Papers

We prove that every traceless real or complex matrix $A$ can be written as $A=BC-CB$, with factors of the same size and over the same field satisfying $\|B\|\,\|C\|\le K\|A\|$, where $K$ is an absolute constant. The proof combines an approximate-rank dichotomy with stable commutator representations obtained from quantum expansion.

We introduce the notion of a unitary disk-like $n$-category, which is a disk-like $n$-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like $n$-category is defined to be the local field data of a fully extended $(n+1)$D unitary TQFT, we propose a complete finite unitary disk-like $n$-category as the definition of a finite $(n+1)$-Hilbert space for all $n$. For $n=1$ and $n=2$ we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For $n=1$ we recover a unitary refinement of Schommer-Pries' classification of oriented $(1+1)$D TQFTs in terms of $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-algebras, and for $n=2$ we categorify this to classify oriented $(2+1)$D unitary TQFTs by $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like $n$-categories.

We study the action of quantum channels on states on a von Neumann algebra $\mathcal{M}$ with conserved quantities encoded by a subalgebra $\mathcal{N}$. We define a phase as an equivalence class of normal states on $\mathcal{M}$ under the action of quantum channels preserving $\mathcal{N}$. First, we establish a relative quasi-entropic characterization of phase equivalence using the Petz recovery map. We further introduce the operator algebra (OA) symmetry of a state and define an equivalence relation between OA symmetries via bishifts of biprojections. When $\mathcal{N}\subseteq\mathcal{M}$ is an irreducible finite-index subfactor of type II$_1$, we prove that phase equivalence implies OA symmetry equivalence. Consequently, OA symmetry breaking provides a sufficient mechanism for detecting phase transitions. Furthermore, combining finite-index subfactor theory with quantum Fourier analysis, we prove that the cardinality of each phase is finite and obtain an explicit upper bound depending only on the Jones index.

Understanding the capabilities and limits of quantum systems is central to quantum information processing. This includes determining which correlations can arise in networks of quantum systems distributed by independent sources. Yet for general quantum networks, no systematic, dimension-free method is known to bound these correlations arbitrarily well. We resolve this problem for all quantum tree networks through a new source-transfer model, which we prove equivalent to a mixed quantum model. The mixed model places no restriction on dimension and reduces to the standard tensor-product model in finite dimensions; the source-transfer model instead describes these correlations through states on $C^*$-algebras and completely positive maps composed recursively along the tree. We prove the equivalence by reconstructing local measurements as Radon-Nikodym derivatives in the spatial tensor products prescribed by the network. Building on this characterization, we establish two convergent outer hierarchies of semidefinite programs (SDPs): a new source-transfer hierarchy and the inflation-NPA hierarchy. For the source-transfer hierarchy, we extend noncommutative real algebraic geometry to multi-state polynomials with completely positive maps, prove a reconstruction theorem establishing its convergence, and give a sufficient stopping criterion for extracting finite-dimensional realizations. For the inflation-NPA hierarchy, we reconstruct source-transfer realizations from weak-operator limits of averages over source copies produced by quantum inflation. Consequently, the game-value promise problem for quantum tree networks in the mixed model is $\mathsf{coRE}$-complete.

Motivated by Ozawa's question of whether every maximal algebraic two-sided ideal in a \(C^*\)-algebra must be closed, we study the existence of maximal algebraic two-sided ideals in nonunital \(C^*\)-algebras. We formulate singular-distribution estimates intrinsically in terms of lower semicontinuous \(2\)-quasitraces and develop a novel criterion, called a compact quasitracial comparison pair. These are used to control membership in algebraic ideals and to establish the nonexistence of maximal ideals in nonunital \(C^*\)-algebras under a closed finite-scale face condition. This nonexistence result applies to a wide class of simple \(C^*\)-algebras. Among others such as certain hereditary \(C^*\)-subalgebras of some stable algebras, certain \(C^*\)-algebras with purely infinite corona algebras, and certain crossed products, this class includes [i] stable simple \(C^*\)-algebras with an increasing projection approximate identity \((p_n)\) and a stably finite full corner having finite radius of comparison (e.g., all simple separable stable AF algebras); [ii] all nonunital, simple, separable, stably finite, \(\mathcal Z\)-stable \(C^*\)-algebras \(A\) that have an approximate identity consisting of increasing projections \((p_n)\) and for which the simplex \(\operatorname{QT}_2^1(A,p_1)\) of normalized traces at \(p_1\) has finitely many extreme points. We also demonstrate a large class of \(C^*\)-algebras \(E\), constructed from extensions of the \(C^*\)-algebras above, such that \(E\) still has no maximal ideals. In particular, for these classes, Ozawa's question can be settled in an unexpected manner.

Let \(G\) be an infinite residually finite countable discrete group with Kazhdan's property~\((T)\). Assume that its finite-dimensional irreducible unitary representations admit an exhaustive ordering with nondecreasing dimensions tending to infinity and bounded consecutive dimension ratios. From this data we construct a separable unital residually finite-dimensional \(C^*\)-algebra with a faithful quasidiagonal tracial state \(\tau=\frac14\mu_1+\frac34\mu_2\), where \(\mu_1\) is not quasidiagonal. For the resulting algebra, quasidiagonal traces do not form a face of the tracial state space. The construction has two copies of a weighted representation ladder joined at their top levels. We prove that every quasidiagonal trace gives equal weight to the two copies. The proof combines uniform Kazhdan spectral projections, exact ranks of rounded finite-rank compressions, and a weighted Hilbert–Schmidt comparison that controls all representation blocks.

We construct a bounded Hochschild one-cocycle which detects amenability of tracial states on unital C*-algebras. For an arbitrary trace the construction uses a faithful representation containing the GNS representation as a direct summand. When \(\tau\) is faithful, \(H_\tau=L^2(A,\tau)\), and \(J\) is the canonical conjugation, the cocycle has the particularly simple form \[ \delta_\tau(a)(x) =J\pi_\tau(a^*)Jx-xJ\pi_\tau(a^*)J. \] It is inner in a natural Banach \(A\)-bimodule if and only if \(\tau\) is amenable. We also give the corresponding reformulation of embeddability into an ultrapower of the hyperfinite \(\mathrm{II}_1\) factor. Finally, for faithful traces we study a C*-algebraic variant of the Christensen–Sinclair two-cocycle and prove, with all admissibility details, that amenability makes this cocycle a coboundary.

We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every $\varepsilon\in(0,\log2)$, there exists a finite-dimensional quantum channel with a real Stinespring isometry such that the same maximally entangled input witnesses a tensor-square entropy gap of at least $\log2-\varepsilon$ for all $p\in[0,\infty]$. The output dimension can be chosen to be $O(\varepsilon^{-3})$ as $\varepsilon\downarrow0$. The construction uses direct products of free groups: tensorized Haagerup estimates control the one-copy outputs, while commutation between distinct factors forces exact Bell-branch collisions at two copies. Strong convergence gives both an existential realization through finite-dimensional representations of right-angled Artin groups followed by realification, and a Haar-orthogonal model whose success probability tends to one as the matrix dimension grows. We also determine the exact Bell quotient, prove asymptotically sharp regular-radius bounds, and show that the cubic output-dimension scale is optimal within the present purity–rank certificate.

Let $E$ be a Hilbert C*-module and $\mathcal{L}(E)$ the C*-algebra of all bounded adjointable operators on $E$. An operator $T \in \mathcal{L}(E)$ is a Kasparov cycle if $T^*T - 1$ and $TT^*-1$ are compact operators on $E$. If $\mathcal{L}(E)\rightarrow \mathcal{L}(E)$ is a prime C*-algebra, and $\varphi:\mathcal{L}(E)\rightarrow \mathcal{L}(E)$ is a linear map which is unital and surjective up to compact operators, and preserves Kasparov cycles in both directions, then the induced map $\psi:\mathcal{L}(E)/\mathcal{K}(E) \rightarrow \mathcal{L}(E)/\mathcal{K}(E)$ is either a $*$-automorphism or a $*$-anti-automorphism. Our work extends the main result of [J. Math. Anal. Appl. 354 (2009), 625-629] to the set of Kasparov cycles, showing that the assumption “$\mathcal{L}(E)/\mathcal{K}(E)$ has real rank zero” is redundant in our results.

Let $A$ and $B$ be nonzero unital $C^*$-algebras with units $1_A$ and $1_B$, respectively, and let $T\colon A\to B$ be a bijection satisfying \[ \|T(a+b)\|=\|T(a)+T(b)\|, \qquad \|T(ab)\|=\|T(a)T(b)\|, \qquad T(a^*)=T(a)^* \] for all $a,b\in A$. We prove that there exist a central symmetry $u$ in $B$ and a real $*$-isomorphism $\Phi\colon A\to B$ such that \[ T(a)=u\Phi(a) \qquad(a\in A). \] Moreover, $u$ and $\Phi$ are uniquely determined by $T$. Conversely, if $u\in B$ is a central symmetry and $\Phi\colon A\to B$ is a real $ * $-isomorphism, then $T=u\Phi$ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map $uT$ to preserve the full product, not merely the Jordan product.

Let $\mathcal M$ be a diffuse semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$. We prove that, for every nonzero Banach space $Y$, the projective tensor product $L_1(\mathcal M,\tau)\widehat{\otimes}_{\pi}Y$ has the operator Daugavet property. Moreover, the witnessing operators may always be chosen contractive. This extends the operator Daugavet phenomenon from atomless vector-valued $L_1$-spaces to the semifinite noncommutative setting and yields further Daugavet-type consequences for projective symmetric tensor products, all without approximation assumptions.

It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size $n$ of the matrices and the dimension $d$ of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed $n$ the optimal upper bound tends to infinity as $d\to \infty$. In some cases we are able to determine the optimal bound.

We study generalized amenability of locally compact quantum groups $\mathbb{G}=(\scr M, \Delta, \phi, \psi)$. Let $C_r^*(\mathbb{G})$ and $VN(\mathbb{G})$ denote, respectively, the C*-algebra and von Neumann algebra generated by the quantum left regular representation of $\mathbb{G}$. With natural covariance assumptions and a traceability condition on $VN(\mathbb{G})$, we show that if $C_r^*(\mathbb{G})$ is approximately amenable, then $\mathbb{G}$ admits an invariant quantum mean. We show further that the same conclusion holds if the predual algebra $\scr M_*$ has a bounded approximate identity and is approximately amenable.

We give a variational comparison proof of the sharp logarithmic Sobolev inequality with constant $2$ for the word-length Poisson semigroup on free group von Neumann algebras. Our approach extends the Lagrange method for cyclic groups in [Yao26] to a framework in finite tracial von Neumann algebras. The inequality was recently established by Xie and Zhang in [XZ26] using a completely different approach.

Let $\mathcal{S}^ r$ denote the Schatten–von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur–Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be real with $\|\lambda\|_{\ell^\infty}\leq1$. We determine the pairs $0<p,q\leq\infty$ for which $S_{\Psi_{f,\lambda}}:\mathcal{S}^ q\to\mathcal{S}^ p$ is bounded for every such $f$ and $\lambda$. Our principal new positive estimates treat $q=\infty,1$ and $0<p<1$. Precisely, we show that the boundedness holds exactly when \[ \frac1p\leq\frac{\alpha}{r} +\min\!\left\{1,\frac1q\right\}. \] In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.

We prove that groups acting freely and cocompactly on (possibly exotic) affine buildings of type $\tilde{A}_2$ and $\tilde C_2$ are mixed-identity free and have selfless reduced $C^*$-algebras. These results follow from a strong form of ping-pong dynamics that we call 'transversal contractivity'. Our main geometric result regards domesticity properties of elements in the associated polygons at infinity: we prove that, in our context, the opposite geometry of a hyperbolic element is topologically large.

Partial sums of the Taylor expansions of functions from the Neil algebra are uniformly bounded by their uniform norms on the ball of radius $r$ centered at the origin, where $r$ denotes the unique real root of \[ 4x^3 + x - 2 = 0. \] Moreover, this radius $r$ is optimal.

We show that every classifiable simple C*-algebra, in particular the Jiang-Su algebra, admits outer actions of several discrete quantum groups, such as finite quantum groups and duals of certain free compact quantum groups. As a consequence of the construction, we provide new examples of hyperfinite subfactors with the Temperley-Lieb-Jones standard invariant and show that $\operatorname{SU}_q(2)$ can act minimally on some Araki-Woods factors for infinitely many $q$.

Let $\Gamma$ be a countable discrete $C^*$-simple group. Let $\mu$ be a $\Gamma$-invariant Borel probability measure on $\text{Sub}(\Gamma)$, i.e., an IRS. We show that $\mu$-almost every subgroup $H$ is $C^*$-simple.

Let $\mathcal{M}$ be a finite von Neumann algebra with normalized center-valued trace $T_{\mathcal{M}}$. We prove that every $A\in\mathcal{M}$ with $T_{\mathcal{M}}(A)=0$ admits a representation $A=BC-CB$, where $B,C\in\mathcal{M}$ satisfy $\|B\|\|C\|\le K\|A\|$ for an absolute constant $K$ independent of $\mathcal{M}$.

In this paper, we introduce and study a new non Hermitian matrix-valued process built from truncations of Brownian motion on complex positive-definite matrices and of its inverse. We prove that its eigenvalue process is valued in $(1,+\infty)$ and that it forms a non-compact Jacobi particle system. Surprisingly, the same particle system occurs in \cite{BDW} in connection with Brownian motion on a non-compact complex Grassmann manifold. In this respect, we supply evidences showing that this occurrence holds true only at the spectral level and not on the corresponding matrix diffusions. We then study the large-size limit of our matrix model and prove results parallel to those obtained by the second author for the free Jacobi process.

We present a systematic study of $a$-Davis–Wielandt radius estimates for elements in a unital $C^*$-algebra through an Orlicz function approach. We derive novel bounds for the algebraic $a$-Davis–Wielandt radius and further obtain estimates for the classical algebraic Davis–Wielandt radius via polar decomposition and the Moore–Penrose inverse. The results obtained in this work extend, generalize, and unify a variety of established inequalities in the literature as special cases.

In this paper, we further investigate the role of the graded Grothendieck group $K_0^{\gr}$ and its positive cone (the talented monoid) as an effective tool for distinguishing structural types of algebras associated to higher-rank $k$-graphs. We study the prime spectrum (the space of all prime $\Gamma$-order ideals equipped with a Zariski-like topology) of a general commutative $\Gamma$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $\Gamma$-order ideal is finitely generated. As a result we are able to show that the prime spectrum of the talented monoid of a row-finite $k$-graph without sources and with a finite set of vertices is spectral. For any row-finite $k$-graph $\Lambda$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian–Pask algebra $\KP(\Lambda)$ is homeomorphic to both the space of all prime $\mathbb{Z}^k$-order ideals and the space of all prime $\mathbb{Z}^k$-filters of the talented monoid $T_\Lambda$. Another main result of this paper provides a complete topological description of regular $\Gamma$-order ideals of a refinement $\Gamma$-monoid: a $\Gamma$-order ideal $J$ is regular if and only if the corresponding closed (resp., open) set $V(J)$ (resp., $D(J)$) is regular closed (resp., regular open) in the prime spectrum. As an application of these results, we establish a lattice isomorphism between the lattice of all regular $\mathbb{Z}^k$-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian–Pask algebra via a spectrum-theoretic approach.

We show that a positive answer to the centralizer problem for tracial matrix ultraproducts implies the existence of a finitely generated nonhyperlinear group.

Consider the ring of complex-valued continuous functions on a compact Hausdorff space $X$. Cortiñas–Thom computed its nonpositive algebraic $K$-theory, confirming Rosenberg's expectation. We give a motivic refinement of their result: For $i\geq 2j\geq 0$, the $i$th integral motivic cohomology group of weight $j$ is naturally isomorphic to the $i$th Betti cohomology group of $X$.

We give an explicit example of a $W^{*}$-bundle $M$ over a compact, metrizable space $K$, which has each fiber $M_{p}$ a $\textrm{II}_{1}$-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots,a_{n,p}$ do not generate $M_{p}$ as a von Neumann algebra. More generally, if $N$ is a sub-bundle of $M$ with the property that each fiber is hyperfinite, or has a Cartan, or is generated by two commuting diffuse subalgebras, or has diffuse central sequence algebra, or is generated by a single sequential commutation orbit, then given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots,a_{n,p}$ together with $N_{p}$ do not generate $M_{p}$. We discuss implications for the generator problem for von Neumann algebras: e.g. there is no “continuous" way to take countably many generators for a tracial von Neumann algebra with separable predual and produce a single generator, at least if such a procedure works for all von Neumann algebras simultaneously.

We study function fields in one variable over the field with one element $\mathbf{F}_1$. It is proved that the Galois extensions of $\mathbf{Q}$ are isomorphic to the curves over $\mathbf{F}_1$ being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension $\mathbf{F}_{1^m}$ of order $m\ge 1$ of the field $\mathbf{F}_1$ and the $m$-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over $\mathbf{F}_1$ are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.

We show that for any positive operator monotone function $f$ the inequality $\int_{[0,\infty)} f(t) \, \mathrm{d} \left\lVert E^{\Delta_{\varphi,\psi}}(t) \xi_\psi \right\rVert^2 + f'_\infty \varphi(1-s(\psi)) \geq \frac{f(1)}{2} \Bigl( \varphi(1) + \psi(1) - \lVert \varphi - \psi \rVert \Bigr)$ holds, where $\varphi, \psi \in \mathcal{M}_{*,+}$ are two normal positive linear functionals on a von Neumann algebra $\mathcal{M}$, and $\Delta_{\varphi,\psi}$ is the associated relative modular operator. Choosing $f(t) = t^s$ with $s \in [0,1]$, the Powers-Størmer-Ogata inequality is recovered.

We construct a complex spherical fusion category with cyclic Haagerup-Izumi fusion rules for every odd n >= 3, and deduce pseudo-unitary existence. The proof has four parts: explicit real coefficients and their quadratic identities; a contour calculation for a range of cubic Fourier coefficients; algebraic completion of all cubics; and categorical reconstruction. Matrix inversion supplies reflection, and an extension of the dimension-field automorphism supplies the positive-dimensional category.

We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.