192 research outputs found
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 (where and is an algebraically
closed field of characteristic 0). In a more general context where is a
prime algebra satisfying certain technical restrictions, we establish a density
theorem for the associative algebra generated by all inner derivations of .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 and of a Banach
algebra that satisfy \s(g(x)) \subseteq \s(d(x)) for every ,
where \s(\, . \,) stands for the spectrum. It turns out that in some basic
situations, say if , the only possibilities are that , , and,
if is an inner derivation implemented by an algebraic element of degree 2,
also . 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 -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
for all selfadjoint elements from a
-algebra , where and 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 is said to be a zero Jordan product determined Banach
algebra if every continuous bilinear map , where
is an arbitrary Banach space, which satisfies whenever
, are such that , is of the form
for some continuous linear map . We show
that all -algebras and all group algebras 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 be a multilinear polynomial over a field . An
-algebra is said to be -zpd (-zero product determined) if every
-linear functional which preserves zeros
of is of the form for some
linear functional on . We are primarily interested in the question
whether the matrix algebra is -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
with the property that every zero of in
is a zero of . Under the assumption that , we show that and
are linearly dependent
Maps preserving zeros of a polynomial
Let \A be an algebra and let be a multilinear polynomial
in noncommuting indeterminates . We consider the problem of describing
linear maps \phi:\A\to \A that preserve zeros of . Under certain technical
restrictions we solve the problem for general polynomials in the case where
\A=M_n(F). We also consider quite general algebras \A, but only for
specific polynomials .Comment: 11 pages, accepted for publication in Linear Algebra App
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