Multiplier Algebra Research Articles

THE STATISTICAL MULTIPLICATIVE ORDER CONVERGENCE IN RIESZ ALGEBRAS

The statistically multiplicative convergence in Riesz algebras was studied and investigated with respect to the solid topology. In the present paper, the statistical convergence with the multiplication in Riesz algebras is introduced by developing topology-free techniques using the order convergence in vector lattices. Moreover, we give some relations with the other kinds of convergences such as the order statistical convergence, the $mo$-convergence, and the order convergence.

Two lives: Compositions of unimodular rows

The paper lays the foundation for the study of unimodular rows using Spin groups. We show that elementary orbits of unimodular rows (of any length n≥3) are equivalent to elementary Spin orbits on the unit sphere. (This bijection is proved over all commutative rings.) In the special case n=3, this leads to an interpretation of the Vaserstein symbol using Spin groups.In addition, we introduce a new composition law that operates on certain subspaces of the underlying quadratic space (using the multiplication in composition algebras). In particular, the special case of split-quaternions leads to the composition of unimodular rows (discovered by L. Vaserstein and later generalized by W. van der Kallen). Strikingly, with this approach, we now see the possibility of new orbit structures not only for unimodular rows (using octonion multiplication) but also for more general quadratic spaces.

An All-in-One Geometric Algorithm for Cutting, Tearing, and Drilling Deformable Models

Conformal geometric algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, and deformations, such as translations, rotations and dilations as uniform vectors, called multivectors. In this work, we demonstrate the merits of multivector usage with a novel, integrated rigged character simulation framework based on CGA. In such a framework, and for the first time, one may perform real-time cuts and tears as well as drill holes on a rigged 3D model. These operations can be performed before and/or after model animation, while maintaining deformation topology. Moreover, our framework permits generation of intermediate keyframes on-the-fly based on user input, apart from the frames provided in the model data. We are motivated to use CGA as it is the lowest-dimension extension of dual-quaternion algebra that amends the shortcomings of the majority of existing animation and deformation techniques. Specifically, we no longer need to maintain objects of multiple algebras and constantly transmute between them, such as matrices, quaternions and dual-quaternions, and we can effortlessly apply dilations. Using such an all-in-one geometric framework allows for better maintenance and optimization and enables easier interpolation and application of all native deformations. Furthermore, we present these three novel algorithms in a single CGA representation which enables cutting, tearing and drilling of the input rigged model, where the output model can be further re-deformed in interactive frame rates. These close to real-time cut,tear and drill algorithms can enable a new suite of applications, especially under the scope of a medical VR simulation.

Structure of simple multiplicative Hom-Jordan algebras

We study the structure of simple multiplicative Hom-Jordan algebras. We discuss equivalent conditions for multiplicative Hom-Jordan algebras to be solvable, simple, and semi-simple. Moreover, we give a theorem on the classification of simple multiplicative Hom-Jordan algebras and obtain some propositions about bimodules of multiplicative Hom-Jordan algebras.

On function spaces of Lorentz–Sobolev type

We work with Triebel–Lizorkin spaces \(F_{q}^{s}L_{p,r}({\mathbb {R}}^{n})\) and Besov spaces \(B_{q}^{s} L_{p,r}({\mathbb {R}}^{n})\) with Lorentz smoothness. Using their characterizations by real interpolation we show how to transfer a number of properties of the usual Triebel–Lizorkin and Besov spaces to the spaces with Lorentz smoothness. In particular, we give results on diffeomorphisms, extension operators, multipliers and we also show sufficient conditions on parameters for \(F_{q}^{s}L_{p,r}({\mathbb {R}}^{n})\) and \(B_{q}^{s}L_{p,r}({\mathbb {R}}^{n})\) to be multiplication algebras.

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are “close” to one another if and only if their multiplier algebras are “close”, and that this happens if and only if one of the underlying point sets is close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.

On Application of Max-Plus Algebra to Synchronized Discrete Event System

Max-plus algebra is a discrete algebraic system developed on the operations max (<img src=image/13420822_01.gif>) and plus (<img src=image/13420822_03.gif>), where the max and plus operations are defined as addition and multiplication in conventional algebra. This algebraic structure is a semi-ring with its elements being real numbers along with ε=-∞ and e=0. On the other hand, the synchronized discrete event problem is a problem in which an event is scheduled to meet a deadline. There are two aspects of this problem. They include the events running simultaneously and the completion of the lengthiest event at the deadline. A recent survey on max-plus linear algebra shows that the operations max (<img src=image/13420822_01.gif>) and plus (<img src=image/13420822_03.gif>) play a significant role in modeling of human activities. However, numerous studies have shown that there are very limited literatures on the application of the max-plus algebra to real-life problems. This idea motivates the basic algebraic results and techniques of this research. This paper proposed the discrepancy method of max-plus for solving n×n system of linear equations with n≤n, and further show that an nxn linear system of equations will have either a unique solution, an infinitely many solutions or no solution whiles nxn linear system of equations has either an infinitely many solutions or no solution in (<img src=image/13420822_02.gif>). Also, the proposed concept was extended to the job-shop problem in a synchronized event. The results obtained have shown that the method is very efficient for solving n×n system of linear equations and is also applicable to job-shop problems.

Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

Let \(\mathcal {M}_{X(\mathbb {R})}\) be the Banach algebra of all Fourier multipliers on a Banach function space \(X(\mathbb {R})\) such that the Hardy–Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). For two sets \(\varPsi ,\varOmega \subset \mathcal {M}_{X(\mathbb {R})}\), let \(\varPsi _\varOmega\) be the set of those \(c\in \varPsi\) for which there exists \(d\in \varOmega\) such that the multiplier norm of \(\chi _{\mathbb {R}\setminus [-N,N]}(c-d)\) tends to zero as \(N\rightarrow \infty\). In this case, we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if \(\varOmega\) is a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and \(\varPsi\) is an arbitrary unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\), then \(\varPsi _\varOmega\) is a also a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\).

Extensions of the UHF-algebra of 2∞-type by C⁎-algebras with continuous scale

Let A be the UHF-algebra of 2∞-type, B be a non-unital but σ-unital stably finite simple C⁎-algebra with real rank zero and a continuous scale. In this paper, we give a classification of the following essential extensions of A by B⊗A:0→B⊗A→E→A→0, up to unitary equivalence. Denote by Ext(A,B⊗A), the abelian group of unitary equivalence classes of essential extensions of A by B⊗A. There is a group isomorphismΓ:Ext(A,B⊗A)→KK(A,C(B⊗A)) defined by Γ([τ]):=[τ]KK, where C(B⊗A) is the corona algebra of B⊗A.

Study of Exploiting Coarse‐Grained Parallelism in Block‐Oriented Numerical Linear Algebra Routines

AbstractWe have developed streaming implementations of two numerical linear algebra operations that further exploit the block decomposition strategies commonly used in these operations to obtain performance. The implementations formulate algorithms as data flow graphs and use coarse‐grained parallelism to (1) emit a block in the result matrix as soon as it becomes available and (2) compute on multiple blocks in parallel. This streaming design benefits data flow graphs consisting of multiple linear algebra operations as it removes synchronization points between successive operations: a result block from an operation can be used immediately in an algorithm's successor operations without waiting for the full result from the first operation. Early comparisons with OpenBLAS functions on CPUs show comparable performance for computing with large dense matrices and an earliest arrival time of a result block that is up to 50x smaller than the time needed for a full result. More thorough studies can show the impact of such implementations on the performance of systems by chaining multiple linear algebra operations.

CONTINUITY IN FUNDAMENTAL LOCALLY MULTIPLICATIVE TOPOLOGICAL ALGEBRAS

Abstract. In this paper, we first derive specific results concerning the continuity and upper semi-continuity of the spectral radius and spectrum functions on fundamental locally multiplicative topological algebras. We continue our investigation by further determining the automatic continuity of linear mappings and homomorphisms in these algebras.

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

We study simplicity and pure infiniteness criteria for \mathrm{C}^* -algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of R. Exel and D. R. Pitts ["Characterizing groupoid \mathrm{C}^* -algebras of non-Hausdorff étale groupoids", Preprint (2019); [arXiv: 1901.09683](arXiv: 1901.09683)], we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced \mathrm{C}^* -algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic \mathrm{C}^* -inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.

On the unitary group of the multiplier algebra\cr of the Razak algebra

Let $\mathcal W $ be the Razak algebra, which is an example of a continuous scale ${\rm C}^*$-algebra with $K_*(\mathcal W ) = 0$ and unique trace. We show that the group $U(\mathcal M (\mathcal W ))$ is path connected in the norm topology.

Gelfand transforms and boundary representations of complete Nevanlinna–Pick quotients

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna–Pick spaces, examples of which include the Drury–Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson’s notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson–Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.

Equivariant one-parameter formal deformations of Hom-Leibniz algebras

The aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which control deformations of Hom-Leibniz algebras. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equivariant context, under the presence of finite group actions on Hom-Leibniz algebras.

Weak Multiplier Hopf Algebras II: Source and Target Algebras

Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).

A Morita Characterisation for Algebras and Spaces of Operators on Hilbert Spaces

We introduce the notion of $\Delta$ and $\sigma\,\Delta-$ pairs for operator algebras and characterise $\Delta-$ pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of $\Delta$-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that $\sigma\Delta$-Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to $\Delta$-Morita equivalence of their algebraic extensions.

A Larson-Sweedler theorem for Hopf [formula omitted]-categories

The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the ‘many-object’ setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.

Row Contractions Annihilated by Interpolating Vanishing Ideals

Abstract We study similarity classes of commuting row contractions annihilated by what we call higher-order vanishing ideals of interpolating sequences. Our main result exhibits a Jordan-type direct sum decomposition for these row contractions. We illustrate how the family of ideals to which our theorem applies is very rich, especially in several variables. We also give two applications of the main result. First, we obtain a purely operator theoretic characterization of interpolating sequences for the multiplier algebra of the Drury–Arveson space. Second, we classify certain classes of cyclic commuting row contractions up to quasi-similarity in terms of their annihilating ideals. This refines some of our recent work on the topic. We show how this classification is sharp: in general quasi-similarity cannot be improved to similarity. The obstruction to doing so is a scarcity of norm-controlled similarities between commuting tuples of nilpotent matrices, and we investigate this question in detail.

Forcing axioms and coronas of C∗-algebras

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.