Papers

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.

We introduce a topological abelian group $\mathrm{K}_1^\Delta(A)$ for every $\mathrm{C}^\ast$-algebra $A$, which is functorial in $A$ and invariant under stabilisations. $\mathrm{K}_1^\Delta$ can distinguish group actions indistinguishable by equivariant (K)K-theory and traces — we illustrate this on the irrational rotation algebra by exhibiting strongly outer actions of $\mathbb Z$ which, via $\mathrm{K}_1^\Delta$, are not (stably) cocycle conjugate. These actions are moreover trivial in $\mathrm{KK}^{\mathbb Z}$ and on the Hausdorffised unitary algebraic $\mathrm{K}_1$ group $\overline{\mathrm{K}}_1^{\mathrm{alg}}(A)$. The Hausdorffisation $\overline{\mathrm{K}}_1^{\Delta}(A)$ of $\mathrm{K}_1^\Delta(A)$ forms the final part of the total stable invariant $\mathrm{\underline{K}T}_{\mathrm{s}}(A)$, which we also introduce. We prove that $\mathrm{\underline{K}T}_{\mathrm{s}}$ classifies $*$-homomorphisms up to approximate unitary equivalence between certain stable $\mathrm{C}^\ast$-algebras which contain a non-zero projection. $\mathrm{K}_1^\Delta(A)$ is new even in the unital setting, but under further assumptions on $A$ recovers the unitary algebraic $\mathrm{K}_1$ group $\mathrm{K}_1^{\mathrm{alg}}(A)$, whereas the Hausdorffisations $\overline{\mathrm{K}}_1^{\Delta}(A)$ and $\overline{\mathrm{K}}_1^{\mathrm{alg}}(A)$ agree for every unital $A$.

We show that whenever $G$ is a finitely generated group with the rapid decay property (property RD), then the norm on the reduced group C*-algebra $C^*_r(G)$ is computable if (and only if) $G$ has solvable word problem. We also give an example of a finitely presented group $G$ with solvable word problem for which the norm on $C^*_r(G)$ is not computable.

Answering a question of Thomas Sinclair, we show that the standard presentation of the reduced group C*-algebra $C^*_r(G)$ of an infinite hyperbolic group $G$ is computable. The main ingredient in proving this result is showing that the Gromov boundary $\partial G$ of $G$ admits a computably compact metric presentation for which the natural action of $G$ on $\partial G$ is computable.

We construct linearly twisted spectral triples on $C^\ast$-crossed products $\mathfrak{A}\rtimes_\alpha\mathbb{Z}^d$, starting from a spectral triple $(\mathfrak{A},\mathcal{H},D)$ and an action $\alpha$ admitting an adapted weight. The weight compensates for the growth of the base Dirac operator under the dynamics while preserving the canonical differential structure in the $\mathbb{Z}^d$-directions. The construction takes place directly on the original crossed product and does not require the action to be equicontinuous. We prove that, whenever the base spectral triple is metric, the resulting twisted spectral triple is $\kappa$-metric for every $\kappa>0$: its associated Monge–Kantorovich metric metrizes the weak* topology on the state space of the crossed product. Our framework applies, in particular, to every action preserving the domain of the Lipschitz seminorm of the base triple. In this case, an exponential weight may be chosen so that the induced metric geometry restricts exactly to the original metric geometry on the base algebra. In the equicontinuous case, the weight may be chosen constant, the twist disappears, and the standard crossed-product spectral triple is recovered. We also establish a general finite-mode criterion for convergence in the spectral propinquity. As an application, we prove joint continuity, with respect to the deformation parameter and suitable fluctuations of the flat metric, for the twisted spectral triples associated with crossed products of quantum $2$-tori by fixed automorphisms in $\operatorname{SL}(2,\mathbb{Z})$, including Anosov automorphisms.

We extend the theory of Baer, Rickart, and their local, graded, and $*$-analogues for Steinberg algebras. First, we prove that positive definiteness passes from the coefficient field to the Steinberg algebra, and that this, together with graded von Neumann regularity, characterized by Steinberg and van Wyk, yields graded locally Rickart $*$-Steinberg algebras. Then we show that local Baer behavior forces every compact open subset of the unit space to be extremally disconnected. Consequently, when compact open subspaces of the unit space are metrizable, local Baer behavior forces the groupoid to be discrete. In the discrete case, we use the orbit-by-orbit decomposition into finitary matrix algebras over isotropy group algebras to convert annihilator conditions into explicit matrix conditions. For groupoids whose isotropy is trivial or infinite cyclic, these ingredients yield complete local and unital characterizations. In particular, local Baer and graded local Baer conditions are equivalent to discreteness, while the ungraded local Baer $*$-condition additionally requires that every orbit with nontrivial isotropy be a singleton. We then apply the theory to boundary path groupoids. Using the Steinberg algebra model, we obtain the Rickart, Baer, and Baer $*$ characterizations of Leavitt path algebras, including the local and graded variants.

V. Manuilov introduced the universal Roe algebra, namely the uniform Roe algebra associated with the largest uniformly locally finite coarse structure $\mathrm{Univ}$ on $\mathbb{N}$, together with several variants. We study these algebras through embeddability properties of $\mathrm{C}^*$-algebras. Among other results, we show that the universal Roe algebra admits an embedding from $\prod_n\mathbb{M}_n$, answering a question by Farah. Thus, Ozawa's non-embeddability theorem for $\prod_n\mathbb{M}_n$ in uniform Roe algebras of uniformly locally finite metric spaces does not extend to arbitrary uniformly locally finite coarse spaces. Nevertheless, embeddability properties distinguish the variants introduced by Manuilov and the quasi-local algebra of $\mathrm{Univ}$, apart from the multiplier algebra $M(\mathbb{I})$. As a by-product, we also obtain an operator-norm analogue of Birkhoff's Problem 111: the norm-closed convex hull of permutation operators is exactly the set of doubly substochastic operators. Moreover, we show that every separable $\mathrm{C}^*$-algebra embeds into the quasi-local algebra of $\mathrm{Univ}$. Although we do not know whether the analogous statement holds for the universal Roe algebra, we prove that every separable quasidiagonal $\mathrm{C}^*$-algebra embeds into it, and hence into the reduced $\mathrm{C}^*$-algebra of some étale locally compact second countable groupoid.

We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.

We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A, B$ on a real symplectic Hilbert space $(\mathbf{H}, J)$. Under a natural commutativity condition, there exists a (generally unbounded) operator $\Phi$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A = \Phi^t M_A \Phi$, $B = \Phi^t M_B \Phi$, and maps $J$ to a canonical symplectic operator. The result generalizes Williamson's normal form to infinite dimensions and yields a complete spectral invariant for the pair $(A,B)$.

We introduce the notion of a Cartan inverse semigroup and an (algebraic) semi-Cartan subalgebra in an $R$-algebra A, providing a purely algebraic analogue of Cartan semigroups and semi-Cartan subalgebras in a C*-algebra. This extends the theory of quasi-Cartan subalgebras, shifting the focus from designated subalgebras to designated subsemigroups. Our main result characterises twisted Steinberg algebras in terms of the existence of a Cartan inverse semigroup: we show that an $R$-algebra admits a Cartan inverse semigroup if and only if it is isomorphic to a twisted Steinberg algebra. Moreover, when our construction is applied to a twisted Steinberg algebra over an ample Hausdorff groupoid, the constructed twist is naturally isomorphic to the original one.

We introduce and study multi-indexed Schatten spaces which lift the commutative $\ell_{q_1}\![\ell_{q_2}\![\ell_{q_3}\![\dots]]]$ -spaces to the quantum toolbox. They extend Pisier's operator-valued Schatten norms and the recently introduced two-indexed Schatten quasi-norms. For every ordered tuple $\mathcal{Q}=(q_1,\dots,q_n)$ of non-zero indices we define a quasi-norm (for positive indices) or an anti-norm (for negative indices) on operators on an $n$-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition $|\frac{1}{q_i}-\frac{1}{q_j}|\leq1$. Beyond establishing desirable properties of these quasi- and anti-norms via factorization formulas that combine operator space and matrix-analytic techniques, we lift several central tools and results of the seminal [Devetak, Junge, King, Ruskai, CMP 2006] to the multi-indexed and quasi-normed setting. These include a general non-commutative Minkowski inequality relating swapped systems and two identity-removal equalities. As a consequence, complete hypercontractivity and complete reverse hypercontractivity tensorize for all compatible indices, including negative ones. Concerning quantum information theory we prove that the completely bounded minimal and the maximal output conditional entropies are additive under tensor products of quantum channels for all $\alpha\geq\frac12$. This extends the case $\alpha>1$ of [Fawzi, Kochanowski, Rouzé, Van Himbeeck, CMP 2026] and the unconditional results of [Devetak, Junge, King, Ruskai, CMP 2006]. Beyond tensor products, we establish chain rules for $\tilde H^\uparrow_\alpha$, $\alpha\in[\frac12,1)$, for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval $\alpha\in[1/2,1)$.

We study the QWEP property of twisted crossed products and its relation to Fourier multiplier transference on noncommutative $L_p$-spaces. We show that QWEP is not preserved by crossed products of QWEP von Neumann algebras by hyperlinear groups. Our examples include trace-preserving actions of a torsion-free residually finite Kazhdan group on a diffuse abelian algebra and on the hyperfinite type II$_1$ factor. We also obtain non-QWEP crossed products with nontrivial scalar cocycles. For scalar twists, we prove that a finite trace-preserving crossed product is QWEP if and only if its ordinary crossed product and its twisted group von Neumann algebra are QWEP. In the positive direction, we establish a restriction theorem for scalar twists of regular wreath products. These results provide QWEP inputs for Fourier multiplier transference and identify a completely bounded multiplier norm obstruction to QWEP. Finally, operator-valued cocycles give finite QWEP crossed products whose acting groups are not hyperlinear.

Let $(\mathcal{M},\tau)$ be a semi-finite von Neumann algebra. We prove that $L_1(\mathcal{M})$ has Talagrand type $(1,\psi_{1,1})$, where $\psi_{1,1}(t)=t/\log(e+t)$. As an consequence, the Schatten trace class $S_1$ has Talagrand type $(1,\psi_{1,1})$, thus settled an open problem of Question 2 in Cordero-Erausquin and Eskenazis (2023). Our proof is based on the noncommutative Mazur maps initiated by Ricard.

We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $\Phi$, then, at any point of its range, the Levi form of $\Phi$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.

We establish a weak type $(1,1)$ estimate for the Bochner–Riesz mean operator at the critical index $\delta=\frac{d-1}{2}$ on the quantum torus $\mathbb{T}^d_\theta$, where the associated kernel is rough.

We show that $\mathbb{Z}_2$ is a quantum normal subgroup of $SU_q(2)\times SU_q(2)$ and that the quotient group is isomorphic to $SO_q(4)$, thereby establishing that $SU_q(2)\times SU_q(2)$ is a double cover of $SO_q(4)$.

We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra–Itzykson–Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices $A,B$ with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for $(BA)^*(BA)\otimes\mathbf{1}_N$ has two limiting descriptions at speed $N^2$ for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for $(A^*A)\otimes(B^*B)$. For every fixed number of factors $k\geq3$, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number $k$ of factors, global Haar vectors at speed $N^k$ and product-state vectors at speed $N$ yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the $S$-transform.

The space of $n\times n$ density matrices is the full state space of an $n$-level quantum system. While it has been asserted in the literature on quantum information theory that decomposing this state space into equirank pieces yields a Whitney stratification, rigorous proofs have remained absent. Using methods from invariant theory, this work provides a proof that the rank decomposition of the matrix algebra constitutes a (b)-regular stratification in the sense of Whitney. Furthermore, we demonstrate how the Uhlmann connection - which describes the adiabatic holonomy of mixed states and reduces to the Berry connection on pure states - can be geometrically treated as a stratified connection on a stratified vector bundle. Finally, our appendix provides a self-contained exposition of stratified fiber bundles and the new concept of stratified connections, which may be of independent interest.

Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$-subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus–Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.

We show that the canonical embeddings of the Jiang–Su algebra $\mathcal Z$ into the universal unital free product $\mathcal Z \ast_{\mathbb C}\mathcal Z$ are not approximately unitarily equivalent. This answers one of the $99$ problems in the list of Schafhauser, Tikuisis, and White. By contrast, we prove that any two such embeddings into a simple, unital, monotracial $\mathrm{C}^\ast$-algebra with strict comparison by the unique trace are strongly asymptotically unitarily equivalent.

We use the notion and properties of topological graphs to construct a quantum CW complex for quantum real projective spaces. This construction mirrors the CW complex construction in the classical case. Moreover, we introduce multipullback even-spheres as boundaries of noncommutative "cylinders" and show that they admit a gluing akin to the gluing of balls to the ends of a hollow closed cylinder.

Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $\Gamma$-independent semicircular elements. This characterizes their limiting spectral distribution and yields improved bounds on their spectral norm, a key quantity in the analysis of algorithms for Tensor PCA. Finally, we show that, in an appropriate double limit, independent Kikuchi matrices converge to the $q$-Gaussian system, another central object in noncommutative probability.

We develop a method for proving matrix concentration inequalities by identifying random matrices with associated deterministic operators acting on suitable Fock spaces and controlling norms of these operators and their restrictions to low-order subspaces. Applying this method, we obtain new strengthenings of the non-commutative Khintchine inequality that improve on the state of the art, in particular sharpening recent inequalities due to Bandeira, Boedihardjo, and van Handel (2023) quantifying intrinsic freeness of random matrices. Our proofs of these results are based on relatively simple operator algebra arguments and involve neither Gaussian interpolation nor explicit combinatorics of tracial moments. Further, our techniques apply equally well to several models of non-commutative random variables of interest in the literature, such as operator series constructed from $q$-Gaussian and $\Gamma$-independent systems of operators, treating all of these objects with the same method. We obtain new norm bounds for such operators both in the style of the non-commutative Khintchine inequality and in the style of Lehner's operator norm formula.

We compute the Cuntz semigroup of a unital graph C*-algebra by combining Hong and Szymański's realisation of the entire ideal lattice of a graph C*-algebra, Ara, Moreno and Pardo's computation of the Murray-von Neumann semigroup of a unital graph C*-algebra and various Cuntz semigroup techniques. We also investigate certain properties of such Cuntz semigroups, and demonstrate that at the level of the Cuntz semigroup, there is no obstruction to the conjecture that the nuclear dimension of any graph C*-algebra is at most one.

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic \(h\)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

These are the slightly edited notes from a 10-hour course I taught at Institut Henri Poincaré in January-February 2025 in the framework of the thematic semester "Representation theory and noncommutative geometry".

We classify the set of finite simple undirected graphs whose corresponding graph operator system is C*-nuclear. We prove that the operator system associated to the graph is C*-nuclear if and only if the graph is 2-chordal, as defined by McCullough. In particular, we present the first known example of a non-chordal graph whose graph operator system is C*-nuclear, as well as the first known examples of graph operator systems which are not C*-nuclear.

We construct a groupoid model for Rordam's example of a simple, nuclear, separable $C^*$-algebra containing a finite and an infinite projection. This groupoid model arises from a (crossed product of a) carefully chosen model for the inductive limit construction in Rordam's original approach, which we show to yield a Cartan subalgebra of the limit. In particular, this procedure yields a locally compact, second countable, Hausdorff, étale, amenable, minimal and topologically principal groupoid whose unit space is compact, and whose $C^*$-algebra contains an infinite and a non-zero finite projection. Hence, this groupoid is amenable, and yet does not have comparison.

We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+\omega+\omega^*+M\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $\omega,u_1,\ldots,u_{15}$ are specified words, $*$ reverses words, and $M$ is a specified positive integer. Its normalized trace is at least $3/4$ on selfadjoint matrix contraction tuples of every dimension, but equals $-1$ at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus $f$ is a counterexample to the algebraic formulation of Connes' embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in $\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}$ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.

We show that free groups admit amenable actions on unital simple AF-algebras. This settles, for free groups, the last remaining case of the existence problem for amenable actions of non-amenable groups on classifiable simple C*-algebras. The proof relies on the Baire category theorem. This result was obtained with substantial assistance from GPT-6 Astra.