The results on Lie homomorphisms of associative algebras are extended to certain associative superalgebras. It is shown that under appropriate conditions a Lie superautomorphism is a sum of a superautomorphism or the negative of a superantiautomorphism and a central map. In particular we consider the situation when A is a central simple algebra and the grading is induced by an idempotent.

COBISS.SI-ID: 15403609

The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other resultsit is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński's problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.

COBISS.SI-ID: 15583065

Wedderburn's theorem on the structure of finite dimensional (semi)simple algebras is proved by using minimal prerequisites.

COBISS.SI-ID: 15382617

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.

COBISS.SI-ID: 15672665

We introduce a new general technique for solving linear preserver problems. The idea is to localize a given linear preserver F at each non-zero vector. In such a way we get vector-valued linear maps on the space of matrices which inherit certain properties from F. If we can prove that such induced maps have a standard form, then F itself has either a standard form or a very special degenerate form. We apply this technique to characterize linear preservers of full rank.

COBISS.SI-ID: 15743577