Multiplier Algebra Research Articles

A non-newtonian approach to geometric phase through optic fiber via multiplicative quaternions

In this paper, we researched magnetic and electromagnetic curves in multiplicative Euclidean 3-space via multiplicative quaternion algebra. Firstly, we examined the geometric phase representation of the polarized light wave in the optic fiber by multiplicative Frenet frame. Using the quaternionic approaches, we were able to derive the magnetic curve equations and theorems. Then, three particular instances have been illustrated with examples of electromagnetic curves and magnetic field equations. Lastly, we provided an interpretation of the findings. With the help of the results, we were able to present an alternative viewpoint on the construction of trajectories (such as circular or spiral-like ones) that do not exist uniquely in the realm of physics.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Lie nilpotency and Lie solvability of tensor product of multiplicative Lie algebras

In this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. As our main results, we provide upper bounds on the Lie nilpotency class and Lie solvability length of quotient of tensor product of multiplicative Lie algebras.

Complex Structure That Admits Complete Nevanlinna–Pick Spaces of Hardy Type

Abstract In this paper, we will characterize those sets, over which every irreducible complete Nevanlinna–Pick space enjoys that its multiplier and supremum norms coincide. Moreover, we will prove that, if there exists an irreducible complete Nevanlinna–Pick space of holomorphic functions on a reduced complex space $X$ whose multiplier algebra is isometrically equal to the algebra of bounded holomorphic functions (we will say that such a space is of Hardy type in this paper), then $X$ must be biholomorphic to the unit disk minus a zero analytic capacity set. This means that the Hardy space is characterized as a unique irreducible complete Nevanlinna–Pick space of Hardy type.

Sections of Fell bundles over étale groupoids

We construct the reduced and essential C*-algebra of a Fell bundle over an étale groupoid (in full generality, without any second countability, local compactness or Hausdorff assumptions, even on the unit space) directly from sections under convolution. This eliminates the j-map commonly seen in the groupoid and Fell bundle C*-algebra literature. We further show how multiplier algebras and other Banach *-bimodules can again be constructed directly from sections. Finally, we extend Varela's morphisms from C*-bundles to Fell bundles, thus making the reduced C*-algebra construction functorial.

On Inner Derivations of Leibniz Algebras

Leibniz algebras are generalizations of Lie algebras. Similar to Lie algebras, inner derivations play a crucial role in characterizing complete Leibniz algebras. In this work, we demonstrate that the algebra of inner derivations of a Leibniz algebra can be decomposed into the sum of the algebra of left multiplications and a certain ideal. Furthermore, we show that the quotient of the algebra of derivations of the Leibniz algebra by this ideal yields a complete Lie algebra. Our results independently establish that any derivation of a semisimple Leibniz algebra can be expressed as a combination of three derivations. Additionally, we compare the properties of the algebra of inner derivations of Leibniz algebras with the algebra of central derivations.

Nilpotency and capability in multiplicative Lie algebras

This paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect multiplicative Lie algebras.

Bimultipliers of R-algebroids

Group action is determined bythe automorphism group and algebra action is defined by the multiplication algebra. In the study we generalize the multiplication algebra
 by defining multipliers of an R-algebroid M. Firstly, the set of bimultipliers on an R-algebroid is introduced, it is denoted by Bim(M), then it is proved that this set is an R-algebroid,
 it is called multiplication R-algebroid. Using this Bim(M), for an R-algebroid morphism A → Bim(M) it is shown that this morphism gives an R-algebroid action. Then we examine
 some of the properties associated with this action.

Criteria for Davies irreducibility of Markovian quantum dynamics

The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e. the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e. they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators La and the operator iH+∑aLa†La , involving the Hamiltonian H, is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.

Goldie dimension for C*-algebras

AbstractIn this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

Nowhere scattered multiplier algebras

We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$ -algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$ , that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$ -unital $\mathrm {C}^*$ -algebra $A$ of (i) finite nuclear dimension, or (ii) real rank zero, or (iii) stable rank one with $k$ -comparison, is nowhere scattered if and only if $\mathcal {M}(A)$ is.

Obstacle to Learning Algebra in Elementary Schools

This research is based on the results of a preliminary study which identified students' learning barriers in algebraic multiplication and division material in elementary schools. Students do not fully understand the concept of algebraic multiplication and division due to learning obstacles related to limited didactic designs made by teachers. Learning obstacles experienced by students must be anticipated through a learning process that is able to overcome these problems. The researcher developed a didactic learning design using a Realistic Mathematics Education (PMR) approach. The aim of this research is to determine the learning obstacles experienced by students, develop didactical designs for algebraic multiplication and division, describe, test, and describe the designs that have been created. The research method used in this research is Didactical Design Research (DDR). This method consists of three stages, namely didactic situation analysis before learning in the form of Hypothetical Learning Trajectory (HLT) and Pedagogical Didactic Anticipation (ADP), metapedadidactic analysis and retrospective analysis. The data collection technique used is triangulation which combines observation, interviews and documentation. The design development process was carried out in class V with a total of 18 students, and the preliminary study stage was carried out in class IV at SDN 1 Sukamanah with a total of 22 students, and class IV at SDN Sukamulya with a total of 16 students. This research produces data regarding student barriers to multiplication and division material, didactical design in the form of teaching materials, student worksheets (LKPD), to minimize students' learning barriers to multiplication and division material in algebra.

Convergence of the Multiplicative Algebraic Reconstruction Technique for the Inconsistent System of Equations

Convergence of the Multiplicative Algebraic Reconstruction Technique for the Inconsistent System of Equations

Framed cohomological Hall algebras and cohomological stable envelopes

There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver Q to the Yangian Y^{Q}_textrm{MO} by Maulik–Okounkov, whose construction is based on the notion of stable envelopes of Nakajima varieties. In this article, we introduce the cohomological Hall algebra of the moduli stack of framed representations of a quiver Q (framed CoHA), and we show that the equivariant cohomology of the disjoint union of the Nakajima varieties {mathcal {M}}_Q(text {v},text {w}) for all dimension vectors text {v} and framing vectors text {w} has a canonical structure of subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map. As a corollary, we deduce an explicit inductive formula to compute stable envelopes in terms of tautological classes.

On the strict topology of the multipliers of a JB^*-algebra

We introduce the Jordan-strict topology on the multiplier algebra of a JB^*-algebra, a notion which was missing despite the forty years passed after the first studies on Jordan multipliers. In case that a C^*-algebra A is regarded as a JB^*-algebra, the J-strict topology of M(A) is precisely the well-studied C^*-strict topology. We prove that every JB^*-algebra {mathfrak {A}} is J-strict dense in its multiplier algebra M({mathfrak {A}}), and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB^*-algebras admit J-strict continuous extensions to the corresponding type of operators between the multiplier algebras. We characterize J-strict continuous functionals on the multiplier algebra of a JB^*-algebra {mathfrak {A}}, and we establish that the dual of M({mathfrak {A}}) with respect to the J-strict topology is isometrically isomorphic to {mathfrak {A}}^*. We also present a first application of the J-strict topology of the multiplier algebra, by showing that under the extra hypothesis that {mathfrak {A}} and {mathfrak {B}} are sigma -unital JB^*-algebras, every surjective Jordan ^*-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from {mathfrak {A}} onto {mathfrak {B}} admits an extension to a surjective J-strict continuous Jordan ^*-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from M({mathfrak {A}}) onto M({mathfrak {B}}).

Comparing the Effects of Algebra Tiles and Multiplication Frame on Basic Nine Pupils Performance in Binomial Expansion and Factorization of Trinomials

Aim: This study was conducted primarily to compare the effectiveness of two known teaching strategies used in teaching expansion of two linear binomials and factorization of quadratic trinomials. The study employed a quasi-experimental design.
 Methods: The study was conducted in three junior high schools in the Upper East Region of Ghana, using 113 pupils as a sample size. A test was used to collect data for this study, and the data was analyzed using analysis of variance (ANOVA).
 Results: It was found that the post-test mean score of pupils taught with the multiplication frame was significantly different from the post-test mean score of pupils taught with the algebra tiles and the conventional method, but there was however no significant difference in the mean scores of pupils taught with the algebra tiles and those taught with the conventional approach.
 Conclusion: The study concluded that the multiplication frame/box method was more effective than the use of algebra tiles.
 Recommendation: The study recommend multiplication frame/box method to be the choice in teaching expansion of two linear binomials and factorization of quadratic trinomials for maximum gains.

TAMM: Tensor algebra for many-body methods.

Tensor algebra operations such as contractions in computational chemistry consume a significant fraction of the computing time on large-scale computing platforms. The widespread use of tensor contractions between large multi-dimensional tensors in describing electronic structure theory has motivated the development of multiple tensor algebra frameworks targeting heterogeneous computing platforms. In this paper, we present Tensor Algebra for Many-body Methods (TAMM), a framework for productive and performance-portable development of scalable computational chemistry methods. TAMM decouples the specification of the computation from the execution of these operations on available high-performance computing systems. With this design choice, the scientific application developers (domain scientists) can focus on the algorithmic requirements using the tensor algebra interface provided by TAMM, whereas high-performance computing developers can direct their attention to various optimizations on the underlying constructs, such as efficient data distribution, optimized scheduling algorithms, and efficient use of intra-node resources (e.g., graphics processing units). The modular structure of TAMM allows it to support different hardware architectures and incorporate new algorithmic advances. We describe the TAMM framework and our approach to the sustainable development of scalable ground- and excited-state electronic structure methods. We present case studies highlighting the ease of use, including the performance and productivity gains compared to other frameworks.

The corona theorem for bounded analytic functions in Q K type spaces

ABSTRACT For 1 < p < ∞ and a nondecreasing function K on [ 0 , ∞ ) , we give an explicit derivative-free characterization of Q K type space, denoted by Q K p and characterize the boundary version of Q K p . By using these results, we prove the corona theorems both for Q K p ∩ H ∞ and for the multiplier algebra of Q K p under some assumptions of the weight function K.

Calabi–Yau structures for multiplicative preprojective algebras

In this paper, we deal with Calabi–Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.

A Local Douglas formula for Higher Order Weighted Dirichlet-Type Integrals

We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $${\mathcal {H}}_{\pmb \mu },$$ induced by an m-tuple $$\pmb \mu =(\mu _1,\ldots ,\mu _{m})$$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order m, for $$m\geqslant 3,$$ forms an algebra under pointwise product. We also prove that every non-zero closed $$M_z$$ -invariant subspace of $${\mathcal {H}}_{\pmb \mu },$$ has codimension 1 property if $$m\geqslant 3$$ or $$\mu _2$$ is finitely supported. As another application of this local Douglas formula obtained in this article, it is shown that for any $$m\geqslant 2,$$ weighted Dirichlet-type space of order m does not coincide with any de Branges–Rovnyak space $${\mathcal {H}}(b)$$ with equivalence of norms.