Экономические науки/.
Математика 1. Дифференциальные и интегральные уравнения
к.ф.-м.н., доц.
Juraev I.M., преп. Sharipova I.F., преп.Juraev
F.M.
Bukhara
State University, Uzbekistan
Lie triple
derivations of algebras of measurable operators
Abstract
We
prove that every Lie triple derivation on algebras of measurable operators is
in standard form, that is, it can be uniquely decomposed into the sum of a
derivation and a center-valued trace.
Key words: von Neumann
algebras, measurable operator, type I von Neumann algebras, derivation, inner derivation,
Lie triple derivation, center-valued trace.
linear operator 
is called a derivation
if 
for all 
(Leibniz rule). Each
element 
defines a derivation 
on 
given as 
Such derivations 
are said to be inner
derivations. If the element 
implementing the
derivation 
on 
belongs to a larger
algebra 
containing 
(as a proper ideal as
usual) then 
is called a spatial
derivation.
A
linear operator L:A→A is called a Lie triple derivation if L[[x,y],z]=[[L(x),y],z]+[[x,L(y)],z]+[[x,y],L(z)]],
for all x,y,z∈A, where [x,y]=xy−yx.
Denote by Z(A)
the center of A.
A linear operator τ:A→Z(A) is called a
center-valued trace if τ(xy)=τ(yx), ∀x,y∈A.
Let H be a
Hilbert space, B(H) be the algebra of all bounded linear
operators acting in a H, M be a von Neumann subalgebra in B(H),
P(M) be a complete lattice of all ortoprojections in M.
A linear subspace D on H is said to be affiliated
with M (denoted as DηM), if u(D)⊆D for every unitary operator u
from the commutant 
of the algebra M.
A linear operator x
on H with the domain D(x) is said to be affiliated with M (denoted as xηM), if and ux(ξ)=xu(ξ) for every unitary
operator u∈M', and all ξ∈D(x).
A linear subspace D in H is said to be strongly
dens in H with respect to the von Neumann algebra M, if
1) DηM,
2) there exists a sequence of projections 
such that 
and 
is finite in 
for all 
where 
is the identity 
A closed linear operator x, on a H,
is said to be measurable with respect to the von Neumann algebra M,
if xηM, and D(x) is strongly dens in
H. Denote by S(M) the set of all measurable operators
affiliated with M (see. [2,3]) and the center of an algebra S(M)
by Z(S(M)). A von Neumann algebra M is of type I
if it contains a faithful abelian projection e (i.e. eMe is an
abelian(commutative) von Neumann algebra).
If
are projectors in 
then 
Set 
and 
Then 
Let further 
Recall that 
, for 
.
Theorem. Let L:S(M)→S(M)
be a Lie triple differentiation. Then 
, where 
is an associated
derivation and 
is a center-valued trace from 
into 
.
References
[1] S. Albeverio, Sh. A. Ayupov, K. K. Kudaybergenov, Structure
of derivations on various algebras of measurable operators for type I von
Neumann algebras, J. Func. Anal. 256 (2009), 2917-2943.
[2] M. A. Muratov and V. I. Chilin, Algebras of
measurable and locally measurable operators,-Kyiv, Pratsi In-ty matematiki
NAN Ukraini. 69 (2007), 390 pp. (Russian).
[3] C. Robert Miers, Lie triple derivations of von
Neumann algebras, Amer. Math. Soc. 71 (1978), 57-61.