On Spaces of Fourier-Stieltjes Transform of Vector
Measures on Compact Groups
Yaogan MENSAHa,1, V.S.K. ASSIAMOUAb
aDepartment of Mathematics, University of Lom´e, (Togo) and International Chair in Mathematical
Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, (Benin)
bDepartment of Mathematics, University of Lom´e, (Togo)
Abstract
In [1], the author extended to vector measures on a compact non commutative group the notion of Fourier-Stieltjes transform. This brings to light some functional spaces. In this paper, we study some of their topological properties. In particular, we found their dual spaces.
Keywords: Fourier transform, Compact group, Banach space c
2010 Published by Islamic Azad University-Karaj Branch.
1
Introduction
The Fourier transform of a complex valued function on a commutative locally compact
groupG, such as Rn, is again a complex valued function on the character group X of
G. Otherwise, it is a familly (Eσ)σ∈Σ of continuous linear operators Eσ :Hσ → Hσ,
where Σ is the dual object of the compact non commutative groupG, andσ, a class of
irreducible unitary representations ofG in a Hilbert space Hσ.
In case C is replaced by a Banach space A, it is a familly of continuous
sesquilin-ear mappings φ(σ) : Hσ ×Hσ → A. In fact, for each σ ∈ Σ, we choose once and
for all an element Uσ in σ, denote its representation space by H
σ, and fix an
or-thonormal basis (ξσ1, . . . , ξdσσ) of Hσ, wheredσ =dimHσ, as a canonical basis. We put
uσ
ij(t) =hUtσξjσ, ξiσiand introduce the operatorU σ
onHσ such thathU σ
tξσj, ξiσi=uσij(t),
the complex conjugate of uσij(t) The Fourier-Stieltjes transform on G for anA-valued
bounded vector measurem, whereA is a normed space, is given by :
ˆ
m(σ)(ξ, η) =
Z
G
hUσtξ, ηidm(t) (ξ, η)∈Hσ×Hσ.
(For details on vector measures See [4] and [5]). The mappingHσ×Hσ →A,(ξ, η)→
ˆ
m(σ)(ξ, η) is a continuous and sesquilinear [1].
This generates a certain number of interesting linear spaces Sp(Σ, A) that we specify
as follow.
We write Q σ∈Σ
S(Hσ×Hσ, A) =S(Σ, A) whereS(Hσ×Hσ, A) is the space of continuous
sesquilinear mappings fromHσ ×Hσ into A. S(Σ, A) is a linear space with addition
and multiplication by scalars, defined coordinatewise. Forφ∈ S(Σ, A), we put :
kφk∞= sup{kφ(σ)k|σ ∈Σ},
withkφ(σ)k= sup{kφ(σ)(ξ, η)k | kξk ≤1,kηk ≤1}. We denote by
S∞(Σ, A), the space{φ∈ S(Σ, A)|kφk∞<∞},
S00(Σ, A), the space{φ∈ S∞(Σ, A)|{σ∈Σ|φ(σ)6= 0}is finite }
andS0(Σ, A), the space
{φ∈ S∞(Σ, A)|∀ε >0,{σ∈Σ|kφ(σ)k> ε}is finite }.
In [2] it is proved that :
1. The mappingφ→ kφk∞is a norm onS∞(Σ, A), andS∞(Σ, A) is a Banach space
with respect to this norm.
2. S00(Σ, A) is dense inS0(Σ, A).
3. Everyφ(σ)∈ S(Hσ×Hσ, A) is determined by thedσ2 elements aσij =φ(σ)(ξjσ, ξσi)
of A. More precisely, we have :
φ(σ) = Pdσ i,j=1
4. The mappingm →mˆ from M1
(G, A), the space of A-valued bounded measures
on Ginto S∞(Σ, A), is linear, injective and continuous.
2
Main Results
2.1 The spaces Sp(Σ, A) 1≤p≤ ∞
We define :
Sp(Σ, A) ={φ∈ S(Σ, A)| X
σ dσ
X
ij
kφ(σ)(ξjσ, ξσi)kp <∞}, 1≤p <∞,
and S∞(Σ, A) as in the introduction. They are linear spaces for pointwise operations.
We define a norm onSp(Σ, A) by
kφkp=
X
σ∈Σ
dσ X
i,j
kφ(σ)(ξjσ, ξσi)k
1
p .
Theorem 2.1 For each p, 1≤p≤ ∞, the space Sp(Σ, A) is a Banach space.
Proof. The casep=∞ was done in [2] .
Let (φn) be a Cauchy sequence from the space Sp(Σ, A). Then for each σ ∈ Σ, the
sequence (φn(σ))n is a Cauchy sequence from the spaceS(Hσ×Hσ, A) which is known
to be a Banach space. Thus there existsφ(σ)∈ S(Hσ×Hσ, A) such that
lim
n→∞kφn(σ)−φ(σ)k= 0. (1)
Setασij =φ(σ)(ξjσ, ξiσ) and for alln,aσ,nij =φn(σ)(ξσj, ξiσ).
We considerε >0. Since (φn) is a Cauchy sequence, then there existsn0 ∈Nsuch that
∀r, s≥n0,kφr−φskp < ε
1
p (2)
i.e X σ
dσ X
i,j
Lettingstends to infinity in (3), we have :
X
σ dσ
X
i,j
kaσ,rij −αijσkp< ε (4)
i.e kφr−φkp < εpour r≥n0. (5)
We have kφkp = kφ−φr+φrkp
≤ kφ−φrkp+kφrkp
≤ ε+kφrkp <∞.
Henceφ∈ Sp(Σ, A). Finally (5) shows that (φn) converges toφinSp(Σ, A).
2.2 Duality in the spaces Sp(Σ, A)
Theorem 2.2 Let p, q be such that1≤p <∞, 1p +1
q = 1 and A∗ be the dual of A.
Then the space(Sp(Σ, A))∗ is isometric to Sq(Σ, A∗).
Proof. The proof of the casep= 1 (which impliesq=∞) can be found in [9]. Now,
let 1< p <∞. Let T :Sq(Σ, A∗)→(Sp(Σ, A))∗, ϕ7→T ϕ
be defined byhT ϕ, ψi = P σ∈Σ
dσP i,j
hbσ
ij, aσiji, ψ ∈ Sp(Σ, A) where bσij = ϕ(σ)(ξjσ, ξσi)
and aσij = ψ(σ)(ξjσ, ξσi). Then the Theorem is a consequence of the following three
lemmas.
Proof. The linearity ofT is trivial. Let us show that it is bounded.
We have |hT ϕ, ψi| = | P σ∈Σ
dσP i,j
hbσ ij, aσiji|
≤ P
σ∈Σ
dσP i,j
|hbσ ij, aσiji|
≤ P
σ∈Σ
dσP i,j
kbσijkkaσijk
≤ P
σ∈Σ
P i,j d
1
q σkbσijkd
1
p σkaσijk
≤ P
σ∈Σ
P i,j
dσkbσijkq !1
q P σ∈Σ
P i,j
dσkaσijkp !1
p
≤ kϕkqkψkp.
SokT ϕk ≤ kϕkq and therefore T is bounded with kTk ≤1.
Lemma 2.4 The equality kTk= 1 holds.
Proof. From part 1, we havekTk ≤1. Let us show now that kTk ≥1.
Take a∈ A, such that kak = 1. Since a6= 0, we know from Functional analysis that
there existsb∗∈A∗ such thatkb∗k= 1 and hb∗, ai=kak= 1.
Given a fixedτ ∈Σ, we use the Kronecker symbolδij to defineψτ ∈ Sp(Σ, A) by
ψτ(σ)(ξjσ, ξiσ) =aσij = d− 2 p
τ aδij ifσ =τ
0 if σ6=τ
andϕτ inSq(Σ, A∗) by :
ϕτ(σ)(ξjσ, ξiσ) =bσij = d− 2 q
τ b∗δij ifσ=τ
0 if σ6=τ.
We have kϕτkqq = X
σ∈Σ
dσ X
ij
kbσijkq=X σ∈Σ
dσ X
ij
kd−
2
q
and kψτkpp = X σ dσ X ij
kaσijkp =X
σ dσ
X
ij
kd−
2
p
τ aδijkp = 1.
As such, hT ϕτ, ψτi = P σ
dσP ij
hbσij, aσiji
= P
σ dσP
ij
hd−
2
q
τ b∗δij, d
−2
p τ aδiji
= dτ P
i
hd−
2
q τ b∗, d
−2
p τ ai
= d2τ
d
1
q+1p τ
−2
hb∗, ai= 1 =kϕkqkψkp.
HencekTk ≥1. Finally kTk= 1.
Lemma 2.5 The mapping T is surjective.
Proof. In fact letf ∈(Sp(Σ, A))∗. Forτ ∈Σ, let
Vτ ={ψ∈ Sp(Σ, A)|ψ(σ) = 0 if σ6=τ, σ∈Σ}.
For ψ ∈ Vτ, let aτij = ψ(τ)(ξjτ, ξiτ), i, j = 1, . . . , dτ. There exists linear forms bτij ∈
A∗, i, j = 1, . . . , dτ such thathf, ψi =dτP i,j
hbτij, aτiji. In fact, given d2
τ scalars λτij such
thatP ij
λτ ij =
hf,ψi
dτ , there exists bij ∈A
∗ withhb
ij, aτiji = 1 ; denoting bτij =λτijbij, we
have what is required.
Now, let us consider an elementφof S00(Σ, A).
Since S00(Σ, A) is a subset of S2(Σ, A), one can write, according to the Riesz-Fischer
theorem, φ = P τ∈Σ
dτP ij
aτ
ijuˆτij. In fact, there exists a finite subset Σ′ of Σ such that
φ= P τ∈Σ′
dτP ij
aτijuˆτij. Puttingφτ =dτP ij
aτijuˆτij,
we haveφ = P τ∈Σ′
φτ. It is clear that φτ belongs to Vτ because, for σ 6=τ, ˆuτij(σ) = 0
(Schur’s orthogonality property), soφτ(σ) = P
ij
aτijuˆτij(σ) = 0. Thus, there exist linear
formsbτ
ij ∈A∗, i, j= 1, . . . , dτ such that
hf, φτi=dτ X
i,j
Now by linearity off,
hf, φi= X
τ∈Σ′ dτ
X
i,j
hbτij, aτiji.
Definingϕby :
ϕ(τ)(ξjτ, ξiτ) =bijτ ifτ ∈Σ′ and ϕ(τ)(ξτ
j, ξτi) = 0 otherwise,
we have ϕ ∈ S00(Σ, A∗) ⊂ Sq(Σ, A∗) and hf, φi = hT ϕ, φi. This means that the
continuous linear forms f and T ϕ co¨ıncide on S00(Σ, A) which is a dense subset of
Sp(Σ, A).
Hencef =T ϕ.
The three lemmas show thatT is an isometry from Sq(Σ, A∗) onto (Sp(Σ, A))∗.
References
[1] ASSIAMOUA V.S.K. (1989) ”L1(G,A)-multipliers,” Acta Sci. Math., 53, 309-318.
[2] ASSIAMOUA V.S.K., OLUBUMMO A. (1989) ”Fourier-Stieltjes transforms of
vector-valued measures on compact groups,” Acta Sci. Math., 53, 301-307.
[3] ASSIAMOUA V.S.K., MENSAH Y. (2008) ”The Fourier algebra A1(G, A) of
vector-valued functions on compact groups,” Contemporary Problems in
Math-ematical Physics, 223-230.
[4] DIESTEL J., UHL Jr J.J., Vector measures, Amer. Math-Soc, Math surveys,n15,
1977.
[5] DINCULEANU N., Integration on locally compact spaces, Noorhoff International
Publishing, Leyden, 1974.
[6] EFFROS E. (1991) ”A new approach to operator spaces,” Canadian Math. Bull.,
34, 329-337.
[7] HEWITT E., ROSS K.A., Abstract Harmonic Analysis, Volume I and II,
[8] KOSAKI H. (1984) ”Applications of the complex interpolation method to a von
Neumann algebra: non commutativeLp-spaces,” J. Funct. Anal., 56, 29-78.
[9] MENSAH Y., ASSIAMOUA V.S.K. (2009) ”The dual of the Fourier algebra
A1(G, A) of vector valued functions on compact groups,” Afr. Diaspora J. Math.,
Volume 8, Number 1, 28-34.
[10] PISIER G., Non-commutative vector valuedLp-spaces and completely p-summing
