Noncommutative Setting Research Articles

Noncommutative Cauchy Integral Formula

The aim of this paper is to provide and prove the most general Cauchy integral formula for slice regular functions and for $$C^1$$ functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions.

On bi-free De Finetti theorems

We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of n-freeness.

Hardy–Littlewood–Paley inequalities and Fourier multipliers on SU(2)

We prove noncommutative versions of Hardy-Littlewood and Paley inequalities relating a function and its Fourier coefficients on the group SU(2). We use it to obtain lower bounds for the L-p - L-q norms of Fourier multipliers on SU(2) for 1 < p <= 2 <= q < infinity. In addition, we give upper bounds of a similar form, analogous to the known results on the torus, but now in the noncommutative setting of SU(2).

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

In this article, we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures.

On Non-commuting Sets in Certain Finite p-Groups

Let G be a group. A subset X of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y. If |X| ≥ |Y| for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, the bound for the cardinality of a maximal non-commuting set in a finite p-group G is determined, where G is a non-abelian p-group given by a central extension as 1 → ℤpm → G → ℤp× ⋯ × ℤp → 1 and its derived subgroup has order p.

Factorization theory: From commutative to noncommutative settings

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n×n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.

Enabling quaternion derivatives: the generalized HR calculus.

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

Subgaussian 1-cocycles on discrete groups

We prove the $L_p$ Poincar\'e inequalities with constant $C\sqrt{p}$ for $1$-cocycles on countable discrete groups under Bakry--Emery's $\Gamma_2$-criterion. These inequalities determine an analogue of subgaussian behavior for 1-cocycles. Our theorem improves some of our previous results in this direction, and in particular implies Efraim and Lust-Piquard's Poincar\'e type inequalities for the Walsh system. The key new ingredient in our proof is a decoupling argument. As complementary results, we also show that the spectral gap inequality implies the $L_p$ Poincar\'e inequalities with constant $C{p}$ under some conditions in the noncommutative setting. New examples which satisfy the $\Gamma_2$-criterion are provided as well.

A Quantum Finance Model

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and the probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend the classical probability model to a quantum probability model. The classical stochastic calculus is replaced by its quantum counterpart on the Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. As an application, prices of quantum European options are obtained.

Dimensional reduction of the generalized DBI

We study the generalized Dirac–Born–Infeld (DBI) action, which describes a q-brane ending on a p-brane with a (q+1)-form background. This action has the equivalent descriptions in commutative and non-commutative settings, which can be shown from the generalized metric and Nambu-Sigma model. We mainly discuss the dimensional reduction of the generalized DBI at the massless level on the flat spacetime and constant antisymmetric background in the case of flat spacetime, constant antisymmetric background and the gauge potential vanishes for all time-like components. In the case of q=2, we can do the dimensional reduction to get the DBI theory. We also try to extend this theory by including a one-form gauge potential.

A new and simpler noncommutative central sets theorem

Using dynamics, Furstenberg defined the concept of a central subset of positive integers and proved several powerful combinatorial properties of central sets. Later using the algebraic structure of the Stone–Čech compactification, Bergelson and Hindman, with the assistance of B. Weiss, generalized the notion of a central set to any semigroup and extended the most important combinatorial property of central sets to the central sets theorem. Currently the most powerful formulation of the central sets theorem is due to De, Hindman, and Strauss in [3, Corollary 3.10]. However their formulation of the central sets theorem for noncommutative semigroups is, compared to their formulation for commutative semigroups, complicated. In this paper I prove a simpler (but still equally strong) version of the noncommutative central sets theorem in Corollary 3.3.

Noncommutative quantum mechanics of simple matter systems interacting with circularly polarized gravitational waves

The response of a test particle, both for the free case and under the harmonic oscillator potential, to circularly polarized gravitational waves is investigated in a noncommutative quantum mechanical setting. The system is quantized following the prescription in \cite{ncgw1}. Standard algebraic techniques are then employed to solve the Hamiltonian of the system. The solutions, in both cases, show signatures of the coordinate noncommutativity. In the harmonic oscillator case, this signature plays a key role in altering the resonance point and the oscillation frequency of the system.

K-theory and the bridge from motives to noncommutative motives

In this work we present a new approach to the theory of noncommutative motives and use it to explain the different flavors of algebraic K-theory of schemes and dg-categories. The work is divided into three main parts. In the first part we use the techniques of higher algebra developed in [63] to provide a universal characterization for the symmetric monoidal (∞,1)-category underlying the motivic stable A1- homotopy theory of Morel–Voevodsky [107,67]. More precisely, given a symmetric monoidal model category V together with an object X∈V, we characterize the underlying symmetric monoidal (∞,1)-category of the symmetric monoidal model category SpΣ(V,X) introduced by Hovey in [43], by means of a universal property amongst symmetric monoidal (∞,1)- categories. This characterization trivializes the problem of finding motivic monoidal realizations.In the second part we introduce a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of Kontsevich [56,55,54]. The key ingredient is a notion of Nisnevich topology in the noncommutative setting, compatible with the classical notion. This compatibility, together with the universal property proved in the first part, ensures the existence of a canonical monoidal map from the stable motivic theory of Morel–Voevodsky towards these new noncommutative motives that allow us to compare the two theories.In the last part of this paper we explain how this bridge can be used to explain the various flavors of algebraic K-theory of dg-categories. More precisely, we prove that the non-connective K-theory of dg-categories introduced by Schlichting [82] is the (non-commutative) Nisnevich sheafification of connective algebraic K-theory. Then we prove that its further (non-commutative) A1-localization is a tensor unit in our noncommutative motives. As a corollary we obtain a precise proof for an original conjecture of Kontsevich claiming that K-theory gives the correct mapping spaces in noncommutative motives. Our major application is the discovery of a canonical factorization of our motivic bridge through the (∞,1)-category of modules over the commutative algebra object representing homotopy invariant algebraic K-theory of schemes. The results in [77] imply that this bridge is fully faithful over a field k with resolutions of singularities, so that, at the motivic level, no information (below K-theory) is lost by passing to the noncommutative world.

A primer on ideal theoretical operation in non-commutative polynomial rings

Buchberger Theory provides, for commutative polynomial rings, computational tools allowing to perform ideal theoretical operations. Such techniques are easily adapted to deal with similar problems in the non-commutative setting. I give here an elementary primer.

Pick Matricies and Quaternionic Power Series

It is well known that a non-constant complex-valued function f defined on the open unit disk \({\mathbb{D}}\) of the complex plane is an analytic self-mapping of \({\mathbb{D}}\) if and only if Pick matrices \({\left[ (1 - f(z_i)\overline{f(z_j)})/\right.\left.(1 - z_i\overline{z}_j)\right]_{i,j=1}^n}\) are positive semidefinite for all choices of finitely many points \({z_{i} \in \mathbb{D}}\). A stronger version of the “if” part was established by Hindmarsh (Pac J Math 27:527–531, 1968): if all 3 × 3 Pick matrices are positive semidefinite, then f is an analytic self-mapping of \({\mathbb{D}}\). In this paper, we extend this result to the non-commutative setting of power series over quaternions.

Polynomial interpolation over quaternions

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated “on the left” and “on the right”. If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring H[z]. The problem containing both “left” and “right” interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in H[z].

Lagrange interpolation problem for quaternion polynomials

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated “from the left” and “from the right”. If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring H[z]. The problem containing both “left” and “right” interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in H[z].

A non-commutative version of Lépingle–Yor martingale inequality

Let ( f n ) n = 1 N be a stochastic process adapted to the filtration ( ℱ n ) n = 0 N . An inequality of D. Lépingle and M. Yor states that E [ ( ∑ n = 1 N | E n − 1 ( f n ) | 2 ) 1 / 2 ] ≤ 2 E [ ( ∑ n = 1 N | f n | 2 ) 1 / 2 ] . We generalize this inequality to non-commutative martingale setting.

Factorizations of upper triangular matrices

If D is an integral domain in which every element can be factored as a product of irreducible elements, then every element A∈Tn(D), the semigroup of upper triangular matrices with nonzero determinant, can be factored as a product of irreducible elements. We classify both the irreducible elements and units of this cancellative semigroup. Having achieved this, we provide a means of measuring the uniqueness or non-uniqueness of these factorizations in terms of the uniqueness or non-uniqueness of factorizations in D. In addition to using well-studied invariants from factorization theory of commutative semigroups, we introduce a new tool, weak transfer homomorphisms, which are more appropriate for noncommutative settings.

Walsh equiconvergence theorems in the quaternionic setting

In this paper, we prove the quaternionic version of the result of Walsh stating that the difference between the partial sums of the Taylor expansion of an analytic function and its interpolation polynomial at the roots of unity converges in a larger disc than the disc of analyticity of the function. Our result holds for functions of a quaternionic variable which are slice regular in a ball and thus they admit a converging power series expansion. We also prove a generalization of this theorem as well as its converse. Because of the noncommutative setting, the results are nontrivial and require a notion of multiplication of functions (and of polynomials) which does not commute with the evaluation.