Ideals with bounded approximate identities in Fourier algebras

Abstract

We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.

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Journal of Functional Analysis 203 (2003) 286–304
Ideals with bounded approximate identities
in Fourier algebras
B. Forrest,a,1E. Kaniuth,b,* A.T. Lau,c,1and N. Spronka
aDepartment of Pure Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1
bFachbereich Mathematik/Informatik, Universita ¨t Paderborn, D-33095 Paderborn, Germany
cDepartment of Mathematical Sciences, University of Alberta, Edmonton, Alta., Canada T6G 2G1
Received 10 July 2002; accepted 23 September 2002
Abstract
We make use of the operator space structure of the Fourier algebra AðGÞ of an amenable
locally compact group to prove that if H is any closed subgroup of G; then the ideal IðHÞ
consisting of all functions in AðGÞ vanishing on H has a bounded approximate identity. This
result allows us to completely characterize the ideals of AðGÞ with bounded approximate
identities. We also show that for several classes of locally compact groups, including all
nilpotent groups, IðHÞ has an approximate identity with norm bounded by 2, the best possible
norm bound.
r 2002 Elsevier Science (USA). All rights reserved.
Let G be a locally compact group and AðGÞ the Fourier algebra of G as introduced
by Eymard [6]. AðGÞ is the linear subspace of C0ðGÞ consisting of all coordinate
functions of the regular representation of G and can be identified with the predual of
the group von Neumann algebra VNðGÞ generated by left translations on L2ðGÞ:
With the norm given by this duality and pointwise multiplication, AðGÞ becomes a
commutative Banach algebra. The spectrum of AðGÞ is G (point evaluation of
functions in AðGÞ). Leptin [20] has shown that AðGÞ has a bounded approximate
identity precisely when G is amenable.
One of the most interesting open problems in the ideal theory of AðGÞ is to
determine the closed ideals of AðGÞ with bounded approximate identities. Recall that
when G is abelian, AðGÞ is isometrically isomorphic (by means of the inverse Fourier
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*Corresponding author.
E-mail addresses: beforrest@math.uwaterloo.ca (B. Forrest), kaniuth@math.uni-paderborn.de
(E. Kaniuth), tlau@math.ualberta.ca (A.T. Lau), nspronk@math.uwaterloo.ca (N. Spronk).
1Supported by Grants from NSERC Canada.
0022-1236/03/$-see front matter r 2002 Elsevier Science (USA). All rights reserved.
doi:10.1016/S0022-1236(02)00121-0

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transform) to L1ðˆGÞ; the L1-convolution algebra on the dual groupˆG of G: Now, for
an abelian locally compact group G the closed ideals of L1ðGÞ with bounded
approximate identities have been completely characterized by Liu et al. [21]. This
characterization suggests to conjecture that a closed ideal I of AðGÞ has a bounded
approximate identity if and only if I equals
IðEÞ ¼ fuAAðGÞ : uðxÞ ¼ 0 for all xAEg
for some set E in the (closed) coset ring of G: This conjecture has been verified in [8]
for amenable groups with small conjugation invariant neighbourhoods of the
identity (so-called SIN-groups).
In this paper we shall first use the fact that the Fourier algebra has an additional
structure as an operator space to show that if G is amenable and H is any closed
subgroup of G; then IðHÞ has a bounded approximate identity (Theorem 1.5). This
allows us to establish the above conjecture for arbitrary amenable locally compact
groups (Theorem 2.3). Some of these results extend to the more general Herz–Figa ` –
Talamanca algebras ApðGÞ;1opoN (Theorem 4.3).
Using results from Delaporte and Derighetti [2], it can be shown that 2 is the best
possible norm bound for an approximate identity of IðHÞ as long as H is not open in
G (see [16]). In [16] it was shown that the norm bound 2 is achieved if the pair ðG;HÞ
satisfies a certain separation property of positive definite functions. This property is
satisfied in SIN-groups G for all closed subgroups H; but most likely fails in any
other class of locally compact groups (see [16,18]). We proceed to show that,
nevertheless, if G is a locally nilpotent group or an almost connected group with a
compact normal subgroup K such that G=K is nilpotent, then indeed IðHÞ has an
approximate identity with norm bounded by 2 for every closed subgroup H of G
(Theorems 3.3 and 3.7). The question of whether this holds true for all amenable
groups, remains open.
0. Some preliminaries
Let A be a commutative Banach algebra. A net fuag in A is called a bounded
approximate identity if jjuau ? ujj-0 for every uAA and if there exists c40 such that
jjuajjpc for all a: We refer to c as a norm bound of the net fuag:
An operator space is a vector space V together with a family fjj ? jjng of norms on
MnðVÞ; the vector space of n ? n matrices with entries in V such that
(i)
A
0
0
B
nþm
(ii) jj½aij?A½bij?jjnpjj½aij?jj ? jjAjjn? jj½bij?jj for all ½aij?;½bij?AMnðCÞ and AAMnðVÞ:
??
???? ????
????
????
¼ maxfjjAjjn;jjBjjmg for each AAMnðVÞ and BAMmðVÞ; and
Let X and Y be operator spaces and let T : X-Y be linear. For each nAN; we
define TðnÞ: MnðXÞ-MnðYÞ by TðnÞð½aij?Þ ¼ ½TðaijÞ?: T is said to be completely
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bounded if
jjTjjcb¼ sup
nAN
fjjTðnÞjjgoN:
Moreover, T is said to be completely contractive if jjTjjcbp1:
If V is a linear subspace of BðHÞ; the Cn-algebra of bounded operators on a
Hilbert space, then V is an operator space with the matricial norms inherited from
MnðVÞDMnðBðHÞÞ identified with BðHnÞ: In fact, every operator space V is
completely isometric to a vector subspace of BðHÞ for some Hilbert space H
[5, Theorem 2.3.5]. If X and Y are subspaces of BðHÞ; then a linear map T : X-Y
is called completely positive if each TðnÞis a positive map. Concerning all these
notions we refer the reader to [5,23].
Let G be a locally compact group and let PðGÞ denote the set of all continuous
positive definite functions on G: The linear span of PðGÞ; BðGÞ; forms an algebra
called the Fourier–Stieltjes algebra of G: Then BðGÞ can be identified with the dual
space of the group Cn-algebra CnðGÞ (see [6]). Let AðGÞ denote the closed ideal of
BðGÞ generated by all functions with compact support in BðGÞ: As shown in [6],
AðGÞ consists of all functions of the form uðxÞ ¼PN
norm on AðGÞ is given by jjujj ¼ inffPN
generated by the operators rðxÞ; xAG; on L2ðGÞ:
Since AðGÞ is an ideal in BðGÞ and VNðGÞ ¼ AðGÞn; there is a natural action of
BðGÞ on VNðGÞ given by
n¼1ðfn*e gn gnÞðx?1Þ; where
n¼1ðfn*e gn gnðx?1Þg
fn;gnAL2ðGÞ andPN
and AðGÞ can be identified with the predual of the von Neumann algebra VNðGÞ
n¼1jjfnjj2jjgnjj2oN: Hereef f is defined by˜fðxÞ ¼ fðx?1Þ: The
n¼1jjfnjj2jjgnjj2: uðxÞ ¼PN
/u ? T;vS ¼ /T;uvS
for vAAðGÞ and uABðGÞ: We will say that a linear map P: VNðGÞ-VNðGÞ
commutes with the action of AðGÞ if Pðu ? TÞ ¼ u ? PðTÞ for all TAVNðGÞ and
uAAðGÞ:
Since VNðGÞ is a von Neumann algebra, its predual AðGÞ inherits a natural
structure as an operator space with respect to which it becomes a completely
contractive Banach algebra. That is, the multiplication on AðGÞ extends to a
completely contractive map m : AðGÞ# #AðGÞ-AðGÞ with mðu#vÞ ¼ uv: Here
AðGÞ# #AðGÞ denotes the operator space analogue of the projective tensor product.
This allows us to apply operator space techniques to study problems concerning
ideals in AðGÞ: In particular, we shall make direct use of Ruan’s remarkable
generalization of Johnson’s characterization of amenable group algebras, namely
that G is amenable as a locally compact group if and only if AðGÞ is operator
amenable. We refer the reader to [5,24] for details.
Let I be a closed ideal in AðGÞ: Then I>¼ fTAVNðGÞ : /T;uS ¼ 0 for each
uAIg is an AðGÞ-submodule of VNðGÞ: Let H be a closed subgroup of G; and define
VNHðGÞ to be the von Neumann subalgebra of VNðGÞ generated by frðxÞ : xAHg:
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Then IðHÞ>¼ VNHðGÞ (see, for example [16, p. 101]). Moreover, VNHðGÞ is known
to be*-isomorphic to VNðHÞ:
1. The ideals IðHÞ
In this section we shall use the operator space structure of the Fourier algebra
AðGÞ of an amenable group G to show that for each closed subgroup H of G the
ideal IðHÞ has a bounded approximate identity.
Lemma 1.1. Let H be an amenable locally compact group. Then there exists a
completely contractive projection P: BðL2ðHÞÞ-VNðHÞ:
Proof. First note that VNðHÞ is the commutant of frðxÞ : xAHg; where
r: H-BðL2ðHÞÞ is the right regular representation of H: Let m be a left invariant
mean on LNðHÞ: It will be convenient for us to regard m as a finitely additive
measure on H: Now define P : BðL2ðHÞÞ-BðL2ðHÞÞ to be the weak operator
converging integral
Z
PðTÞ ¼
H
rðsÞTrðsÞndmðsÞ
for each TABðL2ðHÞÞ: That is,
/PðTÞf;gS ¼ mðs-/rðsÞTrðsÞnf;gSÞ
for each f;gAL2ðHÞ:
From the invariance of m; it is easy to see that for each TABðL2ðHÞÞ;
rðtÞPðTÞ ¼ PðTÞrðtÞ
for all tAH: It follows that PðTÞAVNðHÞ: It is also clear that if TAVNðHÞ; then
PðTÞ ¼ T:
We have that P is completely positive, since each amplification
PðnÞ: MnðBðL2ðHÞÞÞ-MnðVNðHÞÞ
is given by the weak operator converging integral
PðnÞð½Tij?Þ ¼
Z
H
diagðrðsÞÞ½Tij?diagðrðsÞÞndmðsÞ;
for each ½Tij?AMnðBðL2ðHÞÞÞ; where diagðrðsÞÞ denotes the n ? n diagonal matrix
with all diagonal entries equal to rðsÞ: Finally, since PðIÞ ¼ I; we get that P is
completely contractive.
&
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Assume that H is an amenable group. We claim that if M is a von Neumann
algebra on a Hilbert space H such that M is*-isomorphic to VNðHÞ; then there
is a contractive completely positive expectation ˜P: BðHÞ-M: To see this let
F :VNðHÞ-M be a*-isomorphism. The Arveson-Wittstock Extension Theorem
(see [5, Theorem 4.1.5]) implies that F?1:M-VNðHÞ admits a completely
contractive extension C : BðHÞ-BðL2ðHÞÞ: Moreover, since CðIÞ ¼ I; C is
completely positive [5, Corollary 5.1.2]. We now let ˜P ¼ F 3 P3 C where P is the
projection constructed in Lemma 1.1.
Proposition 1.2. Let G be a locally compact group and let H be an amenable closed
subgroup of G: Then there exists a completely contractive projection from VNðGÞ
onto IðHÞ>:
Proof. We recall that IðHÞ>¼ VNHðGÞ and that VNHðGÞ is*-isomorphic to
VNðHÞ: It follows from Lemma 1.1 and the discussion preceding this proposition
that there exists a completely contractive projection
˜P : BðL2ðGÞÞ-VNHðGÞ ¼ IðHÞ>:
We now simply restrict˜P to VNðGÞ:
&
For G s-compact and H an amenable closed subgroup, the existence of a
continuous projection from VNðGÞ onto VNðHÞ was established by Lohoue ´ [22].
For G not necessarily s-compact and H a closed amenable subgroup, the existence of
a contractive projection from VNðGÞ onto VNðHÞ was obtained by Derighetti [4].
However, for our purposes, we will need the additional fact provided by Proposition
1.2 that the projection can be chosen to be a complete contraction.
Theorem 1.3. Let G be an amenable locally compact group and H a closed subgroup of
G: Then there exists a completely bounded projection P: VNðGÞ-IðHÞ>such that
Pðu ? TÞ ¼ u ? PðTÞ
for all TAVNðGÞ and uAAðGÞ:
Proof. Since G is amenable, the Fourier algebra AðGÞ is an operator amenable
completely contractive Banach algebra. By Proposition 1.2, there is a completely
contractive projection from VNðGÞ onto IðHÞ>: The existence of an invariant
completely bounded projection now follows immediately from [28, Theorem 3].
&
Let G be a locally compact group and let M be an invariant Wn-subalgebra of
VNðGÞ: Let H ¼ fxAG : rðxÞAMg: Then H is a closed subgroup of G and M ¼
IðHÞ>[26, Theorem 3]. In [19] it was shown that if H is normal or, more generally,
G has the H-separation property, then M is invariantly complemented, that is, there
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