Noncommutative Setting Research Articles

Multi-Toeplitz operators and free pluriharmonic functions

We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains Dqm(H)⊂B(H)n, m,n≥1, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions on the interior of the domain Dqm(H). We prove that the set of all weighted multi-Toeplitz operators coincides withA(Dqm)⁎+A(Dqm)‾WOT, where the domain algebra A(Dqm) is the norm-closed unital non-selfadjoint algebra generated by the universal model (W1,…,Wn) of the noncommutative domain Dqm(H). These results are used to study the class of free pluriharmonic functions on Dqm(H)∘. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using noncommutative Cauchy transforms, we provide a free analytic functional calculus for n-tuples of operators, which extends to free pluriharmonic functions. Our study of weighted multi-Toeplitz operators on Fock spaces is a blend of multi-variable operator theory, noncommutative function theory, operator spaces, and harmonic analysis.

Operator Scaling: Theory and Applications

In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over $${\mathbb {Q}}$$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (Ivanyos et al. in Comput Complex 26(3):717–763, 2017) (whether or not randomization is allowed). The algorithm efficiently solves the “word problem” for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004), who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits 2004). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp–Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular, we explain how some of these sources played an important role in the development of Gurvits’ algorithm and in our analysis of it here.

Arbeitsgemeinschaft: Topological Cyclic Homology

Introduced by Bökstedt–Hsiang–Madsen in the nineties, topological cyclic homology is a manifestation of the dual visions of Connes–Tsygan and Waldhausen to extend de Rham cohomology to a noncommutative setting and to replace algebra by higher algebra. The cohomology theory that ensues receives a denominator-free Chern character from algebraic K -theory, used by Hesselholt–Madsen to evaluate the p -adic K -theory of p -adic fields. More recently, Bhatt–Morrow–Scholze have defined a “motivic” filtration of topological cyclic homology and its variants, the filtration quotients of which give rise to their denominator-free p -adic Hodge theory \mathbb A \Omega .

Primeness in Quantales

In this paper we propose a new concept of primeness in quantales. It is proved that this concept coincide with classical definition in commutative quantales, but no longer valid in the noncommutative setting. Also, the notions of strong and uniform strong primeness are investigated.

A $$C^*$$ C ∗ -algebraic approach to the principal symbol II

We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of Sukochev and Zanin (J Oper Theory, 2019) and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes’ trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes’ trace theorem to operators with non-smooth symbols in three examples: the Lie group $$\mathrm {SU}(2)$$ , noncommutative tori and Moyal planes.

The Ricci curvature in noncommutative geometry

Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function of the full Laplacian of the de Rham complex, localized by smooth endomorphisms of the cotangent bundle and their trace. We use this formulation to introduce the Ricci functional in a noncommutative setting and in particular for curved noncommutative tori. This Ricci functional uniquely determines a density element, called the Ricci density, which plays the role of the Ricci operator. The main result of this paper provides an explicit computation of the Ricci density when the conformally flat geometry of the noncommutative two torus is encoded by the modular de Rham spectral triple.

H∞-calculus for semigroup generators on BMO

We prove that the negative generator L of a semigroup of positive contractions on L∞ has bounded H∞(Sη)-calculus on the associated Poisson semigroup-BMO space for any angle η>π/2, provided L satisfies Bakry-Émery's Γ2≥0 criterion. Our arguments only rely on the properties of the underlying semigroup and work well in the noncommutative setting. A key ingredient of our argument is a type of quasi monotone properties for the subordinated semigroup Tt,α=e−tLα,0<α<1, that is proved in the first part of this article.

Bohr Inequalities on Noncommutative Polydomains

The goal of this paper is to study the Bohr phenomenon in the setting of free holomorphic functions on noncommutative regular polydomains $$\mathbf{D_f^m}$$ , $$\mathbf{f}=(f_1,\ldots , f_k)$$ , generated by positive regular free holomorphic functions. These polydomains are noncommutative analogues of the scalar polydomains $$\begin{aligned} {{\mathcal {D}}}_{f_1} ({{\mathbb {C}}})\times \cdots \times {{\mathcal {D}}}_{f_k}({{\mathbb {C}}}), \end{aligned}$$ where each $${{\mathcal {D}}}_{f_i}({{\mathbb {C}}})\subset {{\mathbb {C}}}^{n_i}$$ is a certain Reinhardt domain generated by $$f_i$$ . We characterize the free holomorphic functions on $$\mathbf{D_f^m}$$ in terms of the universal model of the polydomain and extend several classical results from complex analysis to our noncommutative setting. It is shown that the free holomorphic functions admit multi-homogeneous and homogeneous expansions as power series in several variables. With respect to these expansions, we introduce the Bohr radii $$K_{mh}(\mathbf{D_f^m})$$ and $$K_{h}(\mathbf{D_f^m})$$ for the noncommutative Hardy space $$H^\infty (\mathbf{D_{f,\text { rad}}^m})$$ of all bounded free holomorphic functions on the radial part of $$\mathbf{D_f^m}$$ . Several well-known results concerning the Bohr radius associated with classes of bounded holomorphic functions are extended to our noncommutative multivariable setting.

Quasi hyperrigidity and weak peak points for non-commutative operator systems

In this article, we introduce the notions of weak boundary representation, quasi hyperrigidity and weak peak points in the non-commutative setting for operator systems in $$C^*$$ -algebras. An analogue of Saskin’s theorem relating quasi hyperrigidity and weak Choquet boundary for particular classes of $$C^*$$ -algebras is proved. We also show that, if an irreducible representation is a weak boundary representation and weak peak then it is a boundary representation. Several examples are provided to illustrate these notions.

Noncommutative topology and Jordan operator algebras

AbstractJordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.

Inequalities for noncommutative differentially subordinate martingales

The classical differential subordination of martingales, introduced by Burkholder in the eighties, is generalized to the noncommutative setting. Working under this domination, we establish the strong-type inequalities with the constants of optimal order as p→1 and p→∞, and the corresponding endpoint weak-type (1,1) estimate. In contrast to the classical case, we need to introduce two different versions of noncommutative differential subordination, depending on the range of the exponents. For the Lp-estimate, 2≤p<∞, a certain weaker version is sufficient; on the other hand, the strong-type (p,p) inequality, 1<p<2, and the weak-type (1,1) estimate require a stronger version. As an application, we present a new proof of noncommutative Burkholder–Gundy inequalities. The main technical advance is a noncommutative version of the good λ-inequality and a certain summation argument. We expect that these techniques will be useful in other situation as well.

Maximal non-commuting set in 2-generated p-groups of class two

Let [Formula: see text] be a group. A set [Formula: see text] in [Formula: see text] is said to be non-commuting if [Formula: see text] for any two distinct elements [Formula: see text] and [Formula: see text] in [Formula: see text]. If [Formula: see text] for any other non-commuting set [Formula: see text] in [Formula: see text], then [Formula: see text] is said to be a maximal non-commuting set. In this paper, we obtain the cardinality of a maximal non-commuting set in [Formula: see text]-generated [Formula: see text]-groups of class two having finite commutator subgroup.

Power monoids: A bridge between factorization theory and arithmetic combinatorics

We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of Baeth and Smertnig's work on the factorization theory of non-commutative, but cancellative monoids (Baeth and Smertnig (2015) [5]). Then, we bring in power monoids and, applying the abstract machinery developed in the first part, we undertake the study of their arithmetic.More in particular, let H be a multiplicatively written monoid. The set Pfin(H) of all non-empty finite subsets of H is naturally made into a monoid, which we call the power monoid of H and is non-cancellative unless H is trivial, by endowing it with the operation (X,Y)↦{xy:(x,y)∈X×Y}.Power monoids are, in disguise, one of the primary objects of interest in arithmetic combinatorics, and here for the first time we tackle them from the perspective of factorization theory. Proofs lead to consider various properties of finite subsets of N that can or cannot be split into a sumset in a non-trivial way, giving rise to a rich interplay with additive number theory.

Non-commutative ergodic averages of balls and spheres over Euclidean spaces

In this paper, we establish a non-commutative analogue of Calderón’s transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case—operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein–Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener’s pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones’ pointwise ergodic theorem, we use again the transference principle together with the Littlewood–Paley method, which is different from Jones’ original variational method that is still unavailable in the non-commutative setting.

Noncommutative multi-parameter Wiener–Wintner type ergodic theorem

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 setΛd={{λ1k1⋯λdkd}(k1,…,kd)∈Nd:(λ1,…,λd)∈Td}; 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 Λd, 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.

Slice regular semigroups

In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy integral formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford R_n algebras included.

Noncommutative Blackwell–Ross martingale inequality

We establish a noncommutative Blackwell–Ross inequality for supermartingales under a suitable condition which generalizes Khan’s work to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

Noncommutative Riesz transforms – dimension free bounds and Fourier multipliers

We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry’s results in the commutative setting. Our proof is inspired by Pisier’s method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in \mathbb R^n (where Meyer’s conjecture fails) or to the word length of free groups. Lust-Piquard’s work for discrete laplacians on LCA groups is also generalized in several ways. In the context of Fourier multipliers, we will prove that Hörmander–Mikhlin multipliers are Littlewood-Paley averages of our Riesz transforms. This is highly surprising in the Euclidean and (most notably) noncommutative settings. As application we provide new Sobolev/Besov type smoothness conditions. The Sobolev-type condition we give refines the classical one and yields dimension free constants. Our results hold for arbitrary unimodular groups.

Free Holomorphic Functions on the Regular Polyball

In this paper, we study free holomorphic functions on the regular polyball, which was recently introduced by Popescu (Adv Math 279:104–158, 2015, Trans Am Math Soc 368:4357–4416, 2016). The purpose of this paper is to continue the line of Popescu to develop a theory of free holomorphic functions. Some results from the classical holomorphic function theory have free analogues in this noncommutative setting. In particular, we prove the maximum principle, Weierstrass and Montel type theorems for free holomorphic functions. As an application, we construct a metric on $$\mathrm{Hol}(\mathbf{B_n}(\mathcal {H}))$$ such that it becomes a complete space.

Matrix convex sets without absolute extreme points

This article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets we consider are noncommutative sets, KX, formed by taking matrix convex combinations of a single tuple X. In the case that X is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, KX is a closed bounded matrix convex set with no absolute extreme points.A central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning. Matrix extreme points are the strongest type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. This result shows that matrix convex sets may fail to be spanned by their absolute extreme points.