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When a compact quantum group $H$ coacts freely on unital $C^*$-algebras $A$ and $B$, the existence of equivariant maps $A \to B$ may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk-Ulam conjectures of Baum-Dabrowski-Hajac. Among our results, we find that for certain finite-dimensional $H$, there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of $H$. This claim is in stark contrast to the case when $H$ is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of $H$ to be cleft as comodules over the Hopf algebra associated to $H$. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a $\theta$-deformation procedure., Comment: 27 pages. To appear in Journal of Functional Analysis

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple ( A , H , D ) where D is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on domains in R d , d ≥ 2 .

We prove that any weak* continuous semigroup ( T t ) t ⩾ 0 of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ⁎-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative L p -spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's H ∞ functional calculus of the generators of these semigroups on the associated noncommutative L p -spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of R n .

Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B ( l p ) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space l p of all p-summable sequences. The l p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among l p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B ( l 2 ) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.

In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener–Wintner type ergodic theorem for dynamical systems not necessarily commutative. More precisely, we introduce a weight class D , which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set Λ d = { { λ 1 k 1 ⋯ λ d k d } ( k 1 , … , k d ) ∈ N d : ( λ 1 , … , λ d ) ∈ T d } ; then we prove a multi-parameter Bellow and Losert's Wiener–Wintner type ergodic theorem for the class D and for a noncommutative trace preserving dynamical system ( M , τ , T ) , M being a von Neumann algebra. Restricted to Λ d , we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener–Wintner ergodic theorem. The “noncommutativity” and the “multi-parameter” characters induce some difficulties in the proofs. For instance, our argument of proving the uniform convergence for a dense subset turns out to be quite different from the classical case since the “pointwise” argument does not work in the noncommutative setting; also to obtain the uniform convergence in the largest spaces, we need a maximal inequality between the Orlicz spaces, but it cannot be deduced by using classical extrapolation argument directly. Junge and Xu's noncommutative maximal inequalities with the optimal order, together with the atomic decomposition of Orlicz spaces, play the essential role in overcoming the second difficulty.

We will extend earlier transference results due to Neuwirth and Ricard from the context of noncommutative L p -spaces associated with amenable groups to that of noncommutative L p -spaces associated with crossed-products of amenable actions. Namely, if m : G → C is a completely bounded Fourier multiplier on L p , then it extends to the crossed-product with similar bounds provided that the action θ is amenable and trace-preserving. Furthermore, our construction also allows to extend G-equivariant completely bounded operators acting on the space part to the crossed-product provided that the generalized Folner sets of the action θ satisfy certain accretivity property. As a corollary we obtain stability results for maximal L p -bounds over crossed products. We derive, using that stability results, an application to the boundedness of smooth multipliers in the L p -spaces of group algebras.

In this paper, we investigate the boundedness of a square function operator from ergodic theory acting on noncommutative L p -spaces. The main result is a weak type ( 1 , 1 ) estimate of this operator. We also show the ( L ∞ , BMO ) estimate, and thus all the strong type ( L p , L p ) estimates by interpolation. The main new difficulty lies in the fact that the kernel of this square function operator does not enjoy any regularity, while the Lipschitz regularity assumption is crucial in showing such endpoint estimates for the noncommutative Calderon-Zygmund singular integrals.

Originally studied by Gohberg and Krein, the block projection operators admit a natural extension to the setting of quasi-normed ideals and noncommutative integration. Here, we establish several uniform submajorization inequalities for block projection operators. We also show that in the quasi-normed setting, for L p -spaces with 0 p ≤ 1 , a reverse inequality holds.

The goal of the present paper is to provide analogues of Sarason interpolation theorem in the Hardy algebra H ∞ ( D ) and Sz.Nagy-Foias commutant lifting theorem for contractions on Hilbert spaces in the setting of noncommutative Hardy spaces associated with noncommutative regular domains and varieties. This is accompanied by the study of multi-analytic operators with respect to the universal models associated with the regular domains (resp. varieties) and the study of multipliers of noncommutative Bergman spaces. As applications, we obtain Toeplitz-corona theorems for multi-analytic operators, commutant lifting in several variables where the liftings are in certain Schur classes, factorization of multi-analytic operators, and Nevanlinna-Pick interpolation results for multipliers of Bergman spaces over Reinhardt domains in C n . In addition, we obtain Ando type dilations and inequalities on noncommutative bi-domains and varieties.

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution ⋄ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if F j , G j ( 1 ≤ j ≤ N ) are polynomials of z and z ¯ then ∑ j = 1 N T F j T G j − T H is a finite rank operator for some L 1 -function H if and only if ∑ j = 1 N F j ⋄ G j belongs to L 1 and H = ∑ j = 1 N F j ⋄ G j . In the case F j 's are holomorphic and G j 's are conjugate holomorphic, it is shown that H is a solution to a system of first order partial differential equations with a constraint.

We prove a version of Blum–Hanson ergodic theorem for ( L 1 , L ∞ ) -contractions acting in noncommutative symmetric spaces with order continuous norm. A similar result is established for arbitrary contractions on Hardy subspaces of p-convex Schatten ideals of compact operators.

In this paper we study Johnson–Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space E ( M ) of a finite von Neumann algebra M without making any assumption on the Boyd indices of E . We show that we can obtain Johnson–Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when E = L 2 ( 0 , 1 ) . As an application, we partly resolve a problem stated by Randrianantoanina and Wu in [46] . We also show that we can obtain sharp Φ-moment analogues for Orlicz functions satisfying p -convexity and q -concavity for 1 ≤ p ≤ 2 , q = 2 and p = 2 , 2 q ∞ . This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder–Gundy inequalities in symmetric spaces and in the Φ-moment case.

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, free probability, and optimization theory in linear systems engineering). The purpose of this article is to develop a theory of positive kernels and associated reproducing kernel Hilbert spaces for the setting of free noncommutative function theory., Comment: 71 pages

We extend the Revesz and Komlos theorems to arbitrary finite von Neumann algebras, and in doing so solve an open problem of Randrianantoanina, removing the need for hyperfiniteness. The main result is the noncommutative Komlos theorem, which states that every norm-bounded sequence of operators in L 1 ( M ) , for any finite von Neumann algebra M , admits a subsequence, such that for any further subsequence, the Cesaro averages converge bilaterally almost uniformly. This is a natural extension of Komlos' original result to the noncommutative setting. The necessary techniques which facilitate the proof also allow us to extend the Revesz theorem to the noncommutative setting, which gives a similar subsequential law for series over bounded sequences in L 2 ( M ) .

We study states on the universal noncommutative ⁎-algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. Finally, we study a particular class of free Levy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity.

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's $\lambda$-lattices, modular tensor categories etc., Comment: 41 pages, 71 figures

We provide generalizations of Burkholder's inequalities involving conditioned square functions of martingales to the general context of martingales in noncommutative symmetric spaces. More precisely, we prove that Burkholder's inequalities are valid for any martingale in noncommutative space constructed from a symmetric space defined on the interval ( 0 , ∞ ) with the Fatou property and whose Boyd indices are strictly between 1 and 2. This answers positively a question raised by Jiao and may be viewed as a conditioned version of similar inequalities for square functions of noncommutative martingales. Using duality, we also recover the previously known case where the Boyd indices are finite and are strictly larger than 2.

Let M be a type II1 von Neumann factor and let S ( M ) be the associated Murray-von Neumann algebra of all measurable operators affiliated to M . We extend a result of Kadison and Liu [30] by showing that any derivation from S ( M ) into an M -bimodule B ⊊ S ( M ) is trivial. In the special case, when M is the hyperfinite type II1-factor R , we introduce the algebra A D ( R ) , a noncommutative analogue of the algebra of all almost everywhere approximately differentiable functions on [ 0 , 1 ] and show that it is a proper subalgebra of S ( R ) . This algebra is strictly larger than the corresponding ring of continuous geometry introduced by von Neumann. Further, we establish that the classical approximate derivative on (classes of) Lebesgue measurable functions on [ 0 , 1 ] admits an extension to a derivation from A D ( R ) into S ( R ) , which fails to be spatial. Finally, we show that for a Cartan masa A in a hyperfinite II1-factor R there exists a derivation δ from A into S ( A ) which does not admit an extension up to a derivation from R to S ( R ) .

Given a noncommutative (nc) variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we consider the algebra $H^\infty(\mathfrak{V})$ of bounded nc holomorphic functions on $\mathfrak{V}$. We investigate the problem of when two algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are isomorphic. We prove that these algebras are weak-$*$ continuously isomorphic if and only if there is an nc biholomorphism $G : \widetilde{\mathfrak{W}} \to \widetilde{\mathfrak{V}}$ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form $f \mapsto f \circ G$, where $G$ is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras $H^\infty(\mathfrak{B}_d)$ studied by Davidson--Pitts and by Popescu. In particular, we find that $\operatorname{Aut}(H^\infty(\mathfrak{B}_d))$ is a proper subgroup of $\operatorname{Aut}(\widetilde{\mathfrak{B}}_d)$. When $d, Comment: 45 pages. Some details were added and more minor changes

Let $\mathcal{E}$ and $\mathcal{F}$ be symmetrically $\Delta$-normed (in particular, quasi-normed) operator spaces affiliated with semifinite von Neumann algebras $\mathcal{M}_1$ and $\mathcal{M}_2$, respectively. We establish a noncommutative version of Abramovich's theorem \cite{A1983}, which provides the general form of normal order-preserving linear operators $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$ having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem \cite{Huijsmans_W}. By establishing the disjointness preserving property for an order-preserving isometry $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$, we obtain the existence of a Jordan $*$-monomorphism from $\mathcal{M}_1$ into $\mathcal{M}_2$ and the general form of this isometry, which extends and complements a number of existing results. In particular, we fully resolve the case when $\mathcal{F}$ is the predual of $\mathcal{M}_2$ and other untreated cases in [Sukochev-Veksler, IEOT, 2018]., Comment: to appear in JFA

We prove the existence of a 1 / N 2 expansion in unitary multimatrix models which are Gibbs perturbations of the Haar measure, and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem. The recursion obtained is closely related to the “topological recursion” which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems. Our approach consists of two main ingredients: an asymptotic study of the Schwinger–Dyson lattice over noncommutative Laurent polynomials, and uniform control on the cumulants of Gibbs measures on product unitary groups. The required cumulant bounds are obtained by concentration of measure arguments and change of variables techniques.

In this paper, we prove the coarse Novikov conjecture for metric spaces with bounded geometry which are coarsely embeddable into Banach spaces with a geometric condition, called Property (H), introduced by G. Kasparov and G. Yu.

We prove that von Neumann algebras and separable nuclear C ⁎ -algebras are stable for the Banach–Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between C ⁎ -algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach–Mazur cb-distance and the Kadison–Kastler distance. Finally, we show that if two C ⁎ -algebras are close enough for the cb-distance, then they have comparable length.

By a result of Helton and McCullough, open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D_L of a linear matrix inequality (LMI) L(X)>0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called "Perfect" Positivstellensatz. For example, given a generic convex free semialgebraic set D_L we determine all "(strong sense) defining polynomials" p for D_L. This follows from our general result for a given linear pencil L and a finite set I of rows of polynomials. A matrix polynomial p is positive where L is positive and I vanishes if and only if p has a weighted sum of squares representation module the "L-real radical" of I. In such a representation the degrees of the polynomials appearing depend in a very tame way only on the degree of p and the degrees of the elements of I. Further, this paper gives an efficient algorithm for computing the L-real radical of I. Our Positivstellensatz has a number of additional consequences which are presented., 69 pages, includes a table of contents

We study the relationships between Dixmier traces, zeta-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to H\"ormander-Weyl pseudo-differential calculus. In that context, we prove that the Dixmier trace of a pseudo-differential operator coincide with the `Dixmier integral' of its symbol., Comment: 47 pages, final version to appear in JFA

We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges–Rovnyak type spaces H ( b ) contrastively contained in the Drury–Arveson space on the unit ball in C d . The Aleksandrov–Clark measures on the circle are replaced by a family of states on a certain noncommutative operator system, and the backward shift is replaced by a canonical solution to the Gleason problem in H ( b ) . In addition we introduce the notion of a “quasi-extreme” multiplier of the Drury–Arveson space and use it to characterize those H ( b ) spaces that are invariant under multiplication by the coordinate functions.

In this paper, we study noncommutative varieties in polydomains in $B(H)^n$. The goal is to understand the structure of these varieties, determine their elements and classify them up to unitary equivalence. Using noncommutative Berezin transforms associated with each variety, we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains. This includes various commutative cases which are close connected to the theory of holomorphic functions in several complex variables and algebraic geometry., 37 pages, submitted for publication in April, 2013. arXiv admin note: text overlap with arXiv:1304.8043

We formulate and establish a noncommutative version of the well known Helson–Szego theorem about the angle between past and future for subdiagonal subalgebras. We then proceed to use this theorem to characterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated to subdiagonal subalgebras.

In 2010, Lafforgue and de la Salle gave examples of noncommutative L p -spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on S p , denoted AP p , cb Schur , and proved that for p ∈ [ 1 , 4 3 ) ∪ ( 4 , ∞ ] the groups SL ( n , Z ) , with n ⩾ 3 , do not have the AP p , cb Schur . Since for p ∈ ( 1 , ∞ ) the AP p , cb Schur is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also the first examples of exact groups without the AP. Recently, Haagerup and the author proved that also the group Sp ( 2 , R ) does not have the AP, without using the AP p , cb Schur . In this paper, we prove that Sp ( 2 , R ) does not have the AP p , cb Schur for p ∈ [ 1 , 12 11 ) ∪ ( 12 , ∞ ] . It follows that a large class of noncommutative L p -spaces does not have the OAP or CBAP.

In the theory of C⁎-algebras, interesting noncommutative structures arise as deformations of the tensor product, e.g. the rotation algebra Aϑ as a deformation of C(S1)⊗C(S1). We deform the tensor product of two Toeplitz algebras in the same way and study the universal C⁎-algebra T⊗ϑT generated by two isometries u and v such that uv=e2πiϑvu and u⁎v=e−2πiϑvu⁎, for ϑ∈R. Since the second relation implies the first one, we also consider the universal C⁎-algebra T⁎ϑT generated by two isometries u and v with the weaker relation uv=e2πiϑvu. Such a “weaker case” does not exist in the case of unitaries, and it turns out to be much more interesting than the twisted “tensor product case” T⊗ϑT. We show that T⊗ϑT is nuclear, whereas T⁎ϑT is not even exact. Also, we compute the K-groups and we obtain K0(T⁎ϑT)=Z and K1(T⁎ϑT)=0, and the same K-groups for T⊗ϑT.

We make several contributions to our recent program investigating structural properties of algebras of operators on a Hilbert space. For example, we make substantial contributions to the noncommutative peak interpolation program begun by Hay and the first author, Hay and Neal. Another sample result: an operator algebra has a contractive approximate identity iff the linear span of the elements with positive real part is dense. We also extend the theory of compact projections to the most general case. Despite the title, our algebras are often allowed to have no approximate identity.

In this paper, we study the John–Nirenberg inequality for BMO and the atomic decomposition for H1 of noncommutative martingales. We first establish a crude version of the column (resp. row) John–Nirenberg inequality for all 0