Tiểu Luận The derivation problem for group algebras

The derivation problem for group algebras
ByViktor Losert
Abstract
If Gis a locally compact group, then for each derivation Dfrom L
1
(G)
into L
1
(G) there is a bounded measureà∈M(G) withD(a)=a∗àưà∗a
for a∈L
1
(G) (“derivation problem” of B. E. Johnson).
Introduction
LetAbe a Banach algebra, Ean A-bimodule. A linear mapping
D: A→Eis called a derivation,ifD(ab)=aD(b)+D(a)bfor all a, b∈A
([D, Def. 1.8.1]). Forx∈E, we define the inner derivation adx: A→Eby
adx(a)=xaưax(as in [GRW]; adx=ưδxin the notation of [D, (1.8.2)]).
IfGis a locally compact group, we consider the group algebra A=L
1
(G)
andE=M(G), with convolution (note that by Wendel’s theorem [D, Th.
3.3.40],M(G) is isomorphic to the multiplier algebra ofL
1
(G) and also to the
left multiplier algebra). The derivation problemasks whether all derivations
are inner in this case ([D, Question 5.6.B, p. 746]). The question goes back to
J. H. Williamson around 1965 (personal communication by H. G. Dales). The
corresponding problem whenA=Eis a von Neumann algebra was settled
affirmatively by Sakai [Sa], using earlier work of Kadison (see [D, p. 761] for
further references). The derivation problem for the group algebra is linked
to the name of B. E. Johnson, who pursued it over the years as a pertinent
example in his theory of cohomology in Banach algebras. He developed various
techniques and gave affirmative answers in a number of important special cases.
As an immediate consequence of the factorization theorem, the image of
a derivation fromL
1
(G)toM(G) is always contained inL
1
(G). In [JS] (with
A. Sinclair), it was shown that derivations onL
1
(G) are automatically contin-uous. In [JR] (with J. R. Ringrose), the case of discrete groupsGwas settled
affirmatively. In [J1, Prop. 4.1], this was extended to SIN-groups and amenable
groups (serving also as a starting point to the theory of amenable Banach al-gebras). In addition, some cases of semi-simple groups were considered in [J1]
and this was completed in [J2], covering all connected locally compact groups.
222 VIKTOR LOSERT
A number of further results on the derivation problem were obtained in [GRW]
(some of them will be discussed in later sections).
These problems were brought to my attention by A. Lau.
1. The main result
We use a setting similar to [J2, Def. 3.1]. Ω shall be a locally compact
space,Ga discrete group acting on Ω by homeomorphisms, denoted as a left
action(or a leftG-module), i.e., we have a continuous mapping (x, ω)→x◦ω
fromG×Ω to Ω such thatx◦(y◦ω)=(xy)◦ω, e◦ω=ωforx, y∈G, ω∈Ω.
ThenC0(Ω), the space of continuous (real- or complex-valued) functions on Ω
vanishing at infinity becomes a right BanachG-module by (h◦x)(ω)=h(x◦ω)
for h∈C0(Ω),x∈G, ω∈Ω. The spaceM(Ω) of finite Radon measures
on the Borel setsBof Ω will be identified with the dual spaceC0(Ω)

in the
usual way and it becomes a left BanachG-module by x◦à, h=à,h◦x
for à∈M(Ω),h∈C0(Ω),x∈G(in particular,x◦δω=δx◦ωwhenà=δωis
a point measure withω∈Ω ; see also [D,§3.3] and [J2, Prop. 3.2]).
A mapping Φ :G→M(Ω) (or more generally, Φ :G→X, whereXis a left
BanachG-module) is called acrossed homomorphismif Φ(xy)=Φ(x)+x◦Φ(y)
for all x, y∈G([J2, Def. 3.3]; in the terminology of [D, Def. 5.6.35], this is a
G-derivation, if we consider the trivial right action of GonM(Ω) ). Now, Φ
is called bounded if Φ = supx∈G Φ(x) <∞.Forà∈M(Ω), the special
example Φà(x)=àưx◦àis called a principalcrossed homomorphism (this
follows [GRW]; the sign is taken opposite to [J2]).
Theorem 1.1.LetΩbe a locally compact space, Ga discrete group with
a left action ofGonΩby homeomorphisms. Then any bounded crossed ho-momorphismΦfrom Gto M(Ω) is principal. There exists à∈M(Ω) with
à ≤2 Φ such that Φ=Φà.
Corollary 1.2.LetGdenote a locally compact group. Then any deriva-tion D:L
1
(G)→M(G)is inner.
Using [D, Th. 5.6.34 (ii)], one obtains the same conclusion for all deriva-tionsD:M(G)→M(G).
Proof. As mentioned in the introduction, we have D(L
1
(G)) ⊆L
1
(G)
and thenDis bounded by a result of Johnson and Sinclair (see also [D, Th.
5.2.28]). Then by further results of Johnson,Ddefines a bounded crossed
homomorphism Φ fromGtoM(G) with respect to the actionx◦ω=xω xư1
ofGonG([D, Th. 5.6.39]) and (applying our Theorem 1.1) Φ = Φàimplies
D=adà.
Corollary 1.3.LetGdenote a locally compact group, Ha closed sub-group. Then any bounded derivationD:M(H)→M(G)is inner.
THE DERIVATION PROBLEM FOR GROUP ALGEBRAS 223
Again, the same conclusion applies to bounded derivationsD: L
1
(H)→
M(G).
Proof. M(H) is identified with the subalgebra ofM(G) consisting of those
measures that are supported byH(this gives also the structure of anM(H)-module considered in this corollary). As above,Ddefines a bounded crossed
homomorphism Φ fromHto M(G) (for the restriction to Hof the action
considered in the proof of 1.2) and our claim follows.
Corollary 1.4.For any locally compact groupG, the first continuous
cohomology groupH1
(L
1
(G),M(G))is trivial.
Note that
H1
(M(G),M(G)) =H1
(L
1
(G),M(G))
holds by [D, Th. 5.6.34 (iii)].
Proof. Again, this is contained in [D, Th. 5.6.39].
Corollary 1.5.LetGbe a locally compact group and assume thatT∈
VN(G) satisfies T∗uưu∗T∈M(G) for all u∈L
1
(G). Then there exists
à∈M(G)such thatTưàbelongs to the centre ofVN(G).
Proof. This is Question 8.3 of [GRW]. With VN(G) denoting the von
Neumann algebra ofG(see [GRW, §1]),M(G) is identified with the corre-sponding set of left convolution operators onL
2
(G) (see [D, Th. 3.3.19]) and
is thus considered as a subalgebra of VN(G). By analogy, we also use the
notationS∗Tfor multiplication in VN(G). Then adT(u)=T∗uưu∗Tdefines
a derivation fromL
1
(G)toM(G) and (from Corollary 1.2) adT=adàimplies
that TưàcentralizesL
1
(G). Since L
1
(G) is dense in VN(G) for the weak
operator topology, it follows thatTưàis central.
Remark1.6.IfGis a locally compact group with a continuous action on Ω
(i.e., the mappingG×Ω→Ω is jointly continuous; by the theorem of Ellis,
this results from separate continuity), then Theorem 1.1 implies that bounded
crossed homomorphisms from Gto M(Ω) are automatically continuous for
the w*-topology on M(Ω), i.e., for σ(M(Ω),C0(Ω)) (since in this case the
right action of GonC0(Ω) is continuous for the norm topology). This is
a counterpart to [D, Th. 5.6.34(ii)] which implies that bounded derivations
from M(G) to a dual moduleE
are automatically continuous for the strong
operator topology onM(G) and the w*- topology onE
. See also the end of
Remark 5.6.