Papers

We prove that $M_n(\mathbb{C})$ contains no abelian Kadison–Singer algebra for any $n \geq 3$.

A Hermitian completely bounded map $\Phi:\mathscr{A}\to B(\mathcal{H})$ need not have a canonical completely positive analogue of an absolute value. We study this problem through the ordered set $\mathsf{E}(\Phi)$ of completely positive symmetric majorants of $\Phi$, which we call Wittstock envelopes. We show that every Wittstock envelope canonically determines a minimal Wittstock envelope below it. However, minimal envelopes need not be unique. We therefore study the stronger question of when a least completely positive majorant exists. We obtain a criterion, in terms of central supports of spectral projections, for the existence of least majorants below a fixed envelope. In finite dimensions, the Choi representation gives a simple characterization: a least completely positive majorant exists if and only if \(\Phi\) or \(-\Phi\) is completely positive. Finally, we use the envelope-relative modulus to define envelope-relative perspectives for Hermitian completely bounded maps. These constructions extend Okayasu's completely positive map perspectives, reduce to the usual perspective in the completely positive case, and remain dependent in general on the chosen Wittstock envelopes.

Let $A^2(\mathbb D)$ be the unweighted Bergman space and write $Q_m(S)=T_{B_m(S)}$ for the map induced by the $m$th higher-order Berezin transform. Suárez asked in 2005 whether $Q_m(S)$ converges to $S$ in operator norm for every $S$ in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator $S$ such that \[ \sup_{m\geq 0}\|Q_m(S)\|=\infty. \] In particular, along a strictly increasing sequence $(m_n)$ one has \[ \|Q_{m_n}(S)-S\|\longrightarrow\infty. \] The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the $m$th matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal–Hankel theorem showing that the associated weighted Hankel transformation fails to map $\ell^2$ boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.

We characterize the von Neumann algebras that are isomorphic, as $C^*$-algebras, to reduced twisted $C^*$-algebras of locally compact Hausdorff groupoids equipped with continuous Haar systems of full support. They are exactly the subhomogeneous von Neumann algebras, equivalently finite products of matrix algebras over abelian von Neumann algebras. The groupoid can always be chosen compact, principal, and étale, with trivial twist and counting Haar system. We also prove that, for a groupoid in this class, the full or reduced twisted algebra is unital if and only if the groupoid is étale with compact unit space. This criterion reduces the classification for general Haar systems to the étale case. A further obstruction comes from controlled propagation, defined through faithful representations into the norm closure of uniformly sparse matrices. Every reduced twisted étale groupoid algebra and its Borel completion have controlled propagation, including for non-Hausdorff groupoids with locally compact Hausdorff unit space. For every infinite-dimensional Hilbert space $H$, every $*$-homomorphism from a nonzero quotient of $B(H)$ into an algebra with controlled propagation is zero. In particular, neither $B(H)$ nor the Calkin algebra embeds into any of these reduced or Borel groupoid algebras. We also prove that every von Neumann algebra with controlled propagation is finite. No separability or countability assumptions are required.

We develop a time–frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

We exhibit nonamenable groups whose reduced group C*-algebras are not pure. More precisely, if $\Gamma$ is any countably infinite discrete group, then the reduced group C*-algebra of the restricted wreath product $(\mathbb{Z}/2\mathbb{Z}) \wr \Gamma$ has an ideal-quotient isomorphic to $\mathcal{K}(\ell^2(\Gamma))$. It therefore fails to be nowhere scattered and, in particular, is not pure. Taking $\Gamma$ to be any nonamenable group, we thereby obtain a negative answer to a question of Thiel.

Let $\mathcal{H}$ be a complex Hilbert space. We characterize the bijective linear maps $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ that preserve the spectral radius, respectively the numerical radius, on pairs of unitarily similar operators. We also characterize bijective linear maps that transform unitary similarity into similarity. Our results cover both finite and infinite-dimensional Hilbert spaces, with no separability assumption in the latter case.

We establish local Ocneanu rigidity for separable algebra objects in Hom-finite monoidal categories C over an algebraically closed field k. A separability morphism contracts the first two Hochschild cohomology groups without requiring an abelian ambient category. Separable algebra structures and homomorphisms from separable sources have open algebraic-group orbits, and affine Bezout estimates give effective finiteness results. For Frobenius subalgebras, exchange relations replace the source and embedding data by a single self-dual idempotent. We prove that their separable inner-conjugacy classes are open in the exchange locus, yielding a bound of 2^(dim_k(End_C(X))) for any ambient Frobenius algebra X; connectedness gives the same bound on the actual number of subalgebras. As an application, we obtain an effective form of the Etingof-Walton finiteness theorem: every finite-dimensional semisimple Hopf algebra H over C has fewer than 2^(dim_C(H)) left coideal subalgebras. In the unitary setting, we bound E-compatible intermediates of C*-algebra and von Neumann algebra inclusions up to unitary conjugacy, under finite-index and finite-center hypotheses. For irreducible finite-index subfactors N < M, we improve the 9^[M:N] bound of Bakshi-Das-Liu-Ren to fewer than 2^[M:N] intermediate subfactors.

We prove the sharp Hausdorff–Young inequality on the quantum Euclidean space. Our result implies the sharp Hausdorff–Young constants for the Weyl transform, as well as that for Heisenberg groups. The key ingredient is a novel flow related to the mixed-norm of noncommutative Gabor transform. This, meanwhile, implies a new proof of the classical sharp Hausdorff–Young inequality. We then apply the sharp Hausdorff–Young inequality to establish the sharp Young inequality for noncommutative convolution in the range $1\le p,q\le2\le r\le\infty$, with $1/p+1/q=1+1/r$. After appropriate rescaling and trace normalization, this convolution coincides with beam-splitter convolution for bosonic systems, yielding the corresponding Young inequalities with optimal constants in the same range.

The Motzkin subproduct system (SPS) is constructed from the Jones Wenzl idempotents of the Motzkin algebras, which generalizes the Temperley Lieb SPS. The simplest Motzkin SPS, which is not a Temperley Lieb SPS, is constructed from a 3 dimensional Hilbert space. We explicitly describe the spectrum of the corresponding Cuntz Pimsner algebra, which remarkably admits only irreducible representations of dimensions 1 and 2, and can thus be viewed as a mildly quantum space. We moreover analyse representations of the Richard Thompson groups and of the Cuntz algebra that are associated to this spectrum.

Using set-theoretic methods, we prove that every $\mathrm{AW}^*$-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of $\mathrm{AW}^*$-factors, by work of Saitô and Wright. We also show that if there is a model $V$ of $\mathrm{ZFC}$ with an $\mathrm{AW}^*$-algebra that fails to be monotone complete, then in some forcing extension of $V$ there is an $\mathrm{AW}^*$-factor that fails to be monotone complete, implying that any $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-factors are monotone complete yields a $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-algebras are monotone complete. Both results build on transfer principles for Boolean-valued factor representations originally developed by Ozawa. This work was assisted by the Danus LLM orchestration system.

We study three naturally associated notions of regularity for a unital inclusion of von Neumann algebras $\mathcal{B} \subseteq \mathcal{M}$: regularity, groupoid regularity, and unitary regularity. We first observe that regularity and groupoid regularity are unconditionally equivalent by showing that the standard normaliser and the groupoid normaliser of $\mathcal{B}$ in $\mathcal M$ generate the same linear span. The central question addressed in this work is whether regularity implies unitary regularity. For arbitrary von Neumann subalgebras, we demonstrate that this implication fails dramatically; we construct explicit examples to show that every factor (of type $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, and $\mathrm{III}$) contains a regular von Neumann subalgebra that is not unitarily regular. In stark contrast, we prove that the behaviour changes completely in the setting of subfactors. For subfactor inclusions across all these types, regularity always implies unitary regularity, and thus all three notions of regularity coincide. Finally, we establish an analogous result for irreducible unital inclusions of simple $C^*$-algebras admitting a conditional expectation of finite Watatani index. In this setting, we prove that regularity, unitary regularity, being a generalised Cartan subalgebra, and arising from a crossed product by a finite group are all mutually equivalent conditions.

In a tracial von Neumann algebra M, two subsets X and Y are said to be N-independent if alternating products of length at most N of trace zero words from X and Y have trace zero. We show that two approximately 2N-independent finite tuples can be perturbed, in operator norm, to be exactly N-independent. Consequently, we are able to generalize Theorem 5.1 and Theorem H of Houdayer-Ioana and resolve Problem 31 of Kunnawalkam Elayavalli in the positive.

The celebrated theorems of Shilov, Arens–Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The étale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

We introduce a left $G$-circulant decomposition for matrices indexed by an arbitrary finite group $G$, extending the diagonal decomposition associated with cyclic groups. We show that, when $A\in M_{|G|}(\mathcal A)$ is free from $M_{|G|}(\mathbb C)$, the components arising from the left $G$-circulant decomposition of $A^t$ form a free family, with the components associated with inverse pairs forming $R$-diagonal pairs. We also describe the distributions of these components in terms of the distribution of $A$. Our results recover the cyclic case and show that different group structures of the same order may lead to different free decompositions of the same matrix.

We establish a pullback theorem for \(C^*\)-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid \(C^*\)-algebras form a pullback diagram in the category of \(\mathbb T\)-\(C^*\)-algebras and \(\mathbb T\)-equivariant \(*\)-homomorphisms, for the gauge actions induced by a \(\mathbb Z\)-valued cocycle and its restrictions. We then prove a collection of boundary-path decomposition theorems for graphs, relative graphs, and topological graphs. We show that admissible decompositions of graphs, together with their analogues in the relative and topological settings, induce corresponding decompositions of boundary path spaces. Combining these decomposition theorems with the groupoid pullback theorem, we recover previously known pullback theorems for graph \(C^*\)-algebras, relative graph \(C^*\)-algebras, and topological graph \(C^*\)-algebras. Thus these pullback phenomena are explained by a single groupoid-theoretic mechanism.

This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman–Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov–Sukochev–Usachev–Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.

We exhibit a simple unital infinite-dimensional C$^*$-algebra $A$ which is not $K_1$-injective, i.e., the canonical map $ U(A)/U_0(A)\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator $u$ of $\pi_4(U(2))\cong\mathbb{Z}/2$, and we use exactly this example as the seed of our construction. The construction proceeds by developing a higher-rank matrix trapping technique in homogeneous algebras generalizing the machinery of Villadsen in higher stable rank simple C$^*$-algebras. A hypothetical homotopy from the homomorphic image of $u$ to the identity in a certain homogeneous algebra is shown, through matrix trapping, to yield a spin bordism from the class of $u$ to that of the identity matrix $\mathrm{1}_2 \in M_2$, and this, in turn, is obstructed by the fact that the bordism class of $u$ is not trivial. We then fit our new arguments into a generalized version of Villadsen's higher stable rank construction to arrive at a simple example.

The twisted $C^*$-tensor product was exhaustively investigated by the authors in two previous papers. In this new context, generalising the usual tensor product and the Fermi one, we analyse the possibility of studying the set of symmetric states, that is those invariant under all finite permutations, in the setting of infinite twisted $C^*$-tensor products. As a preliminary result, we recognise that such an investigation can proceed only when the bicharacter, involved in the construction of such twisted tensor products, is hermitian. Otherwise, there is no natural action of the finitary symmetric group on the infinite twisted chain. Furthermore, even if the investigation of symmetric states is certainly meaningful for all twisted systems based on hermitian bicharacters, it is shown that it can be fruitfully carried out in three cases only. In these cases, we can provide the "genuine" version of the celebrated De Finetti Theorem already established by Hewitt and Savage for the general classical case, Stormer for the usual tensor product and Fidaleo for Fermi models. Very surprisingly, it emerges that one more twisted model can be treated exhaustively: it corresponds to the chain twisted by the so-called Klein four-group, and its Klein bicharacter unique up to equivalence.

We extend the recent results of Arnold on estimates for Kreiss positive semigroups on $L_p$-spaces to the non commutative setting, $1<p<\infty$. We also provide a sharp example. All the results are achieved thanks to the use of the Schur test, including a new version for von Neumann algebras.

Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras ($\mathbb Z_2$-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, i.e., it is stably bounded-spread isomorphic to a tensor product fermionic algebra.

We construct an MF C*-algebra A such that the spatial tensor product of A and the hyperfinite II_1 factor R contains a proper isometry. Consequently, stable finiteness of C*-algebras is not stable under tensor product and the hyperfinite II_1 factor is not a quasidiagonal C*-algebra. The C*-algebra A is non-simple and has only non-faithful tracial states. This result was obtained using OpenAI's Chat GPT Pro 6.0.

Let $(M,\tau)$ be a semifinite von Neumann algebra, let $J$ be a trace-preserving Jordan isomorphism, and let $(n_k)_{k\geq 1}$ be a random increasing sequence of integers obtained by selecting each integer $n\geq 1$ independently with probability $n^{-\alpha}$, where $0<\alpha<\frac12$. We show that, almost surely, for every $x\in L^1(M)$, $ \frac1m\sum_{k=1}^m J^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire's classical random $L^1$ ergodic theorem to the non-commutative setting.

We establish that a JB$^*$-triple $E$ is commutative if and only if its numerical index $n(E)$ is one. As a consequence, $n(E) = 1$ implies $n(E^*) = n(E^{**}) = 1$. This settles a question left open in the literature since 2008.

The purpose of this paper is to classify all the Rota-Baxter operators explicitly on Grassmann algebra in dimension four over the reals.

We show that the \(q\)-Gaussian von Neumann algebras associated with an infinite-dimensional real Hilbert space are pairwise nonisomorphic as \(q\) ranges over \((-1,1)\).

We study the operator Michelson contrast Delta(A) = (kappa(A)-1)/(kappa(A)+1), with kappa(A)=||A|| ||A^{-1}||, for positive invertible operators, and introduce its logarithmic parametrization via arctanh(Delta(A)) = (1/2) ln kappa(A). This defines the Operator Logarithmic Contrast (OLC), extended to all invertible operators via polar decomposition. We prove OLC is subadditive under multiplication: OLC(AB) <= OLC(A)+OLC(B). For positive invertible A, we identify OLC(A) exactly as the projective Thompson distance from the identity ray to the ray of A. We establish a sharp dimensional threshold for equality in subadditivity: for dimensions 2 and 3, equality forces commutativity, while n=4 is the smallest dimension admitting noncommuting equality examples. Moreover, equality still forces commutativity in any total dimension if the operators factor as pure Kronecker products X = tensor X_i and Y = tensor Y_i with each factor dimension <= 3. We derive contraction inequalities, sum bounds, trace estimates for density operators, Lipschitz continuity (including singular limits), and limiting behavior for Cesaro means and infinite products. For density operators, we obtain an exact closed-form bijection between OLC and von Neumann entropy for qubits; prove this bijection does not exist for d >= 3; and establish upper/lower entropy envelopes at fixed condition number, showing the admissible range [S_min(kappa), S_max(kappa)] satisfies ln 2 <= S_max(kappa) < ln 3 for every kappa > 1. The qubit case recovers, as an operator lift, the classical Michelson-Jensen-Shannon equivalence of Bruni, Rossi, and Vitulano, while a counterexample shows this equivalence fails for d > 2 as a direct consequence of the non-bijectivity.

This is an advanced introduction to the various types of normal random variables, with most of the needed preliminaries included. We first discuss the probability basics, standard central limits, and the theory of the usual, real normal variables, with examples, illustrations and numerous formulae. Then we go on a similar discussion regarding the complex normal variables, and the Rayleigh variables too. We then move to arbitrary dimensions, with a discussion regarding the Gaussian vectors, and related probability laws, featuring some functional analysis, and geometry and physics too. Finally, we provide an introduction to the quantum versions of the normal variables, and notably to those coming from free probability and random matrices.

For an action of an inverse semigroup $S$ on a locally compact Hausdorff space $X$, we construct an étale groupoid $S\ltimes X^\mathrm{KS}$, called the KS-groupoid associated with the action. This construction may be viewed as a dynamical analogue of the construction of universal groupoids for inverse semigroups. We provide a dynamical version of Paterson's theorem which states that the full and reduced crossed products (in the sense of Khoshkam and Skandalis) of $C_0(X)$ are canonically isomorphic to the full and reduced groupoid C*-algebras of $S\ltimes X^\mathrm{KS}$, respectively. We further investigate some topological properties, functoriality and universality of KS-groupoids.

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti–Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.