Papers

The minimal operator systems over ${\rm Mat}_k(\mathbb C)_+$, $k\geqslant2$, and the Lorentz cones $\mathcal L_m$, $m\geqslant4$, are not subhomogeneous. For the $2\times2$ cone we construct extreme positive maps of arbitrarily large output dimension. Positive retracts give the remaining cases. Equivalently, for fixed $k\geqslant2$ and $d$, there exist $s>d$ and an entangled positive operator on $\mathbb C^k\otimes\mathbb C^s$ such that every compression of the second factor to $\mathbb C^d$ is separable.

Starting from an arbitrary conformal net, we construct a genus zero functorial conformal field theory whose Hilbert space is the vacuum sector of the net. Specifically, we construct an algebra for the operad of little discs and conformal embeddings with values in the category of Hilbert spaces and bounded linear maps. We work with closed discs, and our conformal embeddings explicitly allow the image of an incoming disc to overlap with the boundary of the outgoing disc. As an application, we show that all conformal nets satisfy the trace class condition: if $L_0$ is the conformal Hamiltonian of the net, then the operators $r^{L_0}$ are trace class whenever $0 \le r < 1$. In particular, the $L_0$-eigenspaces of a conformal net are automatically finite-dimensional.

We compute the second mixed moment of elementary symmetric polynomials in the eigenvalues of the random matrix model $AUBU^*$, where $A$ and $B$ are positive, and $U$ is a Haar-distributed random matrix, associated with finite free multiplicative convolution.

In this paper, we confirm a conjecture of Ozawa and others asserting that every finitely generated, relatively hyperbolic, exact group is bi-exact (in the sense of Ozawa) relative to its natural peripheral structure. As a consequence, every such group gives rise to a prime group von Neumann algebra. As an application, we construct a continuum family of property (T), relatively hyperbolic groups $\{G_i\}_{i\in I}$ such that, for every fixed arbitrary free, ergodic, probability measure-preserving action $G_i \curvearrowright Z_i$, the collection of associated group measure space von Neumann algebras $\{L^\infty(Z_i)\rtimes G_i\}_{i\in I}$ are pairwise non-stably $\ast$-isomorphic.

We prove a free probabilistic Bakry–Émery criterion for logarithmic Sobolev inequalities and hypercontractivity. Our approach relies on a noncommutative version of the carré du champ operator and its iteration, which we are able to define in the free probability setting. This allows us to formulate a positive curvature condition, which is shown to be sufficient for logarithmic Sobolev inequalities and hypercontractivity in this setting. In the setting of free Gibbs measures, we also show that this curvature condition can be expressed in terms of the positivity of the Hessian of the associated potential.

We construct a subadditive weight on the type I$_\infty$ factor that preserves directed suprema but is not $\sigma$-weakly lower semi-continuous. This provides a counterexample to Haagerup's first problem posed in 1975.

In 2006 Neeb asked whether the Hausdorff completion of a continuous inverse algebra must again be a continuous inverse algebra. The noncommutative case remained open, while the commutative case was known to be true. We give a negative answer after twenty years. We construct a Hausdorff metrizable locally m-convex complex continuous inverse algebra whose completion is a Fréchet locally m-convex algebra, where inversion stays continuous but the set of invertible elements is not open. The construction uses finite-support sequences in a dense nil subalgebra of a Jacobson-semisimple Banach algebra. Finite support makes every element nilpotent, giving continuous inversion without a locally uniform bound on nilpotence indices. In the completion, shifting a fixed noninvertible element to later and later coordinates produces noninvertible elements converging to the identity. Thus completion destroys exactly the local spectral stability at the identity. The counterexample is necessarily noncommutative and marks the precise boundary of the commutative completion theorem.

We establish a general structure theorem for local quantum Cuntz–Krieger families associated to finite quantum graphs whose relations are analogous to those of the partial isometries generating an ordinary graph $C^*$-algebra, with the quantum edge correspondence governing composition. We apply this structure theorem to the quantum dephasing graph on $M_n$ to obtain an explicit isomorphism between $O_n$ and the local quantum Cuntz–Krieger algebra. More generally, we obtain a unital embedding of $O_n$ in the local quantum Cuntz–Krieger algebra for the dephasing graph on a finite direct sum of $M_n$.

We prove that the heat semigroup on the circle has optimal complete modified logarithmic Sobolev constant \(1\). The proof is based on a matrix-valued Wirtinger inequality and yields the stronger Bogoliubov–Kubo–Mori Fisher information contraction with rate \(e^{-2t}\). As a consequence, the complete tensorization and transference principle yields the same sharp constant for the heat semigroups on classical and quantum tori.

This paper explores the algebraic invariants and ergodic behavior of the finite-index inclusions forming commuting squares of $\text{II}_1$ factors. We begin by establishing how regularity, intermediate-subfactor lattices, and Weyl groups transfer across the commuting square under relative commutant and irreducibility conditions. Furthermore, we provide a complete characterization, via a novel factorization theorem, of the unitary normalizers of the upper inclusion when the lower inclusion is regular. Applying these findings to crossed-product inclusions induced by discrete groups, we introduce a relative eigenbasis property that preserves regular inclusions. Finally, we contrast this by proving that a relative weak mixing condition on the group action forces the crossed-product inclusion to be singular.

A central question in higher index theory and operator algebras is whether the Roe algebra and the quasi-local algebra associated with a metric space of bounded geometry coincide, or at least have the same $K$-theory. In this paper, we focus on a sparse metric space $X$. We prove the following three main results: (1) For a block-diagonal operator $T$ with uniformly bounded block-rank, $T$ is quasi-local if and only if it is in the Roe algebra. (2) In general, we discover a ghost block-diagonal projection which is quasi-local but not in the Roe algebra. (3) For a sequence of expander graphs with sufficiently large girth, the inclusion of the uniform Roe algebra into the uniform quasi-local algebra induces a non-surjective map on their $K_0$-groups. This yields the first known $K$-theoretic distinction between the uniform Roe algebra and the uniform quasi-local algebra.

In this paper, for various pairs of topological spaces $X$ and $Y$, we introduce reflection functors between the categories $\mathfrak{KK}(X)$ and $\mathfrak{KK}(Y)$ of Kirchberg's ideal-related KK-theory for separable C*-algebras over $X$ and $Y$. We prove that these functors induce an equivalence $\mathfrak{KK}(X)_{\mathrm{loc}} \simeq \mathfrak{KK}(Y)_{\mathrm{loc}}$ between the localizing subcategories introduced by Meyer and Nest, and that this equivalence restricts to an equivalence $\mathcal{B}(X) \simeq \mathcal{B}(Y)$ between the bootstrap categories. Combining these equivalences with a combinatorial argument due to Bernstein, Gelfand, and Ponomarev from the representation theory of quivers, we show that, for a finite $T_0$-space $X$ whose Hasse diagram is an orientation of a tree, the categories $\mathfrak{KK}(X)_{\mathrm{loc}}$ and $\mathcal{B}(X)$ depend only on the underlying tree and not on its orientation. We also prove the analogous results for Dadarlat and Meyer's ideal-related E-theory. Moreover, we prove a rearrangement property of reflection functors, which yields an analogue of Coxeter functors. Finally, we apply reflection functors to prove that filtrated K-theory satisfies the universal coefficient theorem for C*-algebras over finite $T_0$-spaces whose Hasse diagrams are orientations of Dynkin diagrams of type A.

We establish quantitative versions of the law of rare events and binomial approximations in non-commutative probability settings, including the free, Boolean, and monotone convolution frameworks. Our main results provide explicit error bounds in the non-commutative Wasserstein distance for approximations of convolutions of rare countings by non-commutative Poisson and binomial distributions. These bounds extend classical results from the tensor setting to the non-commutative regime. Our approach relies on a discrete Lindeberg-type interpolation scheme combined with algebraic properties of cumulants adapted to each independence notion. The results presented here fill a gap in the literature concerning explicit rates of convergence in non-commutative limit theorems.

For a channel $\Phi$ on $M_d$ written in the generalised Bloch parameterisation $r \mapsto Ar+c$, we prove that entanglement breaking implies $\|A\|_*^2 + \frac{d(d-1)}{2}|c|^2 \le (d-1)^2$, where $\|\cdot\|_*$ denotes the nuclear norm. The bound is attained in every dimension by the completely dephasing channel, and at $d=2$ by an explicit family with $c\neq0$. The qubit case reads $\|A\|_*^2+|c|^2\le1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\|A\|_*\le1$ by retaining the POVM completeness relation $\sum_k f_k=0$, which her argument discards; this recentres $A$ from a second moment into a cross-covariance, and is the entire content of the improvement.

We introduce new properties of Haagerup-type for Fell bundles over discrete groups and discuss their relationship with Haagerup properties for the associated $C^*$-algebras. In particular, our results on Fell bundles associated with twisted unital $C^*$-dynamical systems extend previous results known in the untwisted case.

Consider a free product $(M,\varphi)= (M_1,\varphi_1)* (M_2,\varphi_2)$ of non-trivial von Neumann algebras. Using amalgamated free product techniques, we establish irreducibility and weak spectral gap results for free product subalgebras and the centralizer subalgebra of the free product state. In particular, it is shown that, under a mild dimension constraint, the inclusion of the diffuse summand of these subalgebras into the corresponding corner of $M^\omega$ is irreducible for any free ultrafilter $\omega\in \beta\mathbb{N}\setminus \mathbb{N}$. We also obtain analogous irreducibility results for graph products of von Neumann algebras.

We represent sandwiched and $\alpha$-$z$ Rényi divergences as averages of ordinary relative entropy. The $\alpha$-$z$ Rényi divergence is shown to be an integral over the relative entropy of a canonical family of fixed-ray escort states along the ray $z=c\alpha$. We prove this representation for normal states on an arbitrary von Neumann algebra, using Haagerup non-commutative $L^p$ spaces and interpolation. The formula holds for every $z>0$: for $0<\alpha<1$ it holds when the support of the first state is contained in that of the reference state, and for $\alpha>1$ it holds whenever the divergence is finite. When the lower-order support condition fails, we identify the exact fixed-ray support-boundary term. The representation yields a monotone escort profile and a convex order potential. We use these to reformulate one-shot testing converses, exact sandwiched strong-converse exponents, and work-extraction reliability as signed-area or level-crossing statements, and discuss a restricted two-parameter pair-conversion rate.

For every $1<p\neq2<\infty$, we exhibit an explicit Schur multiplier on $S_p$ whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such $p$, we construct a finitely supported Schur symbol $m_p$ such that \[ \|M_{m_p}\|_{p\to p} < \|M_{m_p}\|_{\mathrm{cb},p}. \] This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.

We study $G$-equivariant Hörmander pseudodifferential operators on a noncompact symmetric space $G/K$. We define a notion of a complete symbol function, called the Harish-Chandra symbol function, on the spherical tempered dual of $G$, for operators that satisfy a rapid off-diagonal decay condition on their Schwartz kernels, and we characterize the class of such operators in terms of their Harish-Chandra symbol function.

We investigate the relation between the complete isometry class of subspaces of noncommutative $\mathrm{L}^p$-spaces and their contractive complementability, where $1 < p < \infty$ with $p \not= 2$. We show that if $P \colon \mathrm{L}^p(\mathcal{M}) \to \mathrm{L}^p(\mathcal{M})$ is a positive contractive projection whose range is completely isometric to another noncommutative $\mathrm{L}^p$-space, then $P$ is necessarily completely positive. This provides a converse to the main result of [ArR24] and the positivity assumption on $P$ is essential. We further establish a rectangular analogue of this result. More precisely, we prove that every closed subspace of a noncommutative $\mathrm{L}^p$-space which is completely isometric to a rectangular noncommutative $\mathrm{L}^p$-space of the form $e\mathrm{L}^p(\mathcal{N})(1-e)$ is the range of a contractively decomposable projection. Combined with the known converse implication, this yields a characterization of the ranges of contractively decomposable projections as precisely the subspaces completely isometric to rectangular $\mathrm{L}^p$-spaces associated with $\mathrm{W}^*$-ternary rings of operators.

Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.

We establish upper and lower bounds for relative entropy contraction of generalized quantum depolarizing channels and semigroups. Our bound provides tight first order asymptotic of the contraction rate in terms of the dimension constant. One side estimate are based on sharp reverse ratio and convexity of relative entropy of two states, which can be derived from the recently introduced Hockey-Stick quantum $f$-divergence, and also independently, Bogoliubov–Kubo–Mori quantum Fisher information metric. The other side follows from the existence of index achieving pure state with respect to a general conditional expectation. Our results extend to the complete entropy contraction rate, tensor stable estimates for product dynamics. Examples include quantum depolarization, dephasing, and compact group symmetrization, with consequences for the decay rate of coherence and asymmetry.

In this article, we introduce and investigate a new subclass $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$. This class contains the absolutely minimum attaining operators and is properly contained in the class of minimum attaining operators. We establish several structural and spectral characterizations of operators in $\mathcal{M}_r(H)$. In particular, we characterize positive operators in $\mathcal{M}_r(H)$ in terms of their spectral representations. We prove that $\mathcal{M}_r(H)$ is dense in $\mathcal{B}(H)$ in the operator norm and, moreover, that the operators in $\mathcal{M}_r(H)$ having a nontrivial invariant half-space are also dense in $\mathcal{B}(H)$ in the operator norm. We further obtain a representation theorem for normal operators in $\mathcal{M}_r(H)$ and establish additional structural properties of this class.

Let $X$ be a Cantor space and let $Q\subset M_{fc}(X)$ be a compact Choquet simplex of atomless full-support probability measures. We introduce the associated Affine Evaluation Map (AEM), which assigns to each clopen set $A\subset X$ the affine function \[ \widehat A(\mu)=\mu(A),\qquad \mu\in Q, \] and study the geometric subset condition obtained by comparing these evaluation functions pointwise on $Q$. For a good geometric AEM we construct the ordered group \[ G_Q=C(X,\mathbb Z)/N_Q, \qquad N_Q=\left\{f\in C(X,\mathbb Z): \int f\,d\mu=0\ \text{for every }\mu\in Q\right\}, \] and show that it is a simple dimension group whose normalized state space is canonically $Q$. We prove that its order interval $[0,u]$ is precisely the clopen scale and that \[ J_Q=N_Q, \] where $J_Q$ is generated by the elementary relations $\mathbf 1_A-\mathbf 1_B$ with $\widehat A=\widehat B$. We also show that the full stabilizer $\mathcal H_Q$ has invariant-measure simplex exactly $Q$. Using the clopen-scale property and the Herman–Putnam–Skau realization theorem, we obtain a dimension-group proof that every good geometric AEM is realized by a minimal Cantor homeomorphism $T$ with $Q=M_T(X)$. For a Cantor minimal system we identify $G_Q$ with the classical dimension group modulo infinitesimals. We further distinguish goodness, fairness, ergodicity, and minimality of the measure stabilizer by explicit examples. Finally, we apply the AEM framework to minimal Cantor actions of countable amenable groups. If $Q=M_G(X)$, then $Q$ canonically defines a proper geometric AEM, and we clarify which parts of the preceding theory depend only on $Q$ and which are specifically $\mathbb Z$-dynamical. In particular, $Q$ is good if and only if there exists a minimal homeomorphism $T$ of $X$ such that \[ M_T(X)=M_G(X). \]

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$. For $A \in \mathcal{B}(\mathcal{H})$, we refer to the sequence $\{|A^{n}|^{1/n}\}_{n\in\mathbb{N}}$ as the normalized power sequence of $A$. In this article, we study the norm convergence property, i.e. convergence of normalized power sequence in the norm topology for operators belonging to type $\mathrm{I}_k$ von Neumann algebras acting on a separable complex Hilbert space. By utilizing continuous upper-triangular forms via unitary conjugations, we construct specific projection-valued families to prove that every operator in a type $\mathrm{I}_k$ von Neumann algebra is decomposable. As a consequence, this immediately establishes that every such operator possesses the norm convergence property, extending recent results known for matrices with complex-valued entries, compact operators on a separable Hilbert space, spectral operators, and Riesz operators. Finally, we provide a counterexample within the type $I_\infty$ factor $\mathcal{B}(l^2(\mathbb{N}))$ to demonstrate that this convergence property generally fails when the dimension $k$ is infinite.

In 1967, Kadison asked whether every type $\mathrm{II}_1$ factor admits an orthonormal basis, with respect to its trace, consisting of unitaries. We resolve this problem in full generality and, more broadly, characterize the diffuse finite von Neumann algebras admitting such bases consisting of symmetries. Let $M$ be a diffuse finite von Neumann algebra with a faithful normal tracial state $\tau$, let $\kappa$ be the density character of $L^2(M,\tau)$, and identify $M$ with its canonical image in $L^2(M,\tau)$. We prove that there exists a family $B\subset {s\in M:s=s^*=s^{-1},\ \tau(s)=0}$ such that ${1}\cup B$ is an orthonormal basis of $L^2(M,\tau)$ if and only if the density character of $L^2(zM,\tau(z)^{-1}\tau|_{zM})$ equals $\kappa$ for every nonzero central projection $z\in Z(M)$. In particular, every type $\mathrm{II}_1$ factor admits an orthonormal basis consisting of unitaries, thereby answering Kadison's question affirmatively. We also provide a Lean 4 formalization of the main results.

Davidson and Kennedy introduced the dilation order in connection with commutative C*-algebras and classical Choquet theory. Its noncommutative counterpart continues to detect the unique extension property of GNS representations. Motivated by the failure of Arveson's hyperrigidity conjecture and the amended theorem of Clouâtre and Thompson, we develop a state-space approach to this rigidity phenomenon. First, we introduce the strong dilation relation on the state space of a C*-algebra and characterize the unique tight extension property through maximality of states. Second, we define the integral subdivision relation and prove that maximality of all pure states in the dilation order implies maximality of every state in the integral subdivision relation. This provides a state space-level amendment of Arveson's hyperrigidity conjecture and yields an alternative proof of the Clouâtre-Thompson theorem. Finally, we show that the strong dilation and integral subdivision relations coincide in the commutative setting and they both agree with the abstract Choquet order associated with a function system.

We characterize the simplicity of reduced crossed product C*-algebras in terms of stabilizer subgroups. Specifically, we prove that if $G$ is a countable group and $X$ is a minimal $G$-flow, then the reduced crossed product C*-algebra $\mathrm{C}(X) \times_\lambda G$ is simple if and only if there is a point in $X$ with a C*-simple stabilizer subgroup. Further, these conditions are equivalent to a generic point in $X$ having a C*-simple stabilizer subgroup. We also provide an example demonstrating that this result does not extend to uncountable groups. This completely resolves a question of Ozawa.

Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable étale groupoid \(G\) and let \(\mathrm{C}^*(G; \mathcal{A})\) be the associated Fell bundle \(\mathrm{C}^*\)-algebra. Suppose \(\sigma^c\) is the real dynamics on \(\mathrm{C}^*(G;\mathcal{A})\) induced by a real-valued \(1\)-cocycle \(c\). We establish an affine homeomorphism between the \(\sigma^c\)-ground state space of \(\mathrm{C}^*(G;\mathcal{A})\) and the state space of \(\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})\), where \(Z\) is the boundary set of \(c\) and \(G(Z)\) is the boundary groupoid. In particular, \(\mathrm{C}^*(G; \mathcal{A})\) admits \(\sigma^c\)-ground states if and only if \(Z\neq \emptyset\). We further investigate the relation between ground states and KMS\(_{\infty}\) states and give a crystallization interpretation of the boundary \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})\) when the cocycle is locally constant. Finally, we apply our results to several classes of examples, including twisted \(\mathrm{C}^*\)-algebras of Deaconu–Renault groupoids and twisted higher-rank graph \(\mathrm{C}^*\)-algebras, obtaining explicit descriptions of their ground states.

We determine the exact strong converse exponent of quantum soft covering under the sandwiched Rényi divergence for all orders $\alpha\in[\frac{1}{2},\infty)$. For $\alpha\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $\alpha\in[1,\infty)$, it is characterized by the order-$\alpha$ sandwiched Rényi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched Rényi divergence for $\alpha\in(2,\infty)$, expressed in terms of the corresponding order-$\alpha$ sandwiched Rényi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $\alpha\in[\frac{1}{2},1)$.