Papers

It is shown that an infinite dimensional, simple, unital, monotracial C*-algebra with strict comparison with respect to its trace is selfless.

We study the Cuntz semigroup of arbitrary rings with stable rank one. We prove that Cuntz subequivalence between countably generated projective right modules $P$, $Q$ is equivalent to $P$ being isomorphic to a pure submodule of $Q$, whilst Cuntz equivalence amounts to isomorphism.~We also prove order-cancellation of finitely generated projective modules (and, more generally, pairs of modules admitting certain complements) from direct sums with countably generated projective modules.

The determinant approximation conjecture states that for a residually finite group $G$ and a chain $G_i$ of finite index normal subgroups with trivial intersection, the Fuglede-Kadison determinant of every matrix over the group algebra $\mathbb{Q} G$ is the limit of the Fuglede-Kadison determinants of the matrix reductions from $\mathbb{Q} G$ to $\mathbb{Q} (G/G_i)$. We disprove the conjecture by exhibiting an explicit element of the integral group ring of the Heisenberg group with determinant equal to $2$ and reduced determinants bounded by $\sqrt[3]{5}$.

In this paper, we present an application of matrix-valued truncated Toeplitz operators to the matrix version of the Nevanlinna-Pick interpolation problem. We then investigate the unitary part of a matrix-valued truncated Toeplitz operator induced by a contractive matrix-valued symbol. We further prove that, for a pure matrix-valued inner function, the corresponding model space is invariant under multiplication operator if and only if the symbol is constant. In addition, we give an algebraic proof of a characterization of the invertibility of matrix-valued asymmetric truncated Toeplitz operators using Toeplitz operators with 2 by 2 block matrix symbol. We also discuss the solvability of matrix-valued asymmetric truncated Toeplitz operators.

Let $B(\mathcal H)$ be the $*$-algebra of all bounded linear operators on a Hilbert space $\mathcal H$. A classical result due to Stamplfi shows that, for any $a\in B(\mathcal H)$, the norm of the inner derivation $\delta_a=[a,\cdot]$ on $B(\mathcal H)$ is given by $\|\delta_a\|_{B(\mathcal H)\to B(\mathcal H)}=2\inf\{ \|a-c{\bf 1}\|_{B(\mathcal H)}:c\in \mathbb{C}\}$. In the present paper, we provide a formula for the norm of a generalized derivation implemented by self-adjoint operators from an ideal in $B(\mathcal H)$, which partially answers a question posed by Fialkow and Loebl in the 1980s. We also consider this question in a more general setting of a symmetrically normed (not necessarily complete or separable) space affiliated with a properly infinite semifinite von Neumann algebra.

This paper applies the Projection Characterization and the Symmetry Bootstrap for two operators $A, B$ (with $A$ unitary or self-adjoint) nearby commuting operators $A', B'$ (with $A'$ unitary or self-adjoint) from arXiv:2412.20795 to problems related to structured almost commuting unitaries, self-adjoint operators that almost anti-commute, and unitaries that almost commute up to a rational phase. We prove several new stability theorems for these types of relations for matrices. We also provide estimates for how close the constructed operators are for several known stability theorems that did not previously have asymptotic estimates.

We introduce discrete $C^*$-Frobenius algebras as categorical data for constructing infinite-index extensions of Möbius covariant nets. The underlying representation is a countable direct sum of simple dualized modules, while the direct sum itself is not necessarily dualizable. Accordingly, rather than requiring a single bounded multiplication morphism, we work with compatible families of bounded left and right multiplication morphisms. Using Bin Gui's theory of categorical extensions, we associate to every discrete $C^*$-Frobenius algebra two (generally non-local) Möbius covariant extensions generated by left and right charged fields, respectively. If the discrete $C^*$-Frobenius algebra is commutative, the two extensions coincide and yield a local Möbius covariant extension. As applications, for every $C^*$-tensor category generated by an invertible object with trivial self-braiding such that its fusion product rule is given by $\mathbb{Z}$, we construct two classes of examples: the simple current extension by $\mathbb{Z}$ and the discrete Longo-Rehren construction.

We show that every countable group $G$ that has Ozawa's property $\rm PHP$ and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. As a consequence, we deduce that all non-trivial countable linear groups with trivial amenable radical have this property. Along the way, we prove that the class of groups with $\rm PHP$ enjoys some permanence properties that are of independent interest.

In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators $S$ and $n$-qubit “magic" Hamiltonians $H$ that together generate the full Lie algebra $\mathfrak{su}(2^n)$, with a special focus on the Ising Hamiltonian $J$. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian $H$. We find that the number of applications of $H$ needed to approximate any unitary up to accuracy $\epsilon$ scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.

We prove that the quantum monad in dimension $2n$ admits no natural transformation to the polymorphism clone of linear equations modulo $n$. Consequently, for every $n\geq 2$, there exists an unsatisfiable system of linear equations over $\mathbb{Z}_n$ whose constraint system game admits a perfect finite-dimensional quantum strategy. As a corollary, we completely characterise pseudo-telepathic constraint languages in finite dimension. The proof combines a result of Harding, Jager, and Smith on group-valued measures on subspaces of Hilbert spaces with the polymorphism-minion characterisation of quantum pseudo-telepathy.

Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $\tau(T)=0$ for all $\tau\in T(\mathcal{A})$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal{A}$ has unique trace; $(2)$ $\mathcal{A}$ is separable and has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^*$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.

We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},\tau)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $\tau$. This resolves in the affirmative an open problem that has circulated in the non-commutative world for more than four decades.

Let $\{P_j\}$ be a sequence of projections onto the closed subspaces $\mathcal{M}_j$ of a Hilbert space $\mathscr{H}$. Consider an infinite sequence $P_{i_1}, P_{i_2}, \ldots$ with each $P_{i_n} \in \{P_1, P_2, \ldots\}$, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence $\{P_{i_n} \cdots P_{i_2} P_{i_1} x\}_{n=1}^{\infty}$ converge strongly or weakly to $Px$ for every $x \in \mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M} = \bigcap_{i=1}^{\infty} \mathcal{M}_i$? In this paper, we present some results concerning random products of countably infinitely many projections $\{P_j\}_{j=1}^{\infty}$ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions $\sigma \colon \mathbb{N} \to \mathbb{N}$, called infinite-periodic functions, and rigorously show that the sequence $\{T_n x\}$ defined by \[ T_1 := P_{\sigma(1)}\quad \mbox{and} \quad T_n := P_{\sigma(n)} T_{n-1} \quad \text{for all } n \geq 2, \] for $x \in \mathscr{H}$, converges weakly to $Px$, where $P$ is the projection onto $\bigcap_{j=1}^{\infty} \mathcal{R}(P_j)$. We also provide some technical examples to illustrate our results.

Let $(\mathscr{E}, \langle \cdot, \cdot\rangle)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak{A}$. The space of adjointable operators on $\mathscr{E}$ is denoted by $\mathcal{L}\left(\mathscr{E}\right)$. The sets of all states and pure states on $\mathfrak{A}$ are denoted by $\mathcal{S}\left(\mathfrak{A}\right)$ and $\mathcal{P}\left( \mathfrak{A}\right)$, respectively. For $\tau\in\mathcal{S}\left( \mathfrak{A}\right)$, let us define $\mathcal{N}^{\mathscr{E}}_{\tau}:=\left\lbrace x\in\mathscr{E}:\tau\left( \left\langle x,x\right\rangle\right)=0 \right\rbrace$. The Hilbert completion of ${\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ is denoted by $\mathscr{E}_{\tau}$. For $T\in\mathcal{L}(\mathscr{E})$, the operator $T_{\mathscr{E}_\tau}\in\mathbb{B}\left( \mathscr{E}_\tau\right)$, is defined by $T_{\mathscr{E}_\tau}\left(x+\mathcal{N}^{\mathscr{E}}_{\tau}\right)=Tx+\mathcal{N}^{\mathscr{E}}_{\tau}$. In this paper, we show that $\mathscr{E}_{\tau}={\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ when $\mathfrak{A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $\pi^{\mathscr{E}}_k(\cdot)$ for $k\geq1$. We prove that $\pi^{\mathscr{E}}_k(T)\leq\sup_{\tau\in\mathcal{P}\left( \mathfrak{A}\right)}\left\|T_{\mathscr{E}_\tau}\right\|_{\left(k\right)}\leq{\pi}^{\mathscr{E}^{\sharp}}_k(T_{\mathscr{E}^{\sharp}})$ for every $T\in\mathcal{L}\left(\mathscr{E}\right)$, where the space $\mathscr{E}^{\sharp}$ is constructed as the extension of $\mathscr{E}$ by the embedding of $\mathfrak{A}$ into its enveloping von Neumann algebra $\mathfrak{A}^{**}$.

This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^*$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $*$-conjugate-automorphisms $\sharp$ on $C^*$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set $\mathbb{L}_c(\mathscr{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^*$-module $\mathscr{E}$ possesses the structure of a Hilbert $\mathbb{L}(\mathscr{E})$-module. We also introduce the key notion of $\sharp$-conjugation on a Hilbert $C^*$-module and fully characterize such modules in terms of real forms and real $C^*$-algebras. Moreover, we investigate the geometric properties of $\sharp$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^*$-module equipped with a $\sharp$-conjugation $C$, thereby extending the celebrated Garcia–Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913–3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $\sharp$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $\langle Cx,y\rangle = \langle x,Cy\rangle^*$ would force the underlying $C^*$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.

In this note, we prove that every primitive KMS-symmetric quantum Markov semigroup on a finite-dimensional matrix algebra satisfies a modified logarithmic Sobolev inequality (MLSI). We also construct a primitive quantum Markov semigroup without KMS symmetry that fails MLSI, and a primitive KMS-symmetric quantum Markov semigroup that fails complete MLSI (CMLSI). The latter construction also provides a non-primitive KMS-symmetric quantum Markov semigroup without MLSI. For the graph-based KMS-symmetric quantum Markov semigroups studied here, we prove MLSI and CMLSI when the underlying graph is connected and has at least three vertices. Finally, we establish CMLSI for a class of primitive bimodule KMS-symmetric quantum Markov semigroups arising from fermionic systems.

Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.

We study dimension-free Bohnenblust–Hille inequalities for local operators on systems of $K$-level qudits. For the operator space of support at most $d$, uniformly over all tensor-product orthonormal operator bases, we prove a Bohnenblust–Hille inequality with the optimal exponent $2d/(d+1)$ whose asymptotic exponential base in the interaction order is of order $O(\sqrt K)$. Our proof is intrinsic, based on a one-site scalarization and block-transversal decoupling, and does not rely on a scalarization to the cyclic group or on a Remez-type argument. We complement the upper bound by showing that the asymptotic exponential Bohnenblust–Hille base satisfies $cK^{1/4}\le \beta(K)\le C\sqrt K$ with universal constants $c,C>0$. In this sense, the present work may be viewed as continuing the line of work of Slote–Volberg–Zhang from a complementary intrinsic viewpoint. We also revisit their Gell–Mann and Heisenberg–Weyl scalarization procedures. In the Heisenberg–Weyl setting, the prime-dimensional case already yields the optimal interaction exponent, whereas for composite $K$ the scalar total degree used in their reduction leads to a larger exponent. A support-sensitive formulation recovers the optimal interaction exponent for every $K$, while the intrinsic argument gives the stronger dependence on the local dimension at the level of the asymptotic exponential base. As further consequences, we determine the sharp scale, up to factors exponential in $d$ with base depending only on $K$, of coefficient $\ell_1$-normalization and unconditionality on the exact-support spaces. This yields dimension-free coefficient sparsification, including sparse generalized-Pauli approximation in Heisenberg–Weyl coordinates, as well as normalization and query bounds for canonical LCU/qubitization constructions.

We prove the generalized Lax conjecture for cubics in five variables: their hyperbolicity cones are spectrahedral. Our approach constructs spectrahedral realizations from completely positive maps on Clifford algebras. The construction passes through quaternionic determinantal representations of certain extremal cubics. Convexity of the set of Clifford-realizable cubics then yields the result for all cubics in five variables. The approach also covers all previously known cubic cases of the generalized Lax conjecture: forms in at most four variables and symmetric forms in arbitrarily many variables. We further establish a sufficient criterion for Clifford realizability in arbitrary dimension and use it to obtain a neighborhood of the origin in the cubic norm ball consisting of Clifford-realizable cubics. Finally, we prove spectrahedrality for sparse hyperbolic quartics in four variables.

We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra $C(K_{\mathcal A})$ of all complex-valued continuous functions on the Stone space $K_{\mathcal A}$ of a Boolean algebra $\mathcal A$ of cardinality $\omega_1$ is $*$-isomorphic to a masa of the Calkin algebra if and only if $\mathcal A$ does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain $\omega_2$ pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form $C(K)$ for any compact Hausdorff $K$ is $*$-isomorphic to a masa of the Calkin algebra, then no point of $K$ may have character smaller than $\mathfrak p$. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.

We study $\mathbb T^m$-invariant Toeplitz operators on weighted Bergman spaces over the unit ball. A natural unitary transformation represents each such operator as a direct sum of restrictions of Toeplitz operators acting on a polyball. Under an $L^\infty$-valued angular continuity assumption, the corresponding symbols extend from $\mathbb N_0^m$ to a norm-continuous family indexed by the maximal ideal space of the $C^*$-algebra generated by $k$-quasi-radial Toeplitz operators. This extension yields a localization framework for Toeplitz operator algebras. As an application, we consider commutative Banach algebras obtained by adjoining Toeplitz operators whose symbols are invariant under mixed unitary and circle actions. We identify their local quotient algebras on finite-coordinate strata and at infinity and use these identifications to describe their maximal ideal spaces and Gelfand transforms explicitly.

We construct a weakly quasilattice-ordered submonoid of a group whose full semigroup $\mathrm{C}^*$-algebra is nuclear, while the monoid does not embed into any amenable group. Hence, Nica amenability, as well as the nuclearity of its semigroup C*-algebra, do not characterize embeddability into an amenable group for group-embeddable monoids.

We study stationary states for actions of countable discrete groups on unital separable $C^*$-algebras. We prove that the stationary state space associated with an inner action of a subgroup of the unitary group that generates the algebra is a Choquet simplex. We also give a locality criterion covering Bernoulli shifts. The simplex structure yields a canonical decomposition of stationary states into tracial and purely nontracial parts. For inner actions, we characterize extreme stationary states by factoriality of their GNS von Neumann algebras. We further show that a stationary state is tracial if and only if its GNS von Neumann algebra is finite, and that it is purely nontracial if and only if this algebra is of type~$\mathrm{III}$. As applications, for $2\leq d\leq\infty$ the stationary state simplex of $C^*(\mathbb{F}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^*(\Gamma)$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_{\mu}(A)$ to be a Bauer simplex and construct a family $(A_d,\Gamma_d,\mu_d)$ satisfying this condition.

We investigate the operator system structure of finite Hankel matrices and its connection with complex polynomials on the real line. Using polynomial matrix spectral factorization, we establish a unital complete order isomorphism between the operator system of (n + 1) x (n + 1) Hankel matrices and the dual of the operator system of polynomials of degree at most 2n. We also show that the operator system of complex polynomials is minimal, while the Hankel operator system is maximal. As an application, we characterize positivity of block Hankel matrices with entries in a C*-algebra in terms of completely positive maps on the polynomial operator system.

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

In this paper, we investigate the generalization performance of distributed gradient descent algorithms in a reproducing kernel Hilbert space under a robust loss function $l_{\sigma}$. By exploiting the spectral characterization of gradient descent together with the intrinsic properties of robust loss functions, we establish optimal learning rates for the distributed kernel-based robust gradient descent (DKRGD) algorithm with an appropriately chosen scale parameter $\sigma$. The proposed parameter choice of $\sigma$ simultaneously alleviates the saturation phenomenon and guarantees statistical robustness. A key technical contribution is a novel error analysis that provides substantially sharper bounds for products of operators, thereby significantly relaxing existing restrictions on the maximum number of local machines while retaining optimal learning rates. Finally, we develop a communication-efficient strategy that further improves the convergence performance of DKRGD.

We solve two long-standing open problems in von Neumann algebras. First, we classify all flows with full Connes spectrum on the hyperfinite $\mathrm{II}_1$ factor up to cocycle conjugacy. Every such flow is cocycle conjugate to an irrational rotation flow on a noncommutative torus. In particular, there is a unique outer flow with full Connes spectrum up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite $\mathrm{II}_1$ factor. To prove this result, we draw on type $\mathrm{III}$ theory. Notably, we develop bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of the Connes–Størmer transitivity theorem and we generalize the Connes–Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of $\mathbb{R}$. We then go back to the type $\mathrm{III}$ world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type $\mathrm{III}_1$ factors.

We establish the existence of a three-parameter family of compact quantum metric space structures on cocycle twisted crossed products by discrete groups. We are mainly interested in the case where the acting group has exponential/subexponential growth. We prove that the family is jointly continuous with respect to the parameters when the acting group is exact. We obtain quantitative upper and lower bounds for the associated metric dimensions. In particular, the bounds are helpful to prove the failure of lower semicontinuity of the metric dimension with respect to the quantum Gromov-Hausdorff distance. We also prove invariance of metric dimension under zero quantum Gromov-Hausdorff distance.

We study locally finite-dimensional ideals in crossed products of totally disconnected spaces by free actions of the integers and in uniform Roe algebras of exact discrete groups. In the first case, we present a dynamical description of the largest locally finite-dimensional ideal, which turns out to coincide with the intersection of all maximal ideals. In the latter case, we provide a coarse geometric characterization of the locally finite-dimensional compact ideals. Moreover, we show that for crossed products of totally disconnected spaces by free actions of exact groups, the largest type I-ideal is locally finite-dimensional. In the case of uniform Roe algebras, we provide coarse geometric conditions for compact ideals guaranteeing that the ideal is type I and admits an embedding of a UHF-algebra, respectively.

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, $L(\Gamma)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.