each representing the shriek class $\imath_! \in KK(C(X), C_0(Y))$. We compute the unbounded product of $\imath_!^\epsilon$ with the Dirac operator $D_Y$ on $Y$ and show that this represents the $KK$-theoretic factorization of the fundamental class $[X] = \imath_! \otimes [Y]$for all$\epsilon$. In the limit$\epsilon \to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}{\epsilon} T + D_X + \mathcal{O}(\epsilon)$where the ``divergent'' part$T$ is an index cycle representing the unit in $KK(\mathbb{C}, \mathbb{C})$ and the constant ``renormalized'' term is the Dirac operator $D_X$ on $X$. The curvature of $(\imath_!^\epsilon, \nabla^\epsilon)$ is further shown to converge to the square of the mean curvature of $\imath$ as $\epsilon \to 0$.

We give an example of a locally compact effective Hausdorff, minimal ample groupoid such that its rational homology differs from the $K$-theory of its reduced groupoid $C^*$-algebra. Moreover, we prove that such example satisfies Matui's AH-conjecture.

We describe an algebra of observables for a static patch in de Sitter space, with operators gravitationally dressed to the worldline of an observer. The algebra is a von Neumann algebra of Type II$_1$. There is a natural notion of entropy for a state of such an algebra. There is a maximum entropy state, which corresponds to empty de Sitter space, and the entropy of any semiclassical state of the Type II$_1$ algebras agrees, up to an additive constant independent of the state, with the expected generalized entropy $S_{\text{gen}}=(A/4G_N)+S_{\text{out}}$. An arbitrary additive constant is present because of the renormalization that is involved in defining entropy for a Type II$_1$ algebra.

We prove the equivalence of VOA tensor categories and conformal net tensor categories for the following examples: all WZW models; all lattice VOAs; all unitary parafermion VOAs; type $ADE$ discrete series $W$-algebras; their tensor products; their regular cosets. A new proof of the complete rationality of conformal nets is also given.

As an analogue of the topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M, \tau)$ and apply it to generalize the main results of [AHO23], showing that for a trace-preserving action $\Gamma \curvearrowright (A, \tau_A)$ on an amenable tracial von Neumann algebra, any $\Gamma$-invariant amenable intermediate subalgebra between $A$ and $\Gamma \ltimes A$ is necessarily a subalgebra of $\mathrm{Rad}(\Gamma) \ltimes A$. By taking $(A, \tau_A) = L^\infty(X, \nu_X)$ for a free pmp action $\Gamma \curvearrowright (X, \nu_X)$, we obtain a similar result for the invariant subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.

Given a metric space with bounded geometry, one may associate with it the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra, both containing information about the large scale geometry of the metric space. We show that these two Banach algebras are Morita equivalent in the sense of Lafforgue for 1\leq p&lt;\infty. As a consequence, these two Banach algebras have the same $K$-theory. We then define an $\ell^p$ uniform coarse assembly map taking values in the $K$-theory of the $\ell^p$ uniform Roe algebra and show that it is not always surjective.

Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Herv\'e Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework.

We apply quantitative (or controlled) $K$-theory to prove that a certain $L^p$ assembly map is an isomorphism for $p\in[1,\infty)$ when an action of a countable discrete group $\Gamma$ on a compact Hausdorff space $X$ has finite dynamical complexity. When $p=2$, this is a model for the Baum-Connes assembly map for $\Gamma$ with coefficients in $C(X)$, and was shown to be an isomorphism by Guentner, Willett, and Yu.

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper on group theory and quantum mechanics, are $p\times p$ unitary matrices given by the diagonal matrix whose entries are the $p$-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by $\mathfrak u$ and $\mathfrak v$, satisfy$\mathfrak u^p=\mathfrak v^p=1_p$(the$p\times p$ identity matrix) and the commutation relation $\mathfrak u\mathfrak v=\zeta \mathfrak v\mathfrak u$, where$\zeta$is a primitive$p$-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if $u$ and $v$ are any $d\times d$ unitary matrices such that $u^p= v^p=1_d$ and $u v=\zeta vu$, then there exists a unital completely positive linear map $\phi:\mathcal M_p(\mathbb C)\rightarrow\mathcal M_d(\mathbb C)$such that$\phi(\mathfrak u)= u$ and $\phi(\mathfrak v)=v$. We also show, moreover, that any two pairs of $p$-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent. When $p=2$, the Weyl matrices are two of the three Pauli matrices from quantum mechanics. It was recently shown that $g$-tuples of Pauli-Weyl-Brauer unitaries are universal for all $g$-tuples of anticommuting selfadjoint unitary matrices; however, we show here that the analogous result fails for positive integers $p>2$. Finally, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

The Neretin group $\mathcal{N}_{d,k}$ is the totally disconnected locally compact group consisting of almost automorphisms of the tree $\mathcal{T}_{d,k}$. This group has a distinguished open subgroup $\mathcal{O}_{d,k}$. We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether $\mathcal{O}_{d,k}$ is of type I or not.

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk, with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for groupoids which are principal with orbits homeomorphic to a Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

We prove that if a totally disconnected locally compact group admits a topologically free boundary, then the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. We also prove a partial converse of this result.

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.

We classify the homotopy classes of proper Fredholm maps from an infinite dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli-Smale mod 2 degree and the absolute value of the oriented degree.