Noncommutative Analogue Research Articles

On compact extensions of tracial $W^{*}$-dynamical systems

We establish several classification results for compact extensions of tracial W^{*} -dynamical systems and for relatively independent joinings thereof for actions of arbitrary discrete groups. We use these results to answer a question of Austin, Eisner, and Tao and some questions raised by Duvenhage and King. Moreover, combining our results with an earlier classification of weakly mixing extensions by Popa, we can derive non-commutative Furstenberg–Zimmer type dichotomies on the L^{2} -level. Although in general an adequate generalization of the Furstenberg–Zimmer tower of intermediate compact extensions does not seem possible in the von Neumann algebraic framework, we show that there always exists a non-commutative analogue of the finer Host–Kra–Ziegler tower for any ergodic action of a countable abelian group.

Finite spectral triples for the fuzzy torus

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.

Noncommutative Multi-Parameter Subsequential Wiener–Wintner-Type Ergodic Theorem

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class Δ instead of the weight class case. In our subsequence class, every term of k̲∈Δ is one of the three kinds of nonzero density subsequences we consider. As key ingredients, we give the maximal ergodic inequalities of multi-parameter subsequential averages and obtain a noncommutative subsequential analogue of the Banach principle. Then, by combining the critical result of the uniform convergence for a dense subset of the noncommutative Lp(M) space and the noncommutative Orlicz space, we immediately obtain the main theorem.

A geometric Elliott invariant and noncommutative rigidity of mapping tori

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Zd-actions on a compact space can be lifted to an isomorphism of their foliation C⁎-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C⁎-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.

Differential operators on C⁎-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

We introduce the notion of a differential operator on C⁎-algebras. This is a noncommutative analogue of a differential operator on a smooth manifold. We show that the common closed domain of all differential operators is closed under smooth functional calculus. As a corollary, we show that Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Non-abelian discrete Toda chains and related lattices

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations → Somos-N sequences → discrete Painlevé equations.

A geometric representative for the fundamental class in KK-duality of Smale spaces

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module.

Asymptotic growth patterns for class field towers

Let p be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a \mathbb{Z}_{p} -tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain \mathbb{Z}_{p} -extensions in which a given prime above p is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for p -primary class numbers of the number fields in a \mathbb{Z}_{p} -tower.

Universal filtered quantizations of nilpotent Slodowy slices

Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by \mathbb{C}^\times -equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite W -algebra is a universal filtered quantization of the slice, building on the work of Lehn–Namikawa–Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite W -algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finite W -algebra of type \mathsf{B} as a quotient of a shifted Yangian.

Polynomial algebras from Lie algebra reduction chains [formula omitted] [formula omitted][formula omitted][formula omitted

We reexamine different examples of reduction chains g⊃g′ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra g′ leads to polynomial deformations of Lie algebras. These polynomial algebras have already been observed in various contexts, such as in the framework of superintegrable systems. Two relevant chains extensively studied in Nuclear Physics, namely the Elliott chain su(3)⊃so(3) and the chain so(5)⊃su(2)×u(1) related to the Seniority model, are analysed in detail from this perspective. We show that these two chains both lead to three-generator cubic polynomial algebras, a result that paves the way for a more systematic investigation of nuclear models in relation to polynomial structures arising from reduction chains. In order to show that the procedure is not restricted to semisimple algebras, we also study the chain Sˆ(3)⊃sl(2,R)×so(2) involving the centrally-extended Schrödinger algebra in (3+1)-dimensional space–time. The approach chosen to construct these polynomial algebras is based on the use of the Lie-Poisson bracket, so that all the results are presented in the Poisson (commutative) setting. The advantage of considering this approach vs the enveloping algebras is emphasized, commenting on the main formal differences between the polynomial Poisson algebras and their noncommutative analogue. As an illustrative example of the latter, the three-generator cubic algebra associated to the Elliott chain is reformulated in the Lie algebraic (noncommutative) setting.

THE NOWICKI CONJECTURE FOR BICOMMUTATIVE ALGEBRAS

Let K be a field of characteristic zero, and K[Xn,Yn ] be the commutative associative unitary polynomial algebra of rank 2n generated by the set Xn∪Yn={x1,…,xn,y1,…,yn }. It is well known that the algebra K[Xn,Yn ]^δ of constants of the locally nilpotent linear derivation δ of K[Xn,Yn ] sending yi to xi, and xi to 0, is generated by x1,…,xn and the determinants of the form xi yj-xj yi; that was first conjectured by Nowicki in 1994, and later proved by several authors. Bicommutative algebras are nonassociative noncommutative algebras satisfying the identities (xy)z=(xz)y and x(yz)=y(xz). In this study, we work in the 2n generated free bicommutative algebra as a noncommutative nonassociative analogue of the Nowicki conjecture, and find the generators of the algebra of constants in this algebra.

Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

AbstractIn this paper, we study the groups of isometries and the set of bi‐Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

Ring-theoretic blowing down II: Birational transformations

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In a companion paper the authors also described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surfaces containing an elliptic curve. Notably we show that Van den Bergh's quadrics can be obtained from the Sklyanin algebra by suitably blowing up and down, and we also provide a noncommutative analogue of the classical Cremona transform.

Double scaling limits of Dirac ensembles and Liouville quantum gravity

In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory.

Quantum Holonomy Fields

We investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a H-algebra. H-algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random Holonomy that they induce. We propose a general definition of a Holonomy Field with symmetries displaying the structure of a H-algebra and construct such a field starting from a quantum Lévy process on a H-algebra in the category of probability spaces. We call them Quantum Holonomy Fields. We also consider the more abstract case of an H-algebra in a given algebraic category. This yields the notion of a Categorical Holonomy Field. As an application, we define higher dimensional generalizations of the so-called Master Field on the plane.

Callias-type operators associated to spectral triples

Callias-type (or Dirac-Schrödinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.

Random quantum maps and their associated quantum Markov chains

The notion of ‘quantum family of maps’ (QFM) has been defined by Piotr Sołtan as a noncommutative analogue of ‘parameterized family of continuous maps’ between locally compact spaces. A QFM between C*-algebras B, A, is given by a pair $$(C,\phi )$$ where C is a C*-algebra and $$\phi :B\rightarrow A\check{\otimes }C$$ is a $$*$$ -morphism. The main goal of this note, is to introduce the notion of ‘random quantum map’ (RQM), which is a noncommutative analogue of ‘random continuous map’ between compact spaces. We define a RQM between B, A, to be given by a triple $$(C,\phi ,\nu )$$ where $$(C,\phi )$$ is a QFM and $$\nu $$ a state (normalized positive linear functional) on C. Our first application of RQMs takes place in theory of completely positive maps (CPM): RQMs give rise canonically to a class of CPMs which we call implemented CPMs. We consider some partial results about the natural and important problem of characterization of implemented CPMs. For instance, using Stinespring’s Theorem, we show that any CPM from B to A is implemented if A is finite-dimensional. Our second application of RQMs takes place in theory of quantum stochastic processes: We show that iterations of any RQM with $$B=A$$ , gives rise to a quantum Markov chain in a sense introduced by Luigi Accardi.

C∗-extreme points of entanglement breaking maps

In this paper, we study the [Formula: see text]-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of [Formula: see text]-extreme points are discussed. By establishing a Radon–Nikodym-type theorem for a class of EB-maps we give a complete description of the [Formula: see text]-extreme points. It is shown that a unital EB-map [Formula: see text] is [Formula: see text]-extreme if and only if it has Choi-rank equal to [Formula: see text]. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a non-commutative analog of the Krein–Milman theorem for [Formula: see text]-convexity of the set of unital EB-maps.

On the non-commutative multifractional Brownian motion

We define a non-commutative analogue of a real gaussian Volterra-type multifractional Brownian motion (NC-mfBm for short) and show that its trajectories behave locally like non-commutative fractional Brownian motion. We determine the pointwise Hölder exponent as well as a random matrix approximation of this process.