Additive derivations on generalized Arens algebras

Abstract

Given a von Neumann algebra $M$ with a faithful normal finite trace $\tau$ denote by $L^\Lambda(M, \tau)$ the generalized Arens algebra with respect to $M.$ We give a complete description of all additive derivations on the algebra $L^\Lambda(M, \tau).$ In particular each additive derivation on the algebra $L^{\Lambda}(M, \tau),$ where $M$ is a type II von Neumann algebra, is inner.

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arXiv:1010.5117v1 [math.OA] 25 Oct 2010
Additive derivations on generalized Arens
algebras
S. Albeverio1, Sh.A. Ayupov2,∗, R.Z. Abdullaev3, K.K. Kudaybergenov4
October 26, 2010
Abstract
Given a von Neumann algebra M with a faithful normal finite trace τ denote
by LΛ(M,τ) the generalized Arens algebra with respect to M. We give a complete
description of all additive derivations on the algebra LΛ(M,τ). In particular each
additive derivation on the algebra LΛ(M,τ), where M is a type II von Neumann
algebra, is inner.
1Institut fur Angewandte Mathematik, Universitat Bonn, Endenicherllee. 60, D-
53115 Bonn (Germany); SFB 611; HCM; BiBoS; IZKS; CERFIM (Locarno); e-mail
address: albeverio@uni-bonn.de
2Institute of Mathematics and Information Technologies, Uzbekistan Academy of
Sciences, Dormon Yoli str. 29, 100125, Tashkent (Uzbekistan), ICTP (Trieste, Italy),
e-mail: sh ayupov@mail.ru
3Institute of Mathematics and Information Technologies, Uzbekistan Academy of
Science, Dormon Yoli str. 29, 100125, Tashkent, (Uzbekistan) arustambay@yandex.ru
4Karakalpak state university, Ch. Abdirov str. 1, 142012, Nukus (Uzbekistan),
e-mail: karim2006@mail.ru
AMS Subject Classifications (2000): 46L57, 46L50, 46L55, 46L60.
Key words: von Neumann algebras, measurable operator, generalized Arens alge-
bra, additive derivation, inner derivation.
* Corresponding author
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1. Introduction
The present paper continues the series of papers [2]-[9] devoted to the study and de-
scription of derivations on the algebra LS(M) of locally measurable operators affiliated
with a von Neumann algebra M and on its various subalgebras.
Let A be an algebra over the field complex number C. A linear (additive) operator
D : A → A is called a linear (additive) 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 linear
derivation Daon A given as Da(x) = ax − xa, x ∈ A. Such derivations Daare said to
be inner derivations. If the element a implementing the derivation Daon A, belongs to
a larger algebra B, containing A (as a proper ideal, as usual) then Dais called a spatial
derivation.
One of the main problems in the theory of derivations is to prove the automatic
continuity, innerness or spatialness of derivations or to show the existence of non inner
and discontinuous derivations on various topological algebras.
In this direction A. F. Ber, F. A. Sukochev, V. I. Chilin [10] obtained necessary and
sufficient conditions for the existence of non trivial derivations on commutative regu-
lar algebras. In particular they have proved that the algebra L0(0,1) of all (classes of
equivalence of) complex measurable functions on the interval (0,1) admits non trivial
derivations. Independently A. G. Kusraev [16] by means of Boolean-valued analysis
has also proved the existence of non trivial derivations and automorphisms on L0(0,1).
It is clear that these derivations are discontinuous in the measure topology, and there-
fore they are neither inner nor spatial. It was conjectured that the existence of such
exotic examples of derivations deeply depends on the commutativity of the underly-
ing von Neumann algebra M. In this connection we have initiated the study of the
above problems in the non commutative case [2]-[6], by considering derivations on the
algebra LS(M) of all locally measurable operators affiliated with a von Neumann al-
gebra M and on various subalgebras of LS(M). In [2] noncommutative Arens algebras
Lω(M,τ) =
?
p≥1Lp(M,τ) and related algebras associated with a von Neumann algebra
M and a faithful normal semi-finite trace τ have been considered. It has been proved
that every derivation on this algebra is spatial, and, if the trace τ is finite, then all
derivations are inner. In [5] and [6] the mentioned conjecture concerning derivations on
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on the algebra LS(M) has been confirmed for type I von Neumann algebras.
Recently this conjecture was also independently confirmed for the type I case in the
paper of A.F. Ber, B. de Pagter and A.F. Sukochev [11] by means of a representation
of measurable operators as operator valued functions. Another approach to similar
problems in the framework of type I AW∗-algebras has been outlined in the paper of
A.F. Gutman, A.G.Kusraev and S.S. Kutateladze [13].
In [5] we considered derivations on the algebra LS(M) of all locally measurable
operators affiliated with a type I von Neumann algebra M, and also on its subalgebras
S(M) – of measurable operators and S(M,τ) of τ-measurable operators, where τ is a
faithful normal semi-finite trace on M. It was proved that an arbitrary derivation D on
each of these algebras can be uniquely decomposed into the sum D = Da+ Dδwhere
the derivation Dais inner (for LS(M), S(M) and S(M,τ)) while the derivation Dδis
an extension of a derivation δ (possibly non trivial) on the center of the corresponding
algebra.
In the present paper we consider additive derivations on generalized Arens algebras
in the sense of Kunze [15] with respect to a von Neumann algebra with a faithful normal
finite trace.
In section 1 we give some necessary properties of the generalized Arens algebra
LΛ(M,τ).
Section 2 is devoted to study of additive derivations on generalized Arens algebras.
We prove that an arbitrary additive derivation D on the algebra LΛ(M,τ) can be
uniquely decomposed into the sum D = Da+ Dδ, where the derivation Da is inner
while the derivation Dδis an extension of some additive derivation δ on the center of
the algebra LΛ(M,τ). In particular, if M is a type II von Neumann algebra then every
additive derivation on the algebra LΛ(M,τ) is inner.
2. Generalized Arens algebras
Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear
operators on H. Consider a von Neumann algebra M in B(H) with the operator norm
? · ?M. Denote by P(M) the lattice of projections in M.
A linear subspace D in H is said to be affiliated with M (denoted as DηM), if
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u(D) ⊂ D for every unitary u from the commutant
M′= {y ∈ B(H) : xy = yx, ∀x ∈ M}
of the von Neumann algebra M.
A linear operator x on H with the domain D(x) is said to be affiliated with M
(denoted as xηM) if D(x)ηM and u(x(ξ)) = x(u(ξ)) for all ξ ∈ D(x).
Let τ be a faithful normal semi-finite trace on M. We recall that a closed linear
operator x is said to be τ-measurable with respect to the von Neumann algebra M, if
xηM and D(x) is τ-dense in H, i.e. D(x)ηM and given ε > 0 there exists a projection
p ∈ M such that p(H) ⊂ D(x) and τ(p⊥) < ε. The set S(M,τ) of all τ-measurable
operators with respect to M is a unital *-algebra when equipped with the algebraic
operations of strong addition and multiplication and taking the adjoint of an operator
(see [18]).
Consider the topology tτof convergence in measure or measure topology on S(M,τ),
which is defined by the following neighborhoods of zero:
V (ε,δ) = {x ∈ S(M,τ) : ∃e ∈ P(M),τ(e⊥) ≤ δ,xe ∈ M,?xe?M≤ ε},
where ε,δ are positive numbers, and ?.?Mdenotes the operator norm on M.
It is well-known [18] that S(M,τ) equipped with the measure topology is a complete
metrizable topological *-algebra.
Recall [14] that φ is a Young function, if
φ(t) =
t ?
0
ϕ(s)ds,t ≥ 0,
where the real-valued function ϕ defined on [0,∞) has the following properties:
(i) ϕ(0) = 0, ϕ(s) > 0 for s > 0 and lim
s→∞ϕ(s) = ∞,
(ii) ϕ is right continuous,
(iii) ϕ is nondecreasing on (0,∞).
Every Young function is a continuous, convex and strictly increasing function. For
every Young function φ there is a complementary Young function ψ given by the density
ψ(t) = sup{s : φ(s) ≤ t}.
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The complement of ψ is φ again. Further a Young function φ is said to satisfy the
∆2-condition, shortly φ ∈ ∆2, if there exists a k > 0 and T ≥ 0 such that:
φ(2t) ≤ kφ(t)
for all t ≥ T.
Put
Kφ= {x ∈ S(M,τ) : τ(φ(|x|)) ≤ 1}
and
Lφ(M,τ) =
∞
?
n=1
nKφ.
It is known [17] (see also [15]) that Lφ(M,τ) is a Banach space with respect to the
norm
?x?φ= inf
?
λ > 0 :1
λx ∈ Kφ
?
,x ∈ Lφ(M,τ).
We recall from[15] that φ1≺ φ2, if there exist two nonnegative constants c and T
such that φ1(t) ≤ φ2(ct) for all t ≥ T. Let Λ be a generating family of Young functions,
i.e. for φ1,φ2∈ Λ there is a ψ ∈ Λ with φ1,φ2≺ ψ. A generating family Λ of Young
functions is said to be quadratic, if for any φ ∈ Λ there is a ψ ∈ Λ such that the
composition of φ and the squaring function as a Young function is smaller than ψ
regarding the partial order ≺, i.e. there are c > 0 and T ≥ 0 with φ(t2) ≤ ψ(ct) for all
t ≥ T. For a quadratic family Λ of Young functions we define
LΛ(M,τ) =
?
φ∈Λ
Lφ(M,τ).
On the space LΛ(M,τ) one can consider the topology tΛgenerated by the system of
norms {? · ?φ: φ ∈ Λ}.
It is known [15, Proposition 4.1] that if Λ is a quadratic family of Young functions,
then (LΛ(M,τ),tΛ) is a complete locally convex *-algebra with jointly continuous mul-
tiplications.
Note that if Λ = {tp: p ≥ 1} we have that
LΛ(M,τ) = Lω(M,τ) =
?
p≥1
Lp(M,τ).
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Non-commutative Arens algebras Lω(M,τ) were introduced by Inoue [12] and their
properties were investigated in [1]. Generalized Arens algebras were introduced by
Kunze [15].
Let ϕ ∈ Λ be a Young function. Then there exists a Young function φ ∈ Λ and
k > 0 such that
||xy||ϕ≤ k||x||φ||y||φ
(1)
for all x,y ∈ LΛ(M,τ) (see [15]).
Let us remark that, if τ is a finite trace, then t ≺ φ(t) for every Young function,
and for any quadratic family Λ of Young functions we obtain that
LΛ(M,τ) ⊂ Lω(M,τ). (2)
Further, if every φ ∈ Λ satisfies the ∆2-condition then
Lω(M,τ) ⊂ LΛ(M,τ).
It is known [15] that if N is a von Neumann subalgebra of M then
Lφ(N,τN) = S(N,τN) ∩ Lφ(M,τ),
where τNis the restriction of the trace τ onto N.
It should be noted that if M is a finite von Neumann algebra with a faithful normal
semi-finite trace τ, then the restriction τZof the trace τ onto the center Z(M) of M is
also semi-finite.
Further we shall need the description of the center of the algebra LΛ(M,τ) for von
Neumann algebras with a faithful normal finite trace .
Proposition 2.1. Let M be a von Neumann algebra with a faithful normal finite
trace τ and with the center Z(M). Then
Z(LΛ(M,τ)) = LΛ(Z(M),τZ).
Proof. Using the equality
Lφ(N,τN) = S(N,τN) ∩ Lφ(M,τ).
we obtain that
LΛ(N,τN) = S(N,τN) ∩ LΛ(M,τ).
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Hence
LΛ(Z(M),τZ) = S(Z(M),τZ) ∩ LΛ(M,τ) =
= Z(S(M,τ)) ∩ LΛ(M,τ) = Z(LΛ(M,τ)),
i.e.
Z(LΛ(M,τ)) = LΛ(Z(M),τZ).
The proof is complete. ?
3. Derivations on the generalized Arens algebras
In this section we give a complete description of all additive derivations on the
algebra LΛ(M,τ).
Let A be an algebra with the center Z(A) and let D : A → A be an additive
derivation. Given any x ∈ A and a central element a ∈ Z(A) we have
D(ax) = D(a)x + aD(x)
and
D(xa) = D(x)a + xD(a).
Since ax = xa and aD(x) = D(x)a, it follows that D(a)x = xD(a) for any a ∈ A.
This means that D(a) ∈ Z(A), i.e. D(Z(A)) ⊆ Z(A). Therefore given any additive
derivation D on the algebra A we can consider its restriction δ : Z(A) → Z(A).
We shall need some facts about additive derivations δ : C → C. Every such deriva-
tion vanishes at every algebraic number. On the other hand, if λ ∈ C is transcendental
then there is a additive derivation δ : C → C which does not vanish at λ (see [20]).
Let Mn(C) be the algebra of n×n matrices over C. If ei,j, i,j = 1,n, are the matrix
units in Mn(C), then each element x ∈ Mn(C) has the form
x =
n
?
i,j=1
λijeij, λi,j∈ C, i,j = 1,n.
Let δ : C → C be an additive derivation. Setting
Dδ
?
n
?
i,j=1
λijeij
?
=
n
?
i,j=1
δ(λij)eij
(3)
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we obtain a well-defined additive operator Dδon the algebra Mn(C). Moreover Dδis
an additive derivation on the algebra Mn(C) and its restriction onto the center of the
algebra Mn(C) coincides with the given δ.
It is known [21, Theorem 2.2] that if M be a von Neumann factor of type In,n ∈ N
then every additive derivation D on the algebra M can be uniquely represented as a
sum
D = Da+ Dδ,
where Dais an inner derivation implemented by an element a ∈ M while Dδ is the
additive derivation of the form (3) generated by an additive derivation δ on the center
of M identified with C.
Note that if M is a finite-dimensional von Neumann algebra then LΛ(M,τ) = M
for any faithful normal finite trace τ.
Now let M be an arbitrary finite-dimensional von Neumann algebra with the center
Z(M). There exist a family of mutually orthogonal central projections {z1,z2,...,zk}
k?
with the C∗-product of von Neumann factors ziM of type Inirespectively, i.e.
from M with
i=1zi= 1 and n1,n2,...,nk∈ N such that the algebra M is *-isomorphic
M∼= Mn1(C) ⊕ Mn2(C) ⊕ ... ⊕ Mnk(C).
Suppose that D is an additive derivation on M, and δ is its restriction onto its center
Z(M). Since δ(zx) = zδ(x) for all central projection z ∈ Z(M) and x ∈ M then δ
maps each ziZ(M)∼= C into itself, δ generates an additive derivation δion C for each
i = 1,k.
Let Dδibe the additive derivation on the matrix algebra Mni(C),i = 1,k, defined
as in (3). Put
Dδ((xi)k
i=1) = (Dδi(xi)), (xi)k
i=1∈ M. (4)
Then the map Dδis an additive derivation on M.
Lemma 3.1. Let M be a finite-dimensional von Neumann algebra. Each additive
derivation D on the algebra M can be uniquely represented in the form
D = Da+ Dδ,
where Da is an inner derivation implemented by an element a ∈ M, and Dδ is an
additive derivation given (4).
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Proof. Let D be an additive derivation on M, and let δ be its restriction onto Z(M).
Consider an additive derivation Dδon Z(M) of the form (4), generated by an additive
derivation δ. Since additive derivations D and Dδcoincide on Z(M) ,then an additive
derivation of the form D − Dδ is a linear derivation. Hence by Sakai’s theorem [19,
Theorem 4.1.6] D −Dδis an inner derivation. This means that there exists an element
a ∈ M such that Da= D − Dδand therefore D = Da+ Dδ. The proof is complete. ?
Now let M be a commutative von Neumann algebra with a faithful normal finite
trace τ. Given an arbitrary additive derivation δ on LΛ(M,τ) the element
zδ= inf{z ∈ P(M) : zδ = δ}
is called the support of the derivation δ.
Suppose that M is a commutative von Neumann algebra with a faithful normal
finite trace τ and q1,q2,...,qkare atoms in M. Then
LΛ(M,τ)∼= q1C ⊕ q2C ⊕ ... ⊕ qkC ⊕ pLΛ(M,τ),
where p = 1 −
k?
i=1qi.
Now if δi: C → C is an additive derivation then
δ(x) = (δ1(q1x),...,δk(qkx),0), x ∈ LΛ(M,τ) (5)
is also an additive derivation. Note that zδ=?{qi: δi?= 0,1 ≤ i ≤ k}.
Lemma 3.2. Let M be a commutative von Neumann algebra with a faithful normal
finite trace τ. For any non trivial additive derivation δ : LΛ(M,τ) → LΛ(M,τ) there
exists a sequence {an}∞
n=1in M with |an| ≤ 1, n ∈ N, such that
|δ(an)| ≥ nzδ
for all n ∈ N.
In [5, Lemma 2.6] (see also [11, Lemma 4.6]) this assertion was proved for linear
derivations on the algebra S(M), but same the proof is applies also to the case of
additive derivations on LΛ(M,τ).
The following result shows that the above construction (5) is the general form of
additive derivations on the generalized Arens algebras in the commutative case.
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Lemma 3.3. Let M be a commutative von Neumann algebra with a faithful normal
finite trace τ and let δ be an additive derivation on the algebra LΛ(M,τ). Then zδM is
a finite-dimensional algebra.
Proof.Suppose that zδM is infinite-dimensional. Then there exists an infiniti
∞ ?
sequence of mutually orthogonal projections {zn}∞
n=1in M such that
n=1zn= zδ. By
Lemma 3.2 there exists a sequence {an}∞
n=1in M with |an| ≤ 1, n ∈ N, such that
|δ(an)| ≥ 2nτ(zn)−1zδ
(6)
for all n ∈ N. Put
a =
∞
?
n=1
anzn
2n.
Then a ∈ M ⊂ LΛ(M,τ) and
δ(a) = δ
?∞
n=1
?
anzn
2n
?
=
∞
?
n=1
zn
2nδ(an).
From (6) we obtain that
|δ(a)| =
∞
?
n=1
zn
2n|δ(an)| ≥
∞
?
n=1
zn
2n2nτ(zn)−1zδ,
i.e.
|δ(a)| ≥
∞
?
n=1
τ(zn)−1zn.
Thus
τ(|δ(a)|) ≥
∞
?
n=1
τ(zn)−1τ(zn) =
∞
?
n=1
1 = ∞.
This means that δ(a) / ∈ L1(M,τ). Then by (2) we have that δ(a) / ∈ LΛ(M,τ). This
contradiction implies that zδM is a finite-dimensional algebra. The proof is complete.
?
Lemma 3.3 implies the following
Corollary 3.1. Let M be a commutative von Neumann algebra with a faithful
normal finite trace τ such that the Boolean algebra P(M) of all projections of M is
continuous. Then every additive derivation on the algebra LΛ(M,τ) is zero.
Note that the properties of additive derivations on the algebras S(M,τ) and
LΛ(M,τ), where M be a commutative von Neumann algebra with a faithful normal
finite trace τ, are quite opposite. Indeed, if the Boolean algebra P(M) is continuous
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then the algebra S(M,τ) admits a non-zero linear, in particular additive, derivation,
(see [10, Theorem 3.3]), whereas the algebra LΛ(M,τ) in this case does not admit a
non-zero additive derivation (see Corollary 3.1).
Now we consider the noncommutative case.
We shall need following result ([7, Theorem 4.1], see also [9, Theorem 6.8]).
Theorem 3.1. 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 linear
derivation on A is inner.
The following theorem is one of the main results of this paper.
Theorem 3.2. Let M be a type II von Neumann algebra with a faithful normal
finite trace τ. Then every additive derivation on the algebra LΛ(M,τ) is inner.
The proof of the theorem 3.2 follows from Theorem 3.1 and the following assertion.
Lemma 3.4. Let M be a type II von Neumann algebra with a faithful normal finite
trace τ, and suppose that D : LΛ(M,τ) → LΛ(M,τ) is an additive derivation. Then
D|Z(LΛ(M,τ))≡ 0, in particular, D is a linear.
Proof. Let D be an additive derivation on LΛ(M,τ), and let δ be its restriction onto
Z(LΛ(M,τ)).
Since M is of type II there exists a sequence of mutually orthogonal projections
{pn}∞
n=1in M with central covers 1 (i.e.the {pn} are faithful projections). For any
bounded sequence B = {bn}n∈Nin Z(M) define an operator xBby
xB=
∞
?
n=1
bnpn.
Then
xBpn= pnxB= bnpn
(7)
for all n ∈ N.
Take b ∈ Z(M) and n ∈ N. From the identity
D(bpn) = D(b)pn+ bD(pn)
multiplying it by pnon both sides we obtain
pnD(bpn)pn= pnD(b)pn+ bpnD(pn)pn.
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Since pn is a projection, one has that pnD(pn)pn = 0, and since D(b) = δ(b) ∈
Z(LΛ(M,τ)), we have
pnD(bpn)pn= δ(b)pn. (8)
Now from the identity
D(xBpn) = D(xB)pn+ xBD(pn),
in view of (7) one has similarly
pnD(bnpn)pn= pnD(xB)pn+ bnpnD(pn)pn,
i.e.
pnD(bnpn)pn= pnD(xB)pn. (9)
Now (8) and (9) imply
pnD(xB)pn= δ(bn)pn. (10)
Let ϕ ∈ Λ. By (1) there are φ,ψ ∈ Λ and k > 0 such that
||x1x2x3||ϕ≤ k||x1||φ||x2||φ||x3||ψ
for all x1,x2,x3∈ LΛ(M,τ). If we suppose that δ ?= 0 then zδ?= 0. By Lemma 3.2 there
exists a bounded sequence B = {bn}n∈Nin Z(M) such that
|δ(bn)| ≥ ncnzδ
for all n ∈ N, where cn= k||pn||2
φ||pnzδ||−1
ϕ. Then in view of (10) we obtain
k||pn||φ||D(x)||ψ||pn||φ≥ ||pnD(x)pn||ϕ=
= ||δ(bn)pn||ϕ≥ ||ncnpnzδ||ϕ= ncn||pnzδ||ϕ,
i.e.
||D(x)||ψ≥ ncnk−1||pn||−2
φ||pnzδ||ϕ.
Hence
||D(x)||ψ≥ n
for all n ∈ N. This contradiction implies that δ ≡ 0, i.e. D is identically zero on the
center of LΛ(M,τ), and therefore it is linear. The proof is complete. ?
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Now consider an additive derivation D on LΛ(M,τ) and let δ be its restriction onto
its center Z(LΛ(M,τ)). By Lemma 3.3 zδM is a finite-dimensional and z⊥
δδ ≡ 0, i.e.
δ = zδδ.
Let Dδbe the derivation on zδLΛ(M,τ) = zδM defined as in (4) and consider its
extension Dδon LΛ(M,τ) = zδLΛ(M,τ) ⊕ z⊥
δLΛ(M,τ) which is defined as
Dδ(x1+ x2) := Dδ(x1), x1∈ zδLΛ(M,τ),x2∈ z⊥
δLΛ(M,τ). (11)
The following theorem is the main result of this paper, and gives the general form
of derivations on the algebra LΛ(M,τ).
Theorem 3.3. Let M be a von Neumann algebra with a faithful normal finite trace
τ. Each additive derivation D on LΛ(M,τ) can be uniquely represented in the form
D = Da+ Dδ
where Dais an inner derivation implemented by an element a ∈ LΛ(M,τ), and Dδis
an additive derivation of the form (11), generated by an additive derivation δ on the
center of LΛ(M,τ).
Proof. Let D be an additive derivation on LΛ(M,τ), and let δ be its restriction onto
Z(LΛ(M,τ)) = LΛ(Z(M),τZ)). By Lemma 3.3 zδZ(M) is finite-dimensional. Thus zδM
is a C∗-product of a finite number of von Neumann factors of type Inor II. Since by
Lemma 3.4 any additive derivation on LΛ(M,τ), where M is a type II algebra, is linear,
then by Theorem 3.2 it is inner. Therefore zδM is a C∗-product of a finite number of
von Neumann factors of type In.
Now consider an additive derivation Dδon LΛ(M,τ) of the form (11), generated by
a derivation δ. Since the derivations D and Dδcoincide on LΛ(Z(M),τ)) then D − Dδ
is a linear derivation. Hence Theorem 3.2 implies that the derivation D − Dδis inner.
This means that there exists an element a ∈ LΛ(M,τ) such that Da= D − Dδ and
therefore D = Da+ Dδ. The proof is complete. ?
Theorem 3.3 implies that following.
Corollary 3.2. Let M be a von Neumann algebra without type Indirect summands
and with a faithful normal finite trace τ. Then each additive derivation on LΛ(M,τ) is
inner.
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Acknowledgments. The third named author would like to acknowledge the hos-
pitality of the ”Institut f¨ ur Angewandte Mathematik”, Universit¨ at Bonn (Germany).
This work is supported in part by the German Academic Exchange Service – DAAD .
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