On Lie and associative algebras containing inner derivations

We describe subalgebras of the Lie algebra \mf{gl}(n^2) that contain all inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically closed field of characteristic 0). In a more general context where $A$ is a prime algebra satisfying certain technical restrictions, we establish a density theorem for the associative algebra generated by all inner derivations of $A$.Comment: 11 pages, accepted for publication in Linear Algebra App

Lie Superautomorphisms on Associative Algebras, II

Lie superautomorphisms of prime associative superalgebras are considered. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.Comment: 19 pages, accepted for publication in Algebr. Represent. Theor

Identifying derivations through the spectra of their values

We consider the relationship between derivations $d$ and $g$ of a Banach algebra $B$ that satisfy \s(g(x)) \subseteq \s(d(x)) for every $x\in B$, where \s(\, . \,) stands for the spectrum. It turns out that in some basic situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and, if $d$ is an inner derivation implemented by an algebraic element of degree 2, also $g=-d$. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general $C^*$-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition $\|[b,x]\|\leq M\|[a,x]\|$ for all selfadjoint elements $x$ from a $C^*$-algebra $B$, where $a,b\in B$ and $a$ is normal.Comment: 12 page

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.Comment: 8 page

Group gradings on finitary simple Lie algebras

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.Comment: Several typographical errors have been correcte

Zero Jordan product determined Banach algebras

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $\varphi\colon A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\varphi(a,b)=\sigma(ab+ba)$ for some continuous linear map $\sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications

On Herstein's Lie Map Conjectures, II

AbstractThe theory of functional identities is used to study derivations of Lie algebras arising from associative algebras. Definitive results are obtained modulo algebras of “low dimension.” In particular, Lie derivations of [K,K]/([K,K]∩Z), where K is the Lie algebra of skew elements of a prime algebra with involution and Z is its center, are described. This solves the last remaining open problem of Herstein on Lie derivations. For a simple algebra with involution the Lie algebra of all derivations of [K,K]/([K,K]∩Z) is thoroughly analyzed. Maps that act as derivations on arbitrary fixed polynomials are also discussed, and in particular a solution is given for Herstein's question concerning maps of K which act like a derivation on xm, m being a fixed odd integer

f-zpd algebras and a multilinear Nullstellensatz

Let $f=f(x_1,\dots,x_m)$ be a multilinear polynomial over a field $F$. An $F$-algebra $A$ is said to be $f$-zpd ($f$-zero product determined) if every $m$-linear functional $\varphi\colon A^{m}\rightarrow F$ which preserves zeros of $f$ is of the form $\varphi(a_1,\dots,a_m)=\tau(f(a_1,\dots,a_m))$ for some linear functional $\tau$ on $A$. We are primarily interested in the question whether the matrix algebra $M_d(F)$ is $f$-zpd. While the answer is negative in general, we provide several families of polynomials for which it is positive. We also consider a related problem on the form of a multilinear polynomial $g=g(x_1,\dots,x_m)$ with the property that every zero of $f$ in $M_d(F)^{m}$ is a zero of $g$. Under the assumption that $m<2d-3$, we show that $g$ and $f$ are linearly dependent

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201