Papers

In this short article we verify the conjecture proposed by P. M. Sołtan and the author, by proving that a second countable compact quantum group whose scaling automorphisms are inner, must be of Kac type.

Eilers showed that for AD algebras of real rank zero with bounded torsion in $\mathrm{K}_1$, the coefficient transformations $\kappa$ are redundant in the classification by ordered scaled total $K$-theory. In this paper we treat the unbounded torsion case and prove that, in contrast, $\kappa$ becomes necessary. Specifically, we construct two non-isomorphic unital AD algebras of real rank zero, $E_0$ and $E_1$, such that their ordered scaled total $K$-theory invariants agree when the $\kappa$-maps are forgotten, i.e., \[ \bigl( \underline{\mathrm{K}}(E_0), \underline{\mathrm{K}}(E_0)_+, [1_{E_0}] \bigr)_{\underline{\mathrm{K}}_{\langle\kappa\rangle}} \cong \bigl( \underline{\mathrm{K}}(E_1), \underline{\mathrm{K}}(E_1)_+, [1_{E_1}] \bigr)_{\underline{\mathrm{K}}_{\langle\kappa\rangle}} \] but are not isomorphic under the full $\Lambda$-module structure: \[ \bigl( \underline{\mathrm{K}}(E_0), \underline{\mathrm{K}}(E_0)_+, [1_{E_0}] \bigr)_{\Lambda} \not\cong \bigl( \underline{\mathrm{K}}(E_1), \underline{\mathrm{K}}(E_1)_+, [1_{E_1}] \bigr)_{\Lambda} .\] This completes the picture for the necessity of all three operations $\rho$, $\beta$, and $\kappa$ in this context.

For separable $C^*$-algebras $A$ and $B$, the $E$-theory group $E(A,B)$ carries a natural, generally non-Hausdorff, second-countable group topology, Hausdorff exactly when the closure of zero $\overline{\{0\}} \subseteq E(A,B)$ is trivial. The nonzero elements of $\overline{\{0\}}$ are the phantom classes, invisible to every continuous Hausdorff-valued invariant. We identify $\overline{\{0\}}$ as a derived inverse limit over any shape system $(C_n)$ of $SA \otimes \mathcal{K}$, giving a natural Milnor sequence $0 \to {\varprojlim}^1 [C_n, S^2B \otimes \mathcal{K}] \to E(A,B) \to \varprojlim [C_n, SB \otimes \mathcal{K}] \to 0$ for all separable $A$ and $B$, with no UCT or nuclearity hypothesis. For nuclear $A$ it agrees with the Willett-Yu controlled-$KK$ Milnor sequence, and when $A$ satisfies the UCT, nuclear or not, its ${\varprojlim}^1$ term is the pure-extension group $\mathrm{Pext}^1_{\mathbb{Z}}(K_{*+1}(A), K_*(B))$, as in Schochet's fine-structure description of the Kasparov groups.

We consider sums of freely independent self-adjoint random variables that are not necessarily identically distributed. Let $\mu_j$ denote the distribution of the $j$th summand. We assume that they have mean zero and finite absolute moments of order $2+\delta$, where $0<\delta\le 1$. Let $\Delta$ denote the Kolmogorov distance, let $\mu^{(n)}$ be the distribution of the normalized partial sum, let $\omega$ be the standard semicircle law, and let $B_n^2$ be the variance of the partial sum. The purpose of this paper is to prove the Berry–Esseen estimate in the free central limit theorem. Namely, there exists an absolute constant $C>0$ such that, for every $0<\delta\le 1$, \[ \Delta(\mu^{(n)},\omega) \le \frac{C}{B_n^{2+\delta}}\sum_{j=1}^n \int_{\R}|x|^{2+\delta}\,\mu_j(dx), \] Our result not only improves several known estimates for general non-identically distributed random variables, but also establishes exactly the same Berry–Esseen estimate as in classical probability theory. The proof combines truncation, a quantitative estimate for the $R$-transform, a stability analysis of a perturbed semicircle equation, and a Bai-type smoothing inequality.

Using non-$\mathbb{A}^1$-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic $K$-theory of singular rings, such as rings of the form $\mathbb{Z}/p^n$, $\mathbb{Z}[x]/(x^e)$, and $\mathscr{C}(X;\mathbb{C})$ for $X$ a compact Hausdorff space. These refinements are made possible by the use of integral $p$-adic Hodge theory, as a replacement for the standard use of trace methods in $K$-theory.

We prove $a_n(S) \le e\,(n+1)\, s_n(S)$ for every s-number sequence $(s_n)$, every bounded linear operator $S$ between normed spaces, and every $n \in \mathbb{N}_0$, where $a_n$ are the approximation numbers, which are the largest s-numbers. This is sharp up to the constant and settles conjectures of Mityagin, Henkin, Carl and Pietsch dating back to 1963. We also extend it to widths of convex sets and discuss optimality there. The proof is elementary.

We give a complete characterization of regular (in the sense of Kumjian and Renault) unital inclusions of finite-dimensional $C^*$-algebras. For subalgebras $\bigoplus_j( \mathbb{M}_{d_j}(\mathbb{C}) \otimes \mathbb{I}_{p_j})$ of $\mathbb{M}_n(\mathbb{C})$, we show that regularity depends only on equality of the multiplicities $p_j$, while unitary regularity—characterized recently by the first author and Silambarasan—additionally requires equality of the $d_j$; we recover the latter via a streamlined alternative proof. Extending this to inclusions of arbitrary finite-dimensional $C^*$-algebras, encoded by an inclusion matrix $\Lambda$, we show that regularity is equivalent to an explicit row/column condition on $\Lambda$—coinciding with the normalizer matrix introduced by the first author and Silambarasan —so that the device used there to detect unitary regularity is shown to characterize regularity in general; unitary regularity is recovered by a further dimension-equality condition.

We study subhomogeneity of the minimal operator system over a polyhedral cone, or equivalently the size of irreducible Arveson boundary points of free polyhedra. We obtain a complete classification in dimension three for cones (equivalently, dimension two for polytopes): cones with three extreme rays are 1-subhomogeneous, cones with four extreme rays are 2-subhomogeneous, and cones with at least five extreme rays are not subhomogeneous. In the last case, we construct irreducible Arveson boundary points at every even matrix level. We prove that subhomogeneity passes to faces and face quotients. We also determine the size of irreducible Arveson boundary points of free polyhedra over products of two simplices, obtaining a dichotomy between the product of two segments and all remaining cases. As applications, we classify n-dimensional cones with n+1 extreme rays. More generally, when every facet omits at most two extreme rays, we show that subhomogeneity occurs exactly for direct sums of simplicial cones and three-dimensional four-ray cones. We also construct, for a range of fixed dimensions and ray counts, cones exhibiting both subhomogeneous and non-subhomogeneous behavior.

We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.

We establish a simple characterization of amenability of second countable locally compact quantum groups $\mathbb{G}$ in terms of the long-time behavior of convolution semigroups of states on $\mathbb{G}$. We then investigate the deeper problem of combining amenability of $\mathbb{G}$ with other approximation properties of the dual quantum group $\hat{\mathbb{G}}$. Specifically, we characterize the combination of ("strong") amenability of $\mathbb{G}$ with either the failure of property (T) or the presence of the Haagerup property for $\hat{\mathbb{G}}$. These results, which appear to be new even in the classical setting of locally compact groups, produce convolution semigroups on $\mathbb{G}$ whose behavior is controlled both as time goes to zero and as time goes to infinity.

We show that the category of twisted/untwisted representations of the Heisenberg conformal net $\text{Heis}$ is a continuous Tambara-Yamagami category for the group $\mathbb{R}$. We compute the associators and the $\mathbb{Z}/2$-crossed braiding. By taking a $\mathbb{Z}/2$-equivariantization, we obtain an explicit computation of the braided continuous tensor category of representations of the fixed-points conformal net $\text{Heis}^{\mathbb{Z}/2}$. This provides the first explicit computation of a category of representations of a conformal net containing irreducible representations whose tensor product is a direct integral of irreducible representations.

We construct two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$ that are both ICC and have Kazhdan's property (T). Although $\Gamma_1$ and $\Gamma_2$ are not isomorphic as groups, their group von Neumann algebras are isomorphic: $L(\Gamma_1)\cong L(\Gamma_2)$. This provides a counterexample to Connes' rigidity conjecture for ICC property (T) groups. This result was obtained with assistance from GPT-5.6 Sol, independently of and concurrently with work by OpenAI.

In this paper, we obtain the structure of a class of contractive projections, known as generalized tri-circular projections, on some functional Banach spaces on the open unit disk $\mathbb{D}$, which includes Hardy, Bergman and Bloch spaces. We then consider the space $S^p_{\mathcal{K}}$ of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$ such that $f' \in H^p(\mathcal{K})$ and give complete description of generalized tri-circular projections on this space. Here, $\mathcal{K}$ is a complex separable Hilbert space, and $H^p(\mathcal{K})$ denotes the Hardy space of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$.