On a certain class of operator algebras and their derivations

Abstract

Given a von Neumann algebra $M$ with a faithful normal finite trace, we introduce the so called finite tracial algebra $M_f$ as the intersection of $L_p$-spaces $L_p(M, \mu)$ over all $p \geq 1$ and over all faithful normal finite traces $\mu$ on $M.$ Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner.

Full text

Page 1

arXiv:0908.1203v1 [math.OA] 9 Aug 2009
On a certain class of operator
algebras and their derivations
Sh.A. Ayupov1,∗, R.Z. Abdullaev2, K.K. Kudaybergenov3
August 10, 2009
Abstract
Given a von Neumann algebra M with a faithful normal finite
trace, we introduce the so called finite tracial algebra Mf as the in-
tersection of Lp-spaces Lp(M,µ) over all p ≥ 1 and over all faithful
normal finite traces µ on M. Basic algebraic and topological properties
of finite tracial algebras are studied. We prove that all derivations on
these algebras are inner.
1Institute of Mathematics and Information Technologies, Uzbekistan Academy
of Science, Dormon Yoli str. 29, 100125, Tashkent, Uzbekistan
and
Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,
e-mail: sh ayupov@mail.ru
2Institute of Mathematics and Information Technologies, Uzbekistan Academy
of Science, Dormon Yoli str. 29, 100125, Tashkent, Uzbekistan,
e-mail: arustambay@yandex.ru
3Karakalpak state university, Ch. Abdirov str. 1, 142012, Nukus, Uzbek-
istan,
e-mail: karim2006@mail.ru
AMS Subject Classifications (2000): 46L51, 46L52, 46L57, 46L07.
Key words: von Neumann algebra, faithful normal finite trace, non
commutative Lp-spaces, Arens algebra, finite tracial algebra, derivations.
* Corresponding author
1

Page 2

1 Introduction
In the present paper we introduce a new class of algebras, the so called finite
tracial algebras, which are defined as the intersection of non commutative
Lp-spaces Lp(M,µ) [15] over all p ∈ [1,∞) and over all faithful normal fi-
nite (f.n.f.) traces µ on a von Neumann algebra M. Equivalently, a finite
tracial algebra Mfis the intersection of all non commutative Arens algebras
Lω(M,µ) =
?
algebras are metrizable locally convex *-algebras with respect to the topol-
ogy generated by the system of Lp-norms for a fixed trace. Algebraic and
topological properties of Arens algebras have been investigated in the papers
[1]- [3], [7], [10].
In the present paper we study basic properties of finite tracial algebras
with the topology generated by all Lp-norms {? · ?µ
µ runs over all f.n.f. traces on the given von Neumann algebra M. We prove
that a finite tracial algebra Mf is metrizable or reflexive if and only if the
center of the von Neumann algebra M is finite dimensional; in this case Mf
coincides with an appropriate Arens algebra. We also give a necessary and
sufficient condition for Mf to coincide (as a set) with M. But even in this
case one has a new topology on the von Neumann algebra M. We obtain also
a description of the dual space for the algebra Mf.
Finally we prove that every derivation on a solid subalgebra of the Arens
algebra Lω(M,τ) is inner. In particular we obtain that the algebra Mfadmits
only inner derivations.
Throughout the paper we consider a von Neumann algebra M with a
f.n.f trace. Therefore M is a finite von Neumann algebra and thus all closed
densely defined operators affiliated with M are measurable with respect to
M, i. e. the set of all such operators coincides with the algebra S(M) of all
measurable operators and hence also with the algebra LS(M) of all locally
measurable operators affiliated with M ; moreover the center of S(M) =
LS(M) coincides with the set of operators affiliated with the center of M.
p≥1Lp(M,µ), over all f.n.f. traces µ. It is known that Arens
p}, where p ∈ [1,∞) and
2 Preliminaries
Let M be a von Neumann algebra with the positive cone M+and let 1 denote
the identity operator in M.
2

Page 3

A positive linear functional µ is called a finite trace if µ(u∗xu) = µ(x) for
all x ∈ M and each unitary operator u ∈ M.
A finite trace µ is said to be faithful if for x ∈ M+, µ(x) = 0 implies that
x = 0.
A finite trace µ is normal if given any monotone net {xα} increasing to
x ∈ M, one has µ(x) = supµ(xα).
Let τ be a fixed faithful normal finite (f.n.f.) trace on a von Neumann
algebra M. The Radon — Nikodym theorem [13, Theorem 14] implies that
given any f.n.f. trace µ on M there exists a positive operator h ∈ L1(M,τ)
affiliated with the center of M such that µ(x) = τ(hx) for all x ∈ M. This
operator h is called the Radon — Nikodym derivative of the trace µ with
respect to the trace τ and it is denoted asdµ
We recall [13], [15] that given a f.n.f. trace τ on a von Neumann algebra
M the space Lp(M,τ) , p ∈ [1,∞), is defined as
dτ.
Lp(M,τ) = {x ∈ S(M) : |x|p∈ L1(M,τ)}.
The space Lp(M,τ) equipped with the norm ?x?p= (τ(|x|p))1/pis a Banach
space and its dual space coincides with Lq(M,τ) where1p+1q= 1, and the
duality is given by
?x,a? = fa(x) = τ(ax),
for all fa∈ Lp(M,τ)∗, a ∈ Lq(M,τ) (see [15, Theorem 4.4]).
Following [10] consider the intersection
Lω(M,τ) =
?
p∈[1,∞)
Lp(M,τ).
It is known (see also [1], [3], [7]), that Lω(M,τ) is a complete locally convex
∗-algebra with respect to the topology tτgenerated by the system of norms
{? · ?p}p∈[1,∞).
Each operator a ∈
?
q∈(1,∞)
faon (Lω(M,τ),tτ) by the formula fa(x) = τ(ax), and conversely given an
arbitrary continuous linear functional f on the algebra (Lω(M,τ),tτ) there
exists an element a ∈
?
q∈(1,∞)
Lq(M,τ) defines a continuous linear functional
Lq(M,τ) such that f(x) = τ(ax).
3

Page 4

3 Finite Tracial Algebras
Let M be a finite von Neumann algebra. Denote by F the set of all f.n.f.
traces on M and from now on suppose that F ?= ∅.
Consider the space
Mf=
?
µ∈F
?
p∈[1,∞)
Lp(M,µ) =
?
µ∈F
Lω(M,µ).
On the space Mf one can consider the topology t, generated by the system
of norms {? · ?µ
Since each Arens algebra Lω(M,µ), µ ∈ F, is a complete locally convex
topological ∗-algebra in S(M) from the above definition one easily obtains
the following
p: µ ∈ F,p ∈ [1,∞)}.
Theorem 3.1. (Mf,t) is a complete locally convex topological ∗-algebra.
Definition.
algebra with respect to the von Neumann algebra M.
Remark. Finite tracial algebras present examples of so called GW∗-
algebras in the sense of [12].
Recall (see [12]) that a topological ∗-algebra (A,tA) is called GW∗-algebra,
if A has a W∗-subalgebra B with (1+x∗x)−1∈ B for all x ∈ A and the unit
ball of B if tA-bounded.
The finite tracial algebra Mf is a GW∗-algebra. Since M ⊂ Mf it is
sufficient to show that the unit ball in M is t - bounded in Mf.
Let x ∈ M, ?x?∞≤ 1. For µ ∈ F, and 1 ≤ p < ∞ we have
The topological ∗-algebra Mf is called the finite tracial
?x?µ
p= ?x1?µ
p≤ ?x?∞?1?µ
p≤ µ(1)
1
p,
i. e. ?x?µ
of M is t - bounded in Mf. Therefore Mfis a GW∗- algebra.
The algebra Mf contains M but it is a rather small algebra, since it is
contained in all Lp(M,µ) for all p ≥ 1 and f.n.f. traces µ on M. The following
result gives necessary and sufficient conditions for Mfto coincide with M.
p≤ µ(1)
1
pfor all x ∈ M, ?x?∞≤ 1. This means that the unit ball
Theorem 3.2. For a finite von Neumann algebra M the following conditions
are equivalent
i) Mf= M;
ii) M is a finite sum of homogeneous type In, n ∈ N von Neumann alge-
bras.
4

Page 5

The proof of this theorem consists of several auxiliary proposition which
are interesting in their own right. Let us start with the commutative case.
Proposition 3.1. Let M be a von Neumann algebra with a faithful normal
trace and let Z be its center. Then the center of the algebra Mf coincides
with Z, i. e. Z(Mf) = Z. In particular if M is commutative then Mf= M.
Proof. Let M be a von Neumann algebra with a faithful normal finite
trace τ, and τ(1) = 1.
Consider x ∈ Z(Mf), x ≥ 0, and let x =
∞ ?
0
λdeλbe the spectral resolution
of x. Since x ∈ Z(Mf) and M ⊂ Mf, we have that eλ∈ Z for all λ ∈ R.
Passing if necessary to the element ε1 + x we may suppose without loss of
generality that e1= 0.
For n ∈ N set
pn= e(n+1)2 − en2
and
y =
?
n∈N
n2pn.
Since xpn≥ n2pnfor all n ∈ N, we have that 0 ≤ y ≤ x and hence y ∈ Mf.
Let
F = {n ∈ N : tn= τ(pn) ?= 0}
and
h =
?
n∈F
1
n2tnpn∈ Z(S(M)).
Since
m
?
n=1
pn=
m
?
n=1
(e(n+1)2 − en2) =
m
?
n=1
(e(n+1)2 − en2) = e(m+1)2 − e1= e(m+1)2 ↑ 1,
one has that
∞
?
n=1
pn= 1.
Therefore there exists h−1∈ S(M). Further we have
τ(h) =
?
n∈F
1
n2tnτ(pn) =
?
n∈F
1
n2tntn=
?
n∈F
1
n2≤
?
n∈N
1
n2< ∞,
5

Page 6

i.e. h ∈ L1(M,τ).
Put µ(·) = τ(h·). Since y ∈ Mf, it follows that y ∈ L1(M,µ). Therefore
µ(y) < ∞.
On the other hand
1
n2tnpn
n∈N
hy =
?
n∈F
?
n2pn=
?
n∈F
1
tnpn,
and thus
µ(y) = τ(hy) =
?
n∈F
1
tnτ(pn) =
?
n∈F
1
tntn=
?
n∈F
1 = |F|,
where |F| is the cardinality of the set F. Since µ(y) < ∞ this implies that
F is a finite set. Let k = max{n : n ∈ F}. Then τ(pn) = 0 for all n > k, and
since τ is faithful we have that pn= 0 for all n > k, i.e. e(n+1)2 = en2. But
en2 ↑ 1 and thus en2 = 1 for all n > k. This means that 0 ≤ x ≤ (k + 1)21,
i.e. x ∈ Z.
The proof is complete. ?
Proposition 3.2. Let M be a type In, n ∈ N von Neumann algebra. Then
Mf= M.
Proof. By [14, Ch. V, Theorem 1.27] the von Neumann algebra M of type
In (n ∈ N) can be represented as M = Z ⊗ B(Hn), where Z is the center
M and Hnis the n-dimensional Hilbert space. Put FZ = {τ|Z : τ ∈ F}.
Therefore from Proposition 3.1 we obtain
Mf=
?
p∈[1,∞)
?
τ∈F
Lp(M,τ) =
?
p∈[1,∞)
?
µ∈FZ
Lp(Z,µ) ⊗ B(Hn) =
=


?
p∈[1,∞)
?
µ∈FZ
Lp(Z,µ)

⊗ B(Hn) = Zf⊗ B(Hn) =
= Z ⊗ B(Hn) = M,
i.e. Mf= M.
The proof is complete. ?
Proposition 3.3. Let M be a finite von Neumann algebra which is isomor-
phic to the direct sum of an infinite number of homogeneous type In (n ∈ N)
von Neunamm algebras. Then Mf?= M.
6

Page 7

Proof. Suppose that M =
?
k∈K
⊕Mk, where K is an infinite subset of N,
and Mkis a homogeneous type Ikvon Neumann algebra.
Since the set K is infinite, there exists a sequence {kn} ⊂ K such that
kn≥ 2nfor all n ∈ N. We have that
Mkn= Zkn⊗ B(Hkn),
where Zknis the center of Mknand
Nn= 1n⊗ B(H2n) ⊂ Mkn.
Therefore the algebra M contains a subalgebra *-isomorphic to the algebra
N =?
n∈N
Hence, without loss of generality we may assume that M =
⊕Nn.
?
n∈N
⊕Nn,
where Nn= B(H2n) is the algebra of all 2n× 2nmatrices over C. On each
Nnwe consider the unique tracial state (i. e. normalized f.n.f. trace) µnand
define on M the following f.n.f. trace
τ(x) =
?
n∈N
2−nµn(xn),
where x =?
n∈N
⊕xn∈ M. Then every f.n.f. trace µ on M has the form
µ(x) = τ(hx) =
?
n∈N
2−nµn(hnxn) =
?
n∈N
2−nαnµn(xn),
where
h =
?
n∈N
⊕hn=
?
n∈N
⊕αn1n∈ L1(M,τ),
i. e. αn> 0, n ∈ N,?
Take a minimal projection pnin each Nn= B(H2n). Then µn(pn) =1
Consider the unbounded element x =
n∈N2−nαn< ∞.
2n.
?
n∈N
⊕npnin S(M) \ M and let us
prove that x ∈ Mf. For every f.n.f. trace µ on M one has that
µ(xp) =
?
n∈N
2−nαnµn(nppn) =
?
n∈N
2−nαnnp2−n< ∞,
7

Page 8

because np2−n< 1 for sufficiently large n ∈ N. Therefore x ∈ Lp(M,µ) for
all p ≥ 1 and every f.n.f. trace µ ∈ F, i. e. x ∈ Mf.
The proof is complete. ?
Proposition 3.4. Let M be a type II1von Neumann algebra with a f.n.f.
trace τ. Then Mf?= M.
Proof. Suppose that the trace τ is normalized, i. e. τ(1) = 1, and denote
by Φ the canonical center-valued trace on M. Since M is of type II1there
exists a projection p1such that
p1∼ 1 − p1.
Therefore from Φ(p1)+Φ(p⊥
1) = Φ(1) = 1 and Φ(p1) = Φ(p2) we obtain that
Φ(p1) = Φ(p⊥
1) =1
21.
Suppose that we have constructed mutually orthogonal projections p1, p2, ···, pn
in M such that
Φ(pk) =1
2k1, k = 1,n.
n ?
that
pn+1∼ e⊥
Set en=
k=1pk. Then Φ(e⊥
n) =1
2n1. Now take a projection pn+1≤ e⊥
nsuch
n− pn+1,
i. e.
Φ(pn+1) =
1
2n+11.
In this manner we obtain a sequence {pn}n∈N of mutually orthogonal
projections such that
1
2n1, n ∈ N.
It is clear that τ(pn) = τ(Φ(pn)) =1
2n, n ∈ N.
From
∞
?
n=1n=1
Φ(pn) =
||npn||τ
1=
∞
?
τ(npn) =
∞
?
n=1
n
2n< ∞,
it follows that the element x =
∞
?
n=1npnbelongs to L1(M,τ), and it is un-
bounded, i. e. x / ∈ M.
8

Page 9

On the other hand for an arbitrary central element h ∈ L1(M,τ),h > 0,
and n ∈ N we have
τ(hpn) = τ(Φ(hpn)) = τ(hΦ(pn)) = τ(h1
2n1) =
1
2nτ(h).
Therefore for an arbitrary f.n.f. trace µ on M withdµ
dτ= h we have
µ(|x|p) = µ(xp) = τ(hxp) = τ(h
∞
?
n=1
nppn) =
=
∞
?
n=1
npτ(hpn) = τ(h)
∞
?
n=1
np
2n< ∞,
i. e. x ∈ Lp(M,µ) for all p ≥ 1 and every f.n.f. trace µ. Therefore x ∈ Mf\M.
The proof is complete. ?
Proof of Theorem 3.2. The implication (i) ⇒ (ii) follows from Proposi-
tions 3.3 and 3.4, while (ii) ⇒ (i) follows from Propositions 3.2.
The proof is complete. ?
Now let us describe continuous linear functionals on the space (Mf,t).
Theorem 3.3. Given any µ ∈ F, 1 < q < ∞, and a ∈ Lq(M,µ) the
functional ϕ(x) = µ(xa), x ∈ Mf, is a continuous linear functional on
(Mf,t). Conversely for any continuous linear functional ϕ on (Mf,t) there
exist µ ∈ F, 1 < q < ∞, a ∈ Lq(M,µ) such that
ϕ(x) = µ(xa), x ∈ Mf.
Proof. Let µ ∈ F, 1 < q < ∞, a ∈ Lq(M,µ). Put
ϕa(x) = µ(xa), x ∈ Mf.
Take p ∈ R such that1p+1q= 1. Since
|ϕa(x)| = |µ(xa)| ≤ ||a||µ
q||x||µ
p
for all x ∈ Mf, one has that ϕais a continuous linear functional on (Mf,t).
Conversely, let ϕ be a continuous linear functional on (Mf,t). By [16,
Corollary 1 on p.43] there exist µ ∈ F, 1 ≤ p < ∞, c > 0, such that
|ϕ(x)| ≤ c||x||µ
p
9

Page 10

for all x ∈ Mf. Since M ⊂ Mf and M is ? · ?µ
functional ϕ can be uniquely extended onto Lp(M,µ). By [15, Theorem 4.4]
there exists a ∈ Lq(M,µ),1p+1q= 1, such that
p-dense in Lp(M,µ), the
ϕ(x) = µ(xa)
for all x ∈ Lp(M,µ). In particular
ϕ(x) = µ(xa)
for all x ∈ Mf, i.e. ϕ = ϕa.
The proof is complete. ?
If the von Neumann algebra M is a factor then it has a unique (up to
a scalar multiple) f.n.f. trace µ. In this case the finite tracial algebra Mf
coincides with the Arens algebra Lω(M,µ) and the topology t merges to
the topology tµgenerated by the system of norms {? · ?µ
theorem describes the general case where this phenomenon occurs.
Recall some notions from the theory of linear topological spaces. Let E
be a locally convex linear topological space. An absolutely convex absorbing
set in E is called a barrel. If each barrel in E is a neighborhood of zero, then
E is said to be a barreled space.
It is known ([16], Theorem 2, p.200 ) that every reflexive locally convex
space is barreled.
p}p≥1. The following
Theorem 3.4. Let M be a finite von Neumann algebra and suppose that
F ?= ∅ is the family of all f.n.f. traces on M. The following conditions are
equivalent:
(i) Mf= Lω(M,µ) for some (and hence for all) µ ∈ F;
(ii) (Mf,t) is metrizable;
(iii) (Mf;t) is reflexive;
(iv) the center Z of M is finite dimensional, i. e. M =
m ?
i=1Mi, where all
Miare In-factors or II1-factors.
Proof. Suppose that Z is finite dimensional. Then M is a finite direct
sum of factors Mi, i = 1,k. Then for each factor Mithe algebras (Mi)fand
Lω(Mi,µi) coincide and the topology tiis the same as tµi
i. Therefore
Mf= (
n
?
i=1
Mi)f=
n
?
i=1
(Mi)f=
n
?
i=1
Lω(Mi,µi) = Lω(M,µ),
10

Page 11

where µ =
n ?
i=1µi∈ F, i. e. Mf= Lω(M,µ).
Now since the topology tµon the Arens algebra Lω(M,µ) is metrizable
[1] it follows that t = tµis also metrizable.
It is known [2] that for finite traces µ the Arens algebra (Lω(M,µ),tµ) is
reflexive and hence (Mf,t) is also reflexive.
Therefore (iv) implies (i), (ii) and (iii).
(i) ⇒ (iv). Suppose that Mf = Lω(M,µ) for an appropriate µ ∈ F.
Then there exists a sequence of mutually orthogononal projections {pn} in
Z such that pn?= 0 for all n ∈ N. Since the trace µ is finite one has that
∞
?
µ(pnk) ≤1
2kfor all k.
Set
∞
?
k=1
For p ≥ 1 we have
k=1µ(pk) < ∞ and hence there is a subsequence {nk : k ∈ N} such that
x = kpk
µ(|x|p) =
∞
?
k=1
kpµ(pk) ≤
∞
?
k=1
kp1
2k< ∞,
and hence x ∈ Lω(M,µ) = Mf.
On the other hand x is a central element in Mfand Proposition 3.1 implies
that x ∈ Z(Mf) = Z ⊂ M. But it is clear that the element x is unbounded,
i.e. x / ∈ M. The contradiction shows that Z is finite dimensional.
(ii) ⇒ (iv). Suppose that (Mf,t) is metrizable. By Theorem 3.1 it is
complete and hence it is a Fre′chet space. In particular the center of Mf
which coincides with Zfis also a Fre′chet space. By Proposition 3.1 Zf= Z
and hence Z is a Fre′chet space with respect to the induced topology tz= t|Z.
Consider the identity mapping
I : (Z,? · ?∞) → (Z,tz)
where ? · ?∞is the operator norm on Z. From the inequalities
?x?µ
p≤ Cµ
p?x?∞
(where Cµ
the mapping I is continuous. Since (Z,tz) is a Fre′chet space, from Banach
pis an appropriate constant for each p ≥ 1, µ ∈ F) it follows that
11

Page 12

theorem on the inverse operator ([16], Chapter II, Section 5) we obtain that
the inverse mapping
I−1: (Z,tz) → (Z,? · ?∞)
is also continuous. This means that for some p ∈ [1,∞) and an appropriate
µ ∈ F there exists a constant Kµ
psuch that
?x?∞≤ Kµ
p?x?µ
p
(1)
for all x ∈ Z ([16], Theorem 1, p. 42).
Now suppose that dimZ = ∞. There exists a sequence {pn} of projec-
tions in Z such that pn ↑ 1, pn ?= pn+1. Thus p⊥
?p⊥
On the other hand ?p⊥
dimensional.
(iii) ⇒ (iv). Suppose that Mf is reflexive. Then the center Z(Mf) = Z
is also reflexive as a closed subspace of a reflexive space.
The set
B = {x ∈ Z : ||x||∞≤ 1}
n?= 0, τ(p⊥
n?∞→ 0.
n) → 0, i.e.
n?µ
p→ 0. From the inequality (1) we obtain that ?p⊥
n?∞= 1. This contradiction implies that Z is finite
is a barrel in (Z,t) and since Z is reflexive, we have that B is a neighborhood
of zero in Z. Therefore there exist p ≥ 1, µ ∈ F and ε > 0 such that
{x ∈ Z : ?x?µ
p≤ ε} ⊆ B
i.e.
?x?∞≤ ε−1?x?µ
p
for all x ∈ Z. From this as above it follows that Z is finite dimensional.
The proof is complete. ?
Remark.In the von Neumann algebra M the operator topology is
stronger than the topology t, t is stronger than tµ, and tµis stronger than
each Lp-norm topology for any p ≥ 1.
4 Derivations on Finite Tracial Algebras
Derivations on unbounded operator algebras, in particular on various alge-
bras of measurable operators affiliated with von Neumann algebras, appear
to be a very attractive special case of general unbounded derivations on op-
erator algebras.
12

Page 13

Let A be an algebra over the complex number. A linear operator D : A →
A is called a derivation if it satisfies the identity D(xy) = D(x)y + xD(y)
for all x,y ∈ A (Leibniz rule). Each element a ∈ A defines a derivation Da
on A given as Da(x) = ax − xa, x ∈ A. Such derivations Daare said to be
inner derivations.
In [4] we have investigated and completely described derivations on the
algebra LS(M) of all locally measurable operators affiliated with a type
I von Neumann algebra M and on its various subalgebras. Recently the
above conjecture was also confirmed for the type I case in the paper [8] by a
representation of measurable operators as operator valued functions. Another
approach to similar problems in AW∗-algebras of type I was suggested in the
recent paper [9].
In the paper [3] we have proved the spatiality of derivations on the non
commutative Arens algebra Lω(M,τ) associated with an arbitrary von Neu-
mann algebra M and a faithful normal semi-finite trace τ. Moreover if the
trace τ is finite then every derivation on Lω(M,τ) is inner.
In this section we prove that each derivation on a finite tracial algebra is
inner.
The following result is an immediate corollary of [6, Proposition 3.6].
Lemma 4.1. Let M be a von Neumann algebra with a faithful normal trace
τ. Given any derivation D : M → Lω(M,τ) there exists an element a ∈
Lω(M,τ) such that
D(x) = ax − xa, x ∈ M.
Further we need also the following assertion from [8, Proposition 6.17].
Lemma 4.2. Let A be a *-subalgebra of LS(M) such that M ⊆ A and A
is solid (that is, if x ∈ LS(M) and y ∈ A satisfy |x| ≤ |y| then x ∈ A). If
ω ∈ LS(M) is such that [ω,x] ∈ A for all x ∈ A, then there exists ω1∈ A
such that [ω,x] = [ω1,x] for all x ∈ A.
The main result of this section is the following theorem.
Theorem 4.1. Let M be a von Neumann algebra with a faithful normal finite
trace τ. If A ⊆ Lω(M,τ) is a solid *-subalgebra such that M ⊆ A, then every
derivation on A is inver.
Proof. Since A ⊆ Lω(M,τ), by Lemma 4.1 there exits an element a ∈
Lω(M,τ) such that
D(x) = ax − xa,x ∈ M. (2)
13

Page 14

Let us show that in fact
D(x) = ax − xa, for all x ∈ A. (3)
Consider x ∈ A, x ≥ 0. Then (1 + x)−1∈ M. From the Leibniz rule it
follows that for each invertible b ∈ A one has
D(b) = −bD(b−1)b.
Therefore
D(x) = D(1 + x) = −(1 + x)D((1 + x)−1)(1 + x).
On the other hand since (1 + x)−1∈ M the equality (2) implies that
D((1 + x)−1) = a(1 + x)−1− (1 + x)−1a.
Therefore
−(1 + x)D((1 + x)−1)(1 + x) = −(1 + x)[a(1 + x)−1− (1 + x)−1a](1 + x) =
= −(1 + x)a + a(1 + x) = ax − xa,
i.e.
D(x) = ax − xa, x ∈ A, x ≥ 0.
Since each element from A is a finite linear combination of positive ele-
ments, we obtain the equality (3) for arbitrary x ∈ A.
Now since A is a solid *-subalgebra in Lω(M,τ) containing A, Lemma
4.2 implies that the element a implementing the derivation D may be chosed
from the algebra A, i.e.
D(x) = ax − xa, x ∈ A
for an appropriate a ∈ A.
The proof is complete. ?
Since the algebra Mfis a solid *-subalgebra of Lω(M,τ) and contains M,
we obtain the following result.
Corollary 4.1. If M is a von Neumann algebra with a faithful normal trace,
then every derivation on Mf is inner.
14

Page 15

Acknowledgments. Part of this work was done within the framework
of the Associateship Scheme of the Abdus Salam International Centre for
Theoretical Physics (ICTP), Trieste, Italy. The first author would like to
thank ICTP for the kind hospitality and for providing financial support and
all facilities (July-August, 2009).This work is supported in part by the DFG
436 USB 113/10/0-1 project (Germany).
References
[1] Abdullaev R.Z. Isomorphism of Arens algebras, Siberian J. Industrial
Math. 1998, Vol. 1, no 2, p. 3-13.
[2] Abdullaev R.Z. The dual space for Arens algebra, Uzbek. Math. J., 1997,
no 2, p. 3-7.
[3] Albeverio S., Ayupov Sh.A., Kudaybergenov K.K. Non commutative
Arens algebras and their derivations, J. Func. Anal., 253 (2007), no. 1,
p. 287-302.
[4] Albeverio S., Ayupov Sh. A., Kudaybergenov K. K. Structure of deriva-
tions on various algebras of measurable operators for type I von Neu-
mann algebras, J. Func. Anal., 256 (2009), no. 9, p. 2917-2943.
[5] Albeverio S., Sh.A. Ayupov, R.Z. Abdullaev. Arens Spaces associ-
ated with von Neumann Algebras and Normal States, SFB 611, Uni-
versit¨ at Bonn, Preprint, No 381, 2008.(to appear in POSITIVITY,
doi:10.1007/s11117-009-0008-5)
[6] Ayupov Sh. A., Kudaybergenov K. K. Innerness of derivations on subal-
gebras of measurable operators, Lobachevskii J. Math. 29 (2008) 60–67.
[7] Arens R. The space Lω(0;1) and convex topological rings. Bull. Amer.
Math. Soc., 52, 1946, p. 931-935.
[8] Ber A. F., de Pagter B., Sukochev F. A. Derivations in algebras of
operator-valued functions, arXiv.math.OA.0811.0902. 2008.
[9] Gutman A. E., Kusraev A. G., Kutateladze S. S. The Wickstead prob-
lem, Sib. Electron. Mat. Izv. 5: 293–333, 2008.
15