Mathematics / 1.Differential and integral equations
Arzikulov F., Mamatohunova Y.
Andijan state university, Uzbekistan
On purely real lie algebras of
skew-adjoint operators
ABSTRACT. In
the given paper we investigate a real von Neumann algebra 
on
a Hilbert space such that 
and a real Lie algebra 
of
skew-adjoint operators on a Hilbert space H such that 
for the real *-algebra 
, generated by 
in
. These algebras are called a purely real
von Neumann algebra and a purely real Lie algebra respectively.
An analog of Gelfand-Naumark theorem for ultraweakly
closed purely real Lie algebras of skew-adjoint operators on a Hilbert space is
proved. Also, it is proved that the enveloping C* -algebra of such
Lie algebra is a von Neumann algebra if this Lie algebra is reversible and it
is given a condition in which a purely real Lie algebra of skew-adjoint
operators is reversible.
Introduction
A weakly closed real *-subalgebra 
in B(H) is called a purely real von Neumann algebra if 
, where 
. Let 
be
the center of 
. If 
consists of real multiples of 1, i.e. 
, then R is said to be a purely real
factor. Given a purely real von Neumann algebra 
it
is easy to see that the set 
forms a Lie algebra with the multiplication 
. Such Lie algebras and purely real von
Neumann algebras were investigated, in particular, in papers 
by
Sh.Ayupov. In these papers it was considered a spacial case of a more general
algebraic problem: letting 
and 
be
simple or prime rings with involutions, can every (Lie) isomorphism of 
onto 
be
lifted to an (associative) isomorphism of 
onto 
? Here the derived Lie ring 
is
the additive span of all commutators 
,where 
; it is a Lie ideal in Lie ring 
.
The above problem was solved for simple rings with involution of the
first kind were obtained by Martindale 
and for arbitrary kind by Rosen 
. But in the case of prime rings it is not
possible in general to extend a Lie isomorphism between 
and 
to a *-isomorphism between whole 
and 
. Moreover, there are examples of prime
rings with involution such that 
and 
are Lie isomorphic but 
and 
are not *-isomorphic. Concerning purely real von Neumann algebras
one can see that every purely real von Neumann algebra is semiprime: it is
prime if (but not only if) it is a purely real factor. Examples of simple
purely real factors are given by factors of type 
(n < 
), 
, and à-finite type III [4].
In 
it
was proved that if 
and 
are real factors not of types 
and 
then derived Lie algebras
and 
are isomorphic if and only if 
and 
are *-isomorphic.
Also in the paper 
there were proved the following statements: let 
be
a purely real factor except types 
and 
. Then
1. 
2. 
, where Rk is the associative
subalgebra generated by 
.
In the given paper analogues of these statements are proved for real von
Neumann algebras 
, 
, satisfying the conditions 
, 
, where 
, 
are the real von Neumann subalgebras generated by 
, 
respectively.
In the given paper it is introduced weakly closed purely real Lie
algebras on a complex Hilbert space. These Lie algebras are Banach Lie algebras
over the field of real numbers. Banach Lie algebras are investigated in 
, 
. In this paper we give a representation
of all maximal purely real Lie algebras of skew-adjoint elements in the algebra
for a complex Hilbert space 
.
Also, it is proved that the enveloping C*-algebra of such Lie algebra is
a von Neumann algebra if this Lie algebra is reversible. Also, it is found a
condition in which case a purely real Lie algebra of skew-adjoint operators is
reversible.
1.
Maximal purely real Lie algebras of
skew-adjoint operators
In 
it
was proved the following theorem.
Theorem 1.1. Let H be a Hilbert space. Then any maximal
purely real *-algebra on
is
isomorphic to the algebra 
, where 
and 
are Hilbert subspaces of 
such that 
, the identity elements of the algebras 
and 
are mutually orthogonal and their sum is equal to the
identity element of 
, and 
and 
are Hilbert spaces on 
and 
respectively such that 
, where 
Definition 1.2. Let 
be
an Lie algebra of skew-adjoint operators on a Hilbert space 
. The Lie algebra L is called a purely real Lie algebra, if the real *-algebra 
, generated by 
in
satisfies to the condition 
(L) ∩ iR(L) = {0
.
By Zorn’s lemma for any purely real Lie algebra of skew-adjoint operators
on a Hilbert space there exists a maximal purely real Lie algebra of
skew-adjoint operators containing the given Lie algebra. Therefore the
following theorem takes place.
Theorem 1.3. Let H be a Hilbert space. Then any maximal
purely real Lie algebra of skew-adjoint operators on H is isomorphic to the Lie
algebra 
, where 
and 
are Hilbert subspaces of 
such that 
, the identity elements of 
and 
are mutually orthogonal and their sum is equal to the
identity element of 
, and 
, 
are Hilbert spaces on 
and 
respectively such that
, where 
.
Proof. Let
be
a maximal purely real Lie algebra of skew-adjoint operators on a Hilbert space 
and 
be
the real *-algebra, generated by 
. By the definition 
. Then there exists a maximal purely real
algebra 
on
containing the real algebra 
. Since 
is
maximal we have 
.
By theorem 1.1 
, where 
and HH are Hilbert subspaces of 
such that 
, the identity elements of 
and 
are mutually orthogonal and their sum is equal to the identity
element of 
, and 
and 
are Hilbert spaces on 
and 
respectively such that 
, where 
Hence 
. □
Definition
1.4. Let 
be
an Lie algebra of skew-adjoint operators on a Hilbert space 
. The Lie algebra 
is
called reversible, if for any elements 
the following conditions hold
,
.
Theorem 1.5. Let 
be an ultraweakly closed purely real reversible Lie
algebra of skew-adjoint operators on a Hilbert space 
, 
be a real von Neumann algebra, generated
by 
in 
. Then 
. Proof. It is evident that 
.
Let a 
. Then a* = —a and there exists a sequence
such
that 
weakly converges to a and 
where
,
for all 
. Since 
we
have that 
and
. Then
.
Note that in the last sum
if
is an even natural number for any index 
, and
if f
is an add natural number. Hence, for any
index 
by reversibility of the Lie algebra 
. Therefore 
. Since 
is
ultraweakly closed in 
we
have 
.□
Theorem 1.6. Let 
be
an ultraweakly closed reversible Lie algebra of skew- adjoint operators on a
Hilbert space 
, 
be a real von Neumann algebra, generated by 
in 
. Then the following conditions are
equivalen (1)
,
(2)there exist Lie subalgebras 
in
such that
, 
, and 
for some Hilbert spaces 
on 
and 
, respectively, such that 
, where 
.
Proof. 
: By theorem 1.3 
and by theorem 4 in 
there exist central projections 
and 
in
such that 
and 
. At the same time 
. Hence 
and 
, where 
: The converse statement of the theorem is
obvious. □
2.
Reversibility of purely real Lie algebras of
skew-adjoint
operators
Theorem 2.1. Let 
be a purely real Lie algebra of skew-adjoint operators
on a Hilbert space 
be a real von Neumann algebra, generated
by 
in 
. Suppose that 
can be embedded in 
and in this embedding 
and the unit of 
coincides with the unit of 
. Then 
is reversible.
Proof. By
the conditions of the theorem there exists a complete system 
of matrix units in 
where 
and 
, such that 
for all 
.
Then by [8 , lemma 2.8.2] the set 
is
a *-subalgebra of 
and the map
is a *-isomorphism of M3(Ao)
onto Rw (L) + iRw(L).
Let
.
Note that 
, where 
and 
. We prove that 
.
Let
be pairwise distinct indexes in 
. Since 
and
then 
. Hence 
. Therefore 
. Also it is easy to see 
.
Let 
and 
. Then the elements 
belong to 
and
.
Hence 
. Therefore 
is
a real *-algebra.
Let 
. Since 
then 
. It is clear that
Let 
for all 
. Then 
consequently multiplying by 
and 
we get
Hence 
Analogously we have 
, 
. Hence 
for all 
.
Conversely, Let 
, where 
, 
. Then we have to show 
. For the last statement to be proved it
is sufficient to establish 
for all 
. In turn the last statement is true by
the definition of 
. Hence 
. By the last equality L is reversible.
□
3.
The enveloping C*-algebra of a purely real
Lie algebra of
skew-adjoint operators
Theorem 3.1. Let 
be
a weakly closed purely real reversible Lie algebra of skew- adjoint operators
on a Hilbert space 
be a real von Neumann algebra, generated by 
in
. Suppose that 
does not have nonzero direct summands of type I. Then
the enveloping C*-algebra 
of
is
a von Neumann algebra.
Proof. By
theorem 1.5 we have 
, where
and 
is
a JW-algebra with no nonzero direct summands of type I.
Let 
be
a projection in 
. Then 
is
a JW-algebra with no nonzero direct summands of type I. Hence there exist
pairwise orthogonal and pairwise equivalent by symmetry projections 
in
such that 
. Let 
be partial symmetries in 
such that if 
then 
for all 
.
Let 
and if 
then 
for all 
. Then the set 
is
a system of matrix units and
where 
is
a real associative algebra, generated by the set 
. At the same time
where 
is
a real associative algebra, generated by the Lie algebra 
. Also, since 
we
have 
Hence 
and 
. Since 
we
have 
. The projection 
was chosen in 
arbitrarily, and a JC-algebra, generated by the set of all
projections of the algebra 
coincides with 
. Hence
Therefore
At the same time by 
and 
we
have
and 
is
a von Neumann algebra, i.e. 
. Hence 
. □
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