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Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras., 25 pages

We study multiplication modules. The rings are not assumed to be commutative. Several criteria with some applications are given for a direct sum of modules to be a multiplication module.

The aim of this manuscript is to discuss the wormhole geometry in f ( R ) theory of gravity, where R represents the Ricci scalar. For our current work, we consider the spherically symmetric static spacetime with an anisotropic fluid source to discuss the wormhole solutions. We further discuss the existence of wormholes by using two well-known distributions like Gaussian and Lorentzian non-commutative geometry. The stability of wormhole geometry has been discussed through the Tolman–Oppenheimer–Volkoff equation and observed that our wormhole solutions are stable. Furthermore, we analyse the anisotropic parameter to examine the behaviour of the wormhole. It can be observed through the graphical analysis that null energy conditions (NEC) are not satisfied for these distributions. The violation of NEC indicates the existence of dark matter and dark energy, which is the validation of Gaussian and Lorentzian non-commutative wormholes exist in the modified f ( R ) theory of gravity.

In the present article, we construct the super fuzzy Dirac and chirality operators on super fuzzy sphere SF(2|2) . Using the super fuzzy Ginsparg-Wilson algebra, we study the super gauged fuzzy Dirac and chirality operators in instanton sector. We show that they have correct commutative limit in the limit case when noncommutative parameter l tends to infinity.

In this paper we investigate a semi-linear degenerate elliptic equation Δ L u + h ( ξ ) u p + g ( ξ ) u q = 0 related to the generalized Baouendi-Grushin vector fields, where p and q satisfy certain conditions. The idea is to extend the Obata method applying elliptic equations to the degenerate elliptic equation. In consideration of noncommutative relations of the generalized Baouendi-Grushin vector fields, an identity corresponding to the degenerate elliptic equation is established. By using a special truncated function, the identity deformation and accurate estimates, we derive Liouville theorems which show that all nonnegative solutions to the equation are trivial.

We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.

We give an explicit formula showing how the double Poisson algebra introduced in [14] appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on A ⊕ A ⁎ . Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra A and emphasises the special role of the fourth component of pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces ( Rep n A ) G l n for any associative algebra A.

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is a unital algebra with a surjective inner derivation, then every element in $A$ can be written as $f(a_1,\dots,a_n)$ for some $a_i\in A$., Comment: 21 pages, 0 figures, submited to Journal of Algebra

Let k be an algebraically closed field and A a Z -graded finitely generated simple k -algebra which is a domain of Gelfand–Kirillov dimension 2. We show that the category of Z -graded right A-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of U ( sl 2 ) .

Topological structure of translation-invariant noncommutative Yang–Mills theories are studied by means of a cohomology theory, the so-called ⋆-cohomology, which plays an intermediate role between de Rham and cyclic (co)homology theory for noncommutative algebras and gives rise to a cohomological formulation comparable to Seiberg–Witten map.

Recently it has been proved that if σ ∈ GL n ( R ) where R is a commutative ring and n ≥ 3 , then each of the matrices t k l ( σ i j ) ( i ≠ j , k ≠ l ) is a product of eight E n ( R ) -conjugates of σ and σ − 1 . In this article we show that similar results hold true if R is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O -operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.

The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent progress. Applications to cosmology suggest that the Planck scale ought to depend upon dynamics, and possible consequences in the large of the quantum structure in the small are outlined.

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field $k$. Secondly, we prove that the $\mathcal O$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of finite dimensional $A$-modules, in the sense that $\eta_B: B \to \mathcal O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf M$ is considered as a family of $B$-modules., Comment: 9 pages

Factorization in matrix rings has been investigated in a wide variety of contexts. Recently, this study has provided a wealth of information about how factorization works in noncommutative settings. It is often the case that when S is a ring of matrices over a base ring R, multiplicative factorization in R drives the multiplicative factorization in S; that is, the additive structure of R is largely irrelevant to how elements of S factor multiplicatively. Recently, Chen et al. studied factorization of the semiring of upper-triangular matrices with entries in N 0 , the semiring of nonnegative integers. In this work we show that their results fit within the larger context of factorization in upper-triangular matrices over reduced information algebras. We illustrate that factorization in T n ( H 0 ) depends on both the multiplicative and additive structure of H 0 . If H 0 is such an information algebra, we classify the irreducible elements of T n ( H 0 ) and describe the degree to which factorization in T n ( H 0 ) is nonunique by considering elasticities, delta sets, and unions of sets. In addition, we show that unlike many important classes of semigroups whose arithmetic has been studied, T n ( H 0 ) is not transfer Krull. For H 0 = N 0 , we extend previously known results and give explicit descriptions of nonunique factorization in T n ( N 0 ) .

We establish a version of Knorrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let A be a left noetherian AS-regular algebra, let f be a normal and regular element of A of positive degree, and take B = A / ( f ) . Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over B and those over (a noncommutative analog of) its second double branched cover ( B # ) # . Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(-)$, and it describes the $E_2$ term of the K\"unneth spectral sequence for relative $THH$. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors., Comment: Minor revisions. Published version

Let M and N be finitely generated modules over a d-dimensional commutative Noetherian ring R. We show that, if M maps onto N ⊕ ( t + d ) locally, then M maps onto N ⊕ t globally. In a similar manner, we demonstrate that, if M locally admits N ⊕ ( t + d ) as a direct summand, then M globally admits N ⊕ t as a direct summand. We also generalize splitting theorems due to Serre, De Stefani–Polstra–Yao, and Bass by exploiting the j-spectrum of R and by allowing M and N to be modules over a possibly noncommutative module-finite R-algebra.

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then a noncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given noncommutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a given noncommutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases.

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

It is known that the noncommutative Hardy spaces H 1 ( M ) and H 1 max ( M ) do not coincide, in general. In fact, it may happen that H 1 ( M ) ⊈ H 1 max ( M ) . It is an interesting question whether the reverse inclusion holds or not. In this note, motivated by this question, we prove that the validity of inequality ‖ x ‖ H 1 C ( M ) ≤ c ‖ x ‖ H 1 max ( M ) for some c and all martingales x ∈ H 1 max ( M ) gives rise to a certain representation of positive elements of the noncommutative space BMO . Conversely, existence of such a representation for elements of BMO yields the inequality above for positive martingales of H 1 max ( M ) .

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space $V$ given together with a linear map $\Delta : V \to V \otimes V$. In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on $V$, and our main result concerns a certain transformation $\Delta^{*}$ on such families of multilinear functionals., Comment: Version 2: Minor revision, added references

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 noncommutative Khintchine inequalities for all interpolation spaces between $L_p$ and $L_2$ with $p 2$ and $p < 2$. It produces a new proof of Khintchine inequalities for $p, 33 pages, published version

The azimuthal correlation belongs to a class of polarization observables which vanishes at the Born term level in the standard model for the semileptonic rest frame decay of a polarized top quark t ( ↑ ) → b W + → b l + υ l . In this frame, the azimuthal correlation is defined between the planes formed by the vectors ( p → l , p → X b ) and ( p → l , P → t ) . We calculate the azimuthal correlation function of polarized top quark in the framework of the noncommutative standard model for the first time. We find that this observable is nonzero in the leading order of noncommutative space–time. Further the appearance of the oscillatory azimuthal distribution of decay rate provides a practical possibility to test the noncommutativity in particle colliders.

We prove that the negative generator L of a semigroup of positive contractions on L ∞ has bounded H ∞ ( S η ) -calculus on the associated Poisson semigroup-BMO space for any angle η > π / 2 , provided L satisfies Bakry-Emery's Γ 2 ≥ 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 T t , α = e − t L α , 0 α 1 , that is proved in the first part of this article.

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 .

Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of “homomorphic” operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize noncommutative Pick functions.

In this article, we present the thermodynamical properties of the anharmonic oscillator for unrelativistic and relativistic K–G equations in the presence of an external magnetic field in noncommutative plane using q-deformed superstatistics. The explicit form of energy eigenvalues and the wave functions of the considered system are derived in terms of the bi-confluent Heun functions. Subsequently, by employing the effective Boltzmann factor, the thermodynamical properties of the anharmonic oscillator ensemble for both cases are obtained and, some graphs are plotted for the clarity of our results.

Standard table algebras that have an irreducible character of degree two, and with all other irreducible characters of multiplicity one, are classified to exact isomorphism.

Let A be a nonnegatively graded connected algebra over a noncommutative separable k-algebra K, and let M be a bounded below graded right A-module. If we denote by T the A ∞ -coalgebra Tor • A ( K , K ) , we know that there exists an A ∞ -comodule structure on T ′ = Tor • A ( M , K ) over T. The structure of the A ∞ -algebra E = E x t A • ( K , K ) and the corresponding A ∞ -module on E ′ = E x t A • ( M , K ) are just obtained by taking the bigraded dual. In this article we prove that there is partial description of the A ∞ -comodule T ′ over T and of the structure of A ∞ -module E ′ over E, similar to and also generalizing the partial description of the A ∞ -algebra structure on E given by Keller's higher-multiplication theorem in [19] . We also provide a criterion to check if a given A ∞ -comodule structure on T ′ is a model by regarding if the associated twisted tensor product is a minimal projective resolution of M, analogous to a theorem of B. Keller explained by the author of this article in [9] . Finally, we give an application of this result by computing the A ∞ -module structure on E ′ for any generalized Koszul algebra A and any generalized Koszul module M.

Let D be a directed set without maximal element, S be an infinite semigroup and D S be the collection of all functions from D into S . It is shown that for a commutative semigroup S , A ⊆ S is a C -set with respect to N S if and only if A is a C -set with respect to D S . We investigate the Central Sets Theorem for arbitrary semigroups. In fact the Central Sets Theorem is stated with respect to S S for arbitrary semigroups.

We introduce a model of noncommutative geometry that gives rise to the uncertainty relations recently derived from the discussion of a quantum clock. We investigate the dynamics of a free particle in this model from the point of view of doubly special relativity and discuss the geodesic motion in a Schwarzschild background., 7 pages, version accepted for publication on Phys. Lett. A. A discussion of the motion of a particle in a Schwarzschild background has been added

We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space., Comment: 18 pages

Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions., Comment: Minor changes. 26 pages

Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l. In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.

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.

We investigate scalar condensations around noncommutative compact reflecting stars. We find that the neutral noncommutative reflecting star cannot support the existence of scalar field hairs. In the charged noncommutative reflecting star spacetime, we provide upper bounds for star radii. Above the bound, scalar fields cannot exist outside the star. In contrast, when the star radius is below the bound, we show that the scalar field can condense. We also obtain the largest radii of scalar hairy reflecting stars., Comment: 12 pages, 4 figures

For a finite group acting on a polynomial ring, the Chevalley–Shephard–Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was made in work of Kirkman, Kuzmanovich, Zhang, and others to extend this result to Artin–Schelter regular algebras by expanding pseudo-reflections to quasi-reflections. Naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. Our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended Dynkin diagrams of type A, D or E. This work answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. On our way, we will point out additional difficulties in establishing quasi-reflections using the trace and reveal situations which do not occur for Artin–Schelter regular algebras.

Let A be a commutative ring, and let \a = \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and \a-adic completion functors to be nicely behaved is the weak proregularity of \a. In particular, the MGM Equivalence holds. Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem is that in the commutative case, the ideal \a is weakly proregular if and only if the corresponding torsion class T_{\a} is weakly stable. We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: - For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. - The Noncommutative MGM Equivalence, under the same assumptions on T. - A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli-Zhang from 2003., Comment: 49 pages. Final version, a few minor improvements, to appear in J. Algebra

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.

We provide a unified approach to Fraïssé limits in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogs. We obtain in this general framework many known and new results about the Gurarij space and the Poulsen simplex, and at the same time establish their noncommutative analogs. Particularly, we construct noncommutative analogs of universal operators in the sense of Rota.

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformations of that scheme and then expanding this result to $\mathbb{C}^*$-equivariant deformations over schemes with a good $\mathbb{C}^*$-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the $\mathbb{C}^*$-equivariant case. We apply our results to rational surface singularities, generalising the deformed preprojective algebras, and also to symplectic situations where the deformations produced are related to symplectic reflection algebras., Comment: 30 pages

The present paper is devoted to classification of multinormed W ⁎ -algebras in terms of their bounded parts. We obtain a precise description of the bornological predual of a multinormed W ⁎ -algebra, which is reduced to a multinormed noncommutative L 1 -space in the semifinite case. A multinormed noncommutative L 2 -space is obtained as the union space of a commutative domain in a semifinite von Neumann algebra. Multinormed W ⁎ algebras of type I are described as locally bounded decomposable (unbounded) operators associated with a measurable covering.

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 L p -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.

We revisit and generalize the application of a method introduced by Latr\'emoli\`ere and Packer for constructing finitely generated projective modules over the noncommutative solenoid C*-algebras. By realizing them as direct limits of rotation algebras, the method constructs directed systems of equivalence bimodules between rotation algebras that satisfy the necessary compatibility conditions to build Morita equivalence bimodules between the direct limit C*-algebras. In the irrational case, we use a fixed projection in a matrix algebra over the rotation algebra satisfying a key condition to build an equivalence bimodule at each stage following a construction of Rieffel. From this, our main result shows that two irrational noncommutative solenoids are Morita equivalent if and only if such a projection exists. We also make additional observations about the Heisenberg bimodules construction studied by the aforementioned two authors and connect the two constructions., Comment: 24 pages

This paper discusses octonion-valued neural networks (OVNNs) with leakage delay, time-varying delays, and distributed delays, for which the states, weights, and activation functions belong to the normed division algebra of octonions. The octonion algebra is a nonassociative and noncommutative generalization of the complex and quaternion algebras, but does not belong to the category of Clifford algebras, which are associative. In order to avoid the nonassociativity of the octonion algebra and also the noncommutativity of the quaternion algebra, the Cayley–Dickson construction is used to decompose the OVNNs into 4 complex-valued systems. By using appropriate Lyapunov–Krasovskii functionals, with double and triple integral terms, the free weighting matrix method, and simple and double integral Jensen inequalities, delay-dependent criteria are established for the exponential stability of the considered OVNNs. The criteria are given in terms of complex-valued linear matrix inequalities, for two types of Lipschitz conditions which are assumed to be satisfied by the octonion-valued activation functions. Finally, two numerical examples illustrate the feasibility, effectiveness, and correctness of the theoretical results.

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 discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative n -torus T θ n in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups G θ = ( A θ n , Δ ) such that for each θ , G θ is the final object in the category of all compact quantum groups acting on T θ n in a filtration preserving way. We describe in detail the structure of the C*-algebra A θ n and provide a concrete example of its representation in bounded operators. Moreover, we show that A θ n is isomorphic to the C*-completed version of an algebra of multiple noncommutative torus. Finally, we compute the Haar measure of G θ and discuss its representation theory. For θ = 0 , the quantum group G 0 is nothing but the classical group T n ⋊ S n , where S n is the symmetric group.