42 research outputs found
Self-adjoint cyclically compact operators and their applications
This paper is devoted to self-adjoint cyclically compact operators on
Hilbert--Kaplansky module over a ring of bounded measurable functions. The
spectral theorem for such a class of operators are given. We apply this result
to partial integral equations on the space with mixed norm of measurable
functions and to compact operators relative to von Neumann algebras. We will
give a condition of solvability of partial integral equations with self-adjoint
kernel. Moreover, a general form of compact operators relative to a type I von
Neumann algebra is given.Comment: 10 page
Local And 2-Local Derivations On Algebras Of Measurable Operators
The present paper presents a survey of some recent results devoted to
derivations, local derivations and 2-local derivations on various algebras of
measurable operators affiliated with von Neumann algebras. We give a complete
description of derivation on these algebras, except the case where the von
Neumann algebra is of type II. In the latter case the result is obtained
under an extra condition of measure continuity of derivations. Local and
2-local derivations on the above algebras are also considered. We give
sufficient conditions on a von Neumann algebra , under which every local or
2-local derivation on the algebra of measurable operators affiliated with
is automatically becomes a derivation. We also give examples of commutative
algebras of measurable operators admitting local and 2-local derivations which
are not derivations.Comment: accepted to Contem. Math. AMS, 21 pages. arXiv admin note: text
overlap with arXiv:0901.2983, arXiv:0808.014
Local derivations on finite-dimensional Lie algebras
We prove that every local derivation on a finite-dimensional semisimple Lie
algebra over an algebraically closed field of characteristic zero is a
derivation. We also give examples of finite-dimensional nilpotent Lie algebras
with which admit local derivations which
are not derivations.Comment: 14 page
-Local derivations on von Neumann algebras
The paper is devoted to the description of -local derivations on von
Neumann algebras. Earlier it was proved that every -local derivation on a
semi-finite von Neumann algebra is a derivation. In this paper, using the
analogue of Gleason Theorem for signed measures, we extend this result to type
von Neumann algebras. This implies that on arbitrary von Neumann algebra
each -local derivation is a derivation.Comment: 7 page
Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras
This paper is devoted to derivations on the algebra of all measurable
operators affiliated with a finite von Neumann algebra We prove that if
is a finite von Neumann algebra with a faithful normal semi-finite trace
, equipped with the locally measure topology then every
-continuous derivation is inner. A similar result
is valid for derivation on the algebra of -measurable
operators equipped with the measure topology .Comment: Accepted for publication in the Journal of Mathematical Analysis and
Application
-local automorphisms on finite-dimensional Lie algebras
We prove that every -local automorphism on a finite-dimensional
semi-simple Lie algebra over an algebraically closed field of
characteristic zero is an automorphism. We also show that each
finite-dimensional nilpotent Lie algebra with admits a -local automorphism which is not an
automorphism.Comment: 8 page
On Levi-Malcev theorem for Leibniz algebras
The present paper is devoted to provide conditions for the Levi--Malcev
theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not
conjugate by an inner automorphism) in the context of finite-dimensional
Leibniz algebras over a field of characteristic zero. Particularly, in the case
of the field of complex numbers, we consider all possible cases in
which Levi subalgebras are conjugate and not conjugate.Comment: 11 page
Geometric Description of L-Spaces
We describe strongly facially symmetric spaces which are isometrically
isomorphic to L-space
Local derivations on measurable operators and commutativity
We prove that a von Neumann algebra is abelian if and only if the square
of every derivation on the algebra of measurable operators, affiliated
with , is a local derivation. We also show that for general associative
unital algebras this is not true.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1602.0495
Local and 2-local derivations and automorphisms on simple Leibniz algebras
The present paper is devoted to local and 2-local derivations and
automorphism of complex finite-dimensional simple Leibniz algebras. We prove
that all local derivations and 2-local derivations on a finite-dimensional
complex simple Leibniz algebra are automatically derivations. We show that
nilpotent Leibniz algebras as a rule admit local derivations and 2-local
derivations which are not derivations. Further we consider automorphisms of
simple Leibniz algebras. We prove that every 2-local automorphism on a complex
finite-dimensional simple Leibniz algebra is an automorphism and show that
nilpotent Leibniz algebras admit 2-local automorphisms which are not
automorphisms. A similar problem concerning local automorphism on simple
Leibniz algebras is reduced to the case of simple Lie algebras.Comment: 28 page