scholarly journals Unital locally matrix algebras and Steinitz numbers

2019 ◽  
Vol 19 (09) ◽  
pp. 2050180
Author(s):  
Oksana Bezushchak ◽  
Bogdana Oliynyk
Keyword(s):  
Matrix Algebra ◽  
Finite Collection ◽  
Matrix Algebras ◽  
Number Formula ◽  
Unit Formula

An [Formula: see text]-algebra [Formula: see text] with unit [Formula: see text] is said to be a locally matrix algebra if an arbitrary finite collection of elements [Formula: see text] from [Formula: see text] lies in a subalgebra [Formula: see text] with [Formula: see text] of the algebra [Formula: see text], that is isomorphic to a matrix algebra [Formula: see text], [Formula: see text]. To an arbitrary unital locally matrix algebra [Formula: see text], we assign a Steinitz number [Formula: see text] and study a relationship between [Formula: see text] and [Formula: see text].

scholarly journals Primary decompositions of unital locally matrix algebras

2020 ◽  
Vol 10 (01) ◽  
pp. 2050006
Author(s):  
Oksana Bezushchak ◽  
Bogdana Oliynyk
Keyword(s):  
Tensor Product ◽  
Matrix Algebra ◽  
Primary Decomposition ◽  
Locally Finite ◽  
Matrix Algebras ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Number Formula

We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number [Formula: see text] there exists a unital locally matrix algebra [Formula: see text] having the Steinitz number [Formula: see text] and not isomorphic to a tensor product of finite-dimensional matrix algebras.

scholarly journals On Modular Representation Algebras and a Class of Matrix Algebras

1982 ◽  
Vol 33 (3) ◽  
pp. 351-355 ◽  
Author(s):  
J-C. Renaud
Keyword(s):  
Tensor Product ◽  
Cyclic Group ◽  
Prime Order ◽  
Matrix Algebra ◽  
Modular Representation ◽  
Complex Field ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Standard Operations

AbstractLet G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

scholarly journals Zero Triple Product Determined Matrix Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Hongmei Yao ◽  
Baodong Zheng
Keyword(s):  
Linear Operator ◽  
Matrix Algebra ◽  
Triple Product ◽  
Unital Algebra ◽  
Matrix Algebras ◽  
Unital Ring ◽  
Jordan Triple Product

LetAbe an algebra over a commutative unital ringC. We say thatAis zero triple product determined if for everyC-moduleXand every trilinear map{⋅,⋅,⋅}, the following holds: if{x,y,z}=0wheneverxyz=0, then there exists aC-linear operatorT:A3⟶Xsuch thatx,y,z=T(xyz)for allx,y,z∈A. If the ordinary triple product in the aforementioned definition is replaced by Jordan triple product, thenAis called zero Jordan triple product determined. This paper mainly shows that matrix algebraMn(B),n≥3, whereBis any commutative unital algebra even different from the above mentioned commutative unital algebraC, is always zero triple product determined, andMn(F),n≥3, whereFis any field with chF≠2, is also zero Jordan triple product determined.

scholarly journals σ-derivations on generalized matrix algebras

2020 ◽  
Vol 28 (2) ◽  
pp. 115-135
Author(s):  
Aisha Jabeen ◽  
Mohammad Ashraf ◽  
Musheer Ahmad
Keyword(s):  
Commutative Ring ◽  
Matrix Algebra ◽  
Morita Context ◽  
Matrix Algebras ◽  
Generalized Matrix ◽  
Generalized Matrix Algebra

AbstractLet 𝒭 be a commutative ring with unity, 𝒜, 𝒝 be 𝒭-algebras, 𝒨 be (𝒜, 𝒝)-bimodule and 𝒩 be (𝒝, 𝒜)-bimodule. The 𝒭-algebra 𝒢 = 𝒢(𝒜, 𝒨, 𝒩, 𝒝) is a generalized matrix algebra defined by the Morita context (𝒜, 𝒝, 𝒨, 𝒩, ξ𝒨𝒩, Ω𝒩𝒨). In this article, we study Jordan σ-derivations on generalized matrix algebras.

INDUCTION OF SEMIGROUPS OF ENDOMORPHISMS OF ${\mathfrak B} ({\mathfrak h})$ FROM COMPLETELY POSITIVE SEMIGROUPS OF (n × n) MATRIX ALGEBRAS

1999 ◽  
Vol 10 (07) ◽  
pp. 773-790 ◽  
Author(s):  
ROBERT T. POWERS
Keyword(s):  
Matrix Algebra ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Matrix Algebras ◽  
Completely Positive Semigroups

The paper concerns Eo-semigroup of [Formula: see text] induced from unit preserving completely positive semigroups of mapping of an (n × n) matrix algebra into itself. It is shown that Eo-semigroups one obtains are completely spatial and the index of the induced semigroup can be easily computed.

scholarly journals Jordan automorphisms, Jordan derivations of generalized triangular matrix algebra

2005 ◽  
Vol 2005 (13) ◽  
pp. 2125-2132 ◽  
Author(s):  
Aiat Hadj Ahmed Driss ◽  
Ben Yakoub l'Moufadal
Keyword(s):  
Matrix Algebra ◽  
Triangular Matrix ◽  
Jordan Derivation ◽  
Matrix Algebras ◽  
Triangular Matrix Algebra

We investigate Jordan automorphisms and Jordan derivations of a class of algebras called generalized triangular matrix algebras. We prove that any Jordan automorphism on such an algebra is either an automorphism or an antiautomorphism and any Jordan derivation on such an algebra is a derivation.

scholarly journals On the Lie structure of locally matrix algebras

2020 ◽  
Vol 12 (2) ◽  
pp. 311-316
Author(s):  
O. Bezushchak
Keyword(s):  
Lie Algebra ◽  
Matrix Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Matrix Algebras ◽  
Necessary And Sufficient ◽  
Algebra Over A Field

Let $A$ be a unital locally matrix algebra over a field $\mathbb{F}$ of characteristic different from $2.$ We find a necessary and sufficient condition for the Lie algebra $A\diagup\mathbb{F}\cdot 1$ to be simple and for the Lie algebra of derivations $\text{Der}(A)$ to be topologically simple. The condition depends on the Steinitz number of $A$ only.

scholarly journals Nonlinear Jordan Derivable Mappings of Generalized Matrix Algebras by Lie Product Square-Zero Elements

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Xiuhai Fei ◽  
Haifang Zhang
Keyword(s):  
Matrix Algebra ◽  
Matrix Algebras ◽  
Generalized Matrix ◽  
Generalized Matrix Algebra

The aim of the paper is to give a description of nonlinear Jordan derivable mappings of a certain class of generalized matrix algebras by Lie product square-zero elements. We prove that under certain conditions, a nonlinear Jordan derivable mapping Δ of a generalized matrix algebra by Lie product square-zero elements is a sum of an additive derivation δ and an additive antiderivation f . Moreover, δ and f are uniquely determined.

Group Gradings on Matrix Algebras

2002 ◽  
Vol 45 (4) ◽  
pp. 499-508 ◽  
Author(s):  
Yu. A. Bahturin ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Characteristic Zero ◽  
Matrix Algebra ◽  
Closed Field ◽  
Induction Procedure ◽  
Full Matrix ◽  
Full Matrix Algebra ◽  
Matrix Algebras ◽  
Group Gradings

AbstractLet Φ be an algebraically closed field of characteristic zero, G a finite, not necessarily abelian, group. Given a G-grading on the full matrix algebra A = Mn(Φ), we decompose A as the tensor product of graded subalgebras A = B ⊗ C, B ≅ Mp(Φ) being a graded division algebra, while the grading of C ≅ Mq(Φ) is determined by that of the vector space Φn. Now the grading of A is recovered from those of A and B using a canonical “induction” procedure.

Facial Structures for the Positive Linear Maps Between Matrix Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 74-82 ◽  
Author(s):  
Seung-Hyeok Kye
Keyword(s):  
Complete Lattice ◽  
Matrix Algebra ◽  
Convex Set ◽  
Matrix Algebras ◽  
Linear Maps ◽  
The Matrix ◽  
Positive Linear Maps

AbstractLet denote the convex set of all positive linear maps from the matrix algebra Mn(ℂ) into itself. We construct a join homomorphism from the complete lattice of all faces of into the complete lattice of all join homomorphisms between the lattice of all subspaces of ℂn . We also characterize all maximal faces of .

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