We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case.

In this paper, we discuss a refinement of quantum data processing inequality for the sandwiched quasi-relative entropy $\mathcal{S}_2$ on a tracial von-Neumann algebra. The main result is a universal recoverability bound with the Petz recovery map, which was previously obtained in the finite dimensional setup.

We prove that the unitary Drinfeld center of a unitary tensor category is equivalente to the category of unitary bimodules for the canonical W*-algebra object, generalizing Müger's result to the non-fusion case. This is then used to express factorization homology in terms of C*-algebraic extensions of symmetric enveloping algebras and actions of Drinfeld dobules of compact quantum groups.

We propose and investigate a strategy toward a proof of the Riemann Hypothesis based on a spectral realization of its non-trivial zeros. Our approach constructs self-adjoint operators obtained as rank-one perturbations of the spectral triple associated with the scaling operator on the interval $[\lambda^{-1}, \lambda]$. The construction only involves the Euler products over the primes $p \leq x = \lambda^2$ and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of $\zeta(1/2 + i s)$, even for small values of $x$. The theoretical foundation rests on the framework introduced in "Spectral triples and zeta-cycles" (Enseign. Math. 69 (2023), no. 1-2, 93-148), together with the extension in "Quadratic Forms, Real Zeros and Echoes of the Spectral Action" (Commun. Math. Phys. (2025)) of the classical Caratheodory-Fejer theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators converge towards the zeros of $\zeta(1/2 + i s)$ as the parameters $N, \lambda \to \infty$. A rigorous proof of this convergence would establish the Riemann Hypothesis. We further compute the regularized determinants of these operators and discuss the analytic role they play in controlling and potentially proving the above result by showing that, suitably normalized, they converge towards the Riemann $\Xi$ function.

In these notes, we describe an interesting connection between unitary representations of Lie groups and nets of local algebras, as they appear in Algebraic Quantum Field Theory (AQFT). It is based on first translating the axioms for nets of operator algebras parameterized by regions in a space-time manifold into those for nets of real subspaces, and then study this structure from a perspective based on geometry and representation theory of Lie groups.

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 provide a counterexample to the HK-conjecture using the flat manifold odometers constructed by Deeley. Deeley's counterexample uses an odometer built from a flat manifold of dimension 9 and an expansive self-cover. We strengthen this result by showing that for each dimension $d\geq 4$ there is a counterexample to the HK-conjecture built from a flat manifold of dimension $d$. Moreover, we show that this dimension is minimal, as if $d\leq 3$ the HK-conjecture holds for the associated odometer. We also discuss implications for the stable and unstable groupoid of a Smale space.

In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respectively. In the simplest case our result coincides with a sharp version of the classical Hormander Lp-multiplier theorem, which was obtained by Grafakos and Slavikova in [11]. Finally, we present some applications to the evolution equation in non-commutative setting.

We study $\mathrm{W}^*$-categories, and explain the ways in which complete $\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\langle\,\,,\,\rangle_{\mathrm{Hilb}}\,:\,\overline C\times C\,\to\, \mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\mathrm{W}^*$-categories there is an antilinear equivalence $$\dagger:\mathrm{Func}(C,D) \leftrightarrow \mathrm{Func}(D,C)$$ characterised by $\langle c,F^\dagger(d)\rangle_{\mathrm{Hilb}} \simeq \langle F(c),d\rangle_{\mathrm{Hilb}}$, for $c\in C$ and $d \in D$.

We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś's theorem and countable saturation, to this more general setting.

von Neumann algebras have been playing an increasingly important role in the context of gauge theories and gravity. The crossed product presents a natural method for implementing constraints through the commutation theorem, rendering it a useful tool for constructing gauge invariant algebras. The crossed product of a Type III algebra with its modular automorphism group is semifinite, which means that the crossed product regulates divergences in local quantum field theories. In this letter, we find a sufficient condition for the semifiniteness of the crossed product of a type III algebra with any locally compact group containing the modular automorphism group. Our condition surprisingly implies the centrality of the modular flow in the symmetry group, and we provide evidence for the necessity of this condition. Under these conditions, we construct an associated trace which computes physical expectation values. We comment on the importance of this result and and its implications for subregion physics in gauge theory and gravity.

In this article, we consider actions of \mathcal{Z}_+^d, \mathcal{R}_+^d and finitely generated free groups on a von Neumann algebras $M$ and prove a version of maximal ergodic inequality. Additionally, we establish non-commutative analogues of pointwise ergodic theorems for associated actions in the predual when M is finite.

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

We prove cocontinuity of the $\max$-tensor product of C*-categories and develop a framework to perform factorization homology in a C*-setting. In such context, we specialize some results of D. Ben-Zvi, A. Brochier and D. Jordan. As a consequence of our constructions, we realize quantum Hamiltonian reduction in terms of bimodules over a factor $N$. We also provide a GNS-type reconstruction theorem for C*-algebra objects of in categories of bimodules over a II_1-factor, enhancing a realization theorem due to C. Jones and D. Penneys.

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<\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.

The word stable is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and quantum synchronous strategies for non-local games. We observe in particular that simple spectral gap estimates can lead to strong quantitative forms of stability. For example, we prove that the direct product of two (flexibly) Hilbert-Schmidt stable groups is again (flexibly) Hilbert-Schmidt stable, provided that one of them has Kazhdan's property (T). We also provide a simple form and simple analysis of a non-local game with few questions, with the property that synchronous strategies with large value are close to perfect strategies involving large Pauli matrices. This simplifies one of the steps (the question reduction) in the recent announced resolution of Connes' embedding problem by Ji, Natarajan, Vidick, Wright and Yuen.