Some strong limit theorems for weighted sums of measurable operators

The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.

⋆-cohomology, third type Chern character and anomalies in general translation-invariant noncommutative Yang–Mills

A representation of general translation-invariant star products ⋆ in the algebra of [Formula: see text] is introduced which results in the Moyal–Weyl–Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called ⋆-cohomology, with groups [Formula: see text], [Formula: see text], is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg–Witten map for the Moyal case. Employing the Chern–Weil theory via the integral classes of [Formula: see text] a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang–Mills theory. Thereby, we study the mentioned Yang–Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of ⋆-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the ⋆-cohomology.

GENERALISED QUANTUM DETERMINANTAL RINGS ARE MAXIMAL ORDERS

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

A noncommutative geometric LR rule

International audience The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given by Vakil and later generalized by Coskun. In this paper we give a noncommutative version of the geometric LR rule. As a consequence, we establish a geometric explanation for the positivity of noncommutative LR coefficients in certain cases.

Noncommutative Friedmann equations in effective LQC

In this work, we construct a noncommutative version of the Friedmann equations in the framework of effective loop quantum cosmology, extending and applying the ideas presented in a previous proposal by some of the authors. The model under consideration is a flat FRW spacetime with a free scalar field. First, noncommutativity in the momentum sector is introduced. We establish the noncommutative equations of motion and obtain the corresponding exact solutions. Such solutions indicate that the bounce is preserved, in particular, the energy density is the same as in the standard LQC. We also construct an extension of the modified Friedmann equations arising in effective LQC which incorporates corrections due to noncommutativity, and argue that an effective potential is induced. This, in turn, leads us to investigate the possibility of an inflationary era. Finally, we obtain the Friedmann and the Raychaudhuri equations when implementing noncommutativity in the configuration sector. In this case, no effective potential is induced.

Superdense star in a spacetime with minimal length

In this paper, we generalize core–envelope model of superdense star to a noncommutative spacetime and study the modifications due to the existence of a minimal length, predicted by various approaches to quantum gravity. We first derive Einstein’s field equation in [Formula: see text]-deformed spacetime and use this to set up noncommutative version of core–envelope model describing superdense stars. We derive [Formula: see text]-deformed law of density variation, valid up to first-order approximation in deformation parameter and obtain radial and tangential pressures in [Formula: see text]-deformed spacetime. We also derive [Formula: see text]-deformed strong energy conditions and obtain a bound on the deformation parameter.

Spanier–Whitehead K-duality for C∗-algebras

Classical Spanier–Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier–Whitehead [Formula: see text]-duality, which is defined on the category of [Formula: see text]-algebras whose [Formula: see text]-theory is finitely generated and that satisfy the UCT, with morphisms the [Formula: see text]-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier–Whitehead [Formula: see text]-duality.

SUBSTRUCTURAL INQUISITIVE LOGICS

This paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

Noncommutative Reissner–Nordstrøm black hole

In this work we construct a deformed embedding of the Reissner–Nordstrøm (R-N) space–time within the framework of a noncommutative Riemannian geometry. We provide noncommutative corrections to the usual Riemannian expressions for the metric and curvature tensors. For the case of the metric tensor, the expression obtained possesses terms that are valid to all orders in the deformation parameter. Then we calculate the correction to the area of the event horizon of the corresponding noncommutative R-N black hole, obtaining an expression for the area of the black hole, which is correct up to fourth-order terms in the deformation parameter. Finally we include some comments on the noncommutative version on one of the second-order scalar invariants of the Riemann tensor, the so-called Kretschmann invariant, a quantity that is regularly used to extend gravity to the quantum level.