A characterisation of full matrix algebras

.

It is shown that (except for the case where F is the field of two elements) the only finite dimensional algebras over a field F which have the propertv that every non invertible element Can be expressed as a product of idempotents, are the full matrix algebras.     

Jordan derivations of full matrix algebras

Let A be a unital associative ring and M be a 2-torsion free A-bimodule. Using an elementary and constructive method we show that every Jordan derivation from M_n(A) into M_n(M) is a derivation.     

Nonlinear lie-type derivations on full matrix algebras

Let ${\mathcal{R}}$ be a 2-torsion free commutative ring with identity and ${{\rm M}_n(\mathcal{R}) (n\geq 2)}$ be the full matrix algebra over ${\mathcal{R}}$ . In this note, we prove that every nonlinear Lie triple derivation on ${{\rm M}_n(\mathcal{R})}$ is of the standard form, i.e. it can be expressed as a sum of an inner derivation, an additive induced derivation and a functional annihilating all second commutators of ${{\rm M}_n(\mathcal{R})}$ . A open conjecture about Lie n-derivations is posed at the end of this note.     

A geometric criterion for generating the Fukaya category

Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies in the idempotent closure of the chosen collection. The main new ingredients are (1) the construction of operations on the Fukaya category controlled by discs with two outputs, and (2) the Cardy relation.     

The minimal dimension of maximal commutative subalgebras of full matrix algebras

Let F be a field and A a maximal commutative subalgebra of the full matrix algebra M_n(F). It is shown that dim A > (2n)^{$\text{2}\text{3}$} ? 1. It is also shown that if the radical of A has cube zero, then dim A ? [3n^{$\text{2}\text{3}$} ? 4], and that this result is best possible for infinitely many natural numbers n.     

A criterion on the semisimple Brauer algebras

In this paper, we give a necessary and sufficient condition for a Brauer algebra to be semisimple.     

Supersoluble crossed product criterion for division algebras

LetD be a finite-dimensionalF-central division algebra. A criterion is given forD to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroups of the multiplicative groupD* ofD. More precisely, it is shown thatD is a supersoluble (nilpotent) crossed product if and only ifD* contains an abelian-by-supersoluble (abelian-by-nilpotent) generating subgroup.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

一类无限矩阵拓扑代数

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ^β and ,μ^β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.     

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from . The convergence features of these PCG have been naturally studied by means of the Weierstrass–Jackson Theorem [17,36,45], owing to the profound relationship between the spectral features of the matrices , generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f. On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Received October 1, 1996 / Revised version received May 7, 1998     

A linear matrix equation: a criterion for block similarity

In this paper we study a homogeneous linear matrix equation related to the block similarity of rectangular matrices. We obtain the dimension of the vector space of its solutions and we describe these solutions. We give a characterization of the block similarity by rank tests. We extend Roth's criterion to the corresponding non homogeneous equation.