SPECTRAL SYNTHESIS PROBLEM FOR FOURIER ALGEBRAS

Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
SPECTRAL SYNTHESIS PROBLEM
FOR
FOURIER ALGEBRAS
Kunda Chowdaiah
NISER
November 25, 2011
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL

Preliminaries
Spectral synthesis
References
Preliminaries
Banach algebra
A Banach algebra A is a Banach space that is also an algebra and
satisfying the following condition
xy ≤ x y , ∀ x , y ∈ A
.
An algebra A is called commutative if xy = yx ∀x , y ∈ A.
For any Banach algebra A, We Dene
∆(A) = {ϕ : A → C | ϕ is non-zero homomorphism}.
∆(A) ⊆ A∗ with ϕ ≤ 1 ∀ ϕ ∈ ∆(A)
Let A be a commutative Banach algebra,∆(A) with the weak *
topology forms a locally compact space called Gelfand space.

Preliminaries
Spectral synthesis
References
For every x ∈ A we dene x : ∆(A) → C by x (ϕ) = ϕ(x ) then x is
a continuous function which vanish at innity.
x → x is called Gelfand transform.The Gelfand transform is an
algebra homomorphism from A into C0(∆(A)).
Semisimple Banach algebra
Let A be a commutative Banach algebra, A is called semisimple if
{ker ϕ : ϕ ∈ ∆(A)} = {0}
Regular Banach algebra
A commutative Banach algebra A is called regular if given any
closed sub set E of ∆(A) , ϕ0 ∈ ∆(A)E then there exist x ∈ A
such that x (ϕ0) = 1 and x (ϕ) = 0 for all ϕ ∈ E

Preliminaries
Spectral synthesis
References
Denition
Union problem
Results and examples
Spectral synthesis problem
Let A be a regular semisimple commutative Banach algebra.
For any closed ideal I ofA the hull h(I ) of I is dened by
h(I ) = {ϕ ∈ ∆(A) : ϕ(I ) = 0}
Associated to each closed subset E of ∆(A) two distinguished
ideals with hull equal to E namely
I (E ) = {x ∈ A : x (ϕ) = 0 for all ϕ ∈ E }
j (E ) = {x ∈ A : supp(x ) is compact and supp(x ) ∩ E = φ}
J(E ) = j (E )
Then I (E ) is the largest ideal with hull E and J(E ) is the smallest
ideal with hull E .

Preliminaries
Spectral synthesis
References
Denition
Union problem
Denition
Let A be semisimple regular commutative Banach algebra , E be a
closed sub set of ∆(A)
(i) E is called set of spectral synthesis if I (E ) = j (E )
We say that spectral synthesis holds for A if every closed subset of
∆(A) is a set of spectral synthesis.
(ii) E is called Ditkin set if given x ∈ I (E ) there exist a sequence
(yk )k in j (E ) such that xyk → x as k → ∞. i.e., x ∈ xj (E ).
Remark
Every Ditkin set is a set of spectral synthesis.

Preliminaries
Spectral synthesis
References
Denition
Union problem
Tauberian
Let A be semisimple regular commutative Banach algebra ,A is
called Tauberian if the set of all x ∈ A , such that x has compact
support is dense in A.
Remark
A is Tauberian if and only if φ is a set of spectral synthesis A.
Theorem
Let A be a Tauberian, suppose that ∆(A) is discrete. Then
spectral synthesis holds for A if and only if x ∈ x A for each x ∈ A.

Preliminaries
Spectral synthesis
References
Denition
Union problem
Union problem
The union of two Ditkin sets is Ditkin set.
Question
Is the union of two sets of spectral synthesis is a set of spectral
synthesis ?
Suppose that E1 and E2 are closed subsets of ∆(A) such that
E1 ∩ E2 is a Ditkin set. Then E1 ∪ E2 is a set of spectral
synthesis if and only if both E1 and E2 are sets of spectral
synthesis.
This does not remain true if the hypothesis that E1 ∩ E2 be a
Ditkin set is dropped. In fact, the so-called Mirkil algebra M
M = {f ∈ L
2(T) : f |[−π
2 ,π
2 ] is continuous}.
f =
√
2π f 2 + f |[−π
2 , π
2 ] ∞

Preliminaries
Spectral synthesis
References
Denition
Union problem
Let X be a locally compact Hausdor space then Spectral
synthesis holds for C0(X )
Let G be a compact abelian group. Then spectral synthesis
holds for L
1(G ).
[L. Schwartz 1948] The sphere S
n−1 ⊆ R
n
fails to be a set of
spectral synthesis for L
1(R
n
) if n ≥ 3.
A famous theorem due to Malliavin[1959] states that for any
noncompact locally compact abelian group G , spectral
synthesis fails for L
1(G ).
Spectral synthesis fails for the algebra C
1[0, 1] of continuously
dierentiable functions on [0, 1].

Preliminaries
Spectral synthesis
References
Fourier algebra
Results of Eymand(1964)
Fourier algebra
Let G be a locally compact group, G denote the equivalence class
of unitary representations G If π ∈ G and ξ, η ∈ Hπ where Hπ is a
Hilbert space associated to π.
Then the continuous function πξ,η(x ) = π(x )ξ, η is called
coecient function of π
B (G ) = {πξ,η : π ∈ G ; ξ, η ∈ Hπ}
B (G ) is a commutative Banach algebra with respect to point wise
multiplication called the Fourier-Stieltjes algebra of G .
B (G ) ∩ Cc (G ) in B (G ) is called Fourier algebra and is denoted by
A(G ).

Preliminaries
Spectral synthesis
References
Fourier algebra
A(G ) is a semisimple regular Tauberian commutative Banach
algebra.
∆(A(G )) = G Here Gelfand transform is an identity map.
A(G )∗ = VN(G )
Here VN(G ) is the closure of linear span of {λ(x ) : x ∈ G } in
B (L
2(G )) with respect to weak operator topology. Here λ is a
left regular representation of G dened by
λ(x )f (y ) = f (x
−1y ) for any x ∈ G , f ∈ L
2(G )
VN(G ) is called von Neumann algebra of group G

Preliminaries
Spectral synthesis
References
Fourier algebra
We write T , u for the value of T at u. There is a natural action
of A(G ) on VN(G ) given by v .T , u = T , uv .
Let T ∈ VN(G ) the support of T is dened as
suppT = {x ∈ G : λ(x ) is a w-* limit of some uα.T , uα ∈ A(G }

Preliminaries
Spectral synthesis
References
Fourier algebra
Spectral synthesis for A(G), of abelian group
If G be a locally compact abelian group G is the dual group G then
A(G ) = L
1(G ).
Result
If G is discrete abelian group then spectral synthesis holds for A(G ).

Preliminaries
Spectral synthesis
References
Fourier algebra
Spectral synthesis for A(G), of non-abelian group
Denition(Kaniuth and Lau, 2001)
A closed subset E of G is called a set of spectral synthesis for
VN(G ) if T ∈ VN(G ) and suppT ⊆ E implies that T ∈ I (E )⊥.
We say that A(G ) admits VN(G )-spectral synthesis if every closed
subset of G is a set of spectral synthesis for VN(G )
Theorem(Kaniuth and Lau, 2001)
Let E be a closed subset of G. Then E is a set of synthesis
synthesis if and only if E a set of spectral synthesis for VN(G ).
Theorem
Let H be a any closed subgroup of a locally compact group G , then
H is a set of spectral synthesis.

Preliminaries
Spectral synthesis
References
Results of Forrest (1998)
Let K be a compact subgroup of G , G /K denote the the
homogeneous space of left cosets of K . we will denote ˜x for the
left coset xK as a element of G /K
Let ϕ : G → G /K be the quotient map then , given any continuous
map ˜u on G /K we can identify ˜u with continuous function on G
denoted by u = ˜u ◦ ϕ
B (G : K ) = {u ∈ B (G ) : u(x ) = u(xk ), ∀x ∈ G , ∀k ∈ K }
A(G : K ) = {u ∈ B (G : K ) : ϕ(supp(u)) be compact in G /K }− . B(G)

Preliminaries
Spectral synthesis
References
Properties of A(G /K )
Let K1 and K2 be compact subgroups of G , then
A(G : K1) = A(G : K2) i K1 = K2
A(G /K ) ≡ A(G : K )
∆(A(G /K )) = G /K
A(G /K ) is a semisimple regular commutative Banach algebra .

Preliminaries
Spectral synthesis
References
Theorem
Let K be a compact subgroup of G , E ⊆ G /K be a closed subset
for which spectral synthesis fails in A(G /K ) then spectral synthesis
fails for ϕ−1(E ) in A(G ). In particular if spectral synthesis fails for
A(G /K ) then spectral synthesis fails for A(G ).
Corollary
Let K be a compact subgroup of G , then each singleton set
{x } ⊆ G /K is a set of spectral synthesis for A(G /K ).

Preliminaries
Spectral synthesis
References
Suppose that K is a compact subgroup of G . Let
K G /K = {KxK : x ∈ G } denote the space of all double cosets of
K in G .
Example
M(2)= Euclidian Motion group . G = R2 T and K = T
G = SL(2, R), K =
cos θ sin θ
− sin θ cos θ
: θ ∈ [−π, π)
One wants to study the Fourier algebras on K G /K .

Preliminaries
Spectral synthesis
References
If K is a compact subgroup of G then we dene
L
1
(K G /K ) = {f ∈ L
1
(G ) : f (k1xk2) = f (x ) ∀x ∈ G ∀k1, k2 ∈ K }
Gelfand pair
We say that (G , K ) is Gelfand pair if L
1(K G /K ) is abelian under
convolution.
Above two examples are Gelfand pairs.
Remark
Let (G , K ) is a Gelfand pair then Plancherel measure d π and
Inverse Fourier transform I, are available for L
1(K G /K ).
Dene A(K G /K ) = {I(f ) : f ∈ (L
1(K G /K ), d π)}

Preliminaries
Spectral synthesis
References
Theorem (VM, 2008)
A(K G /K ) is a Tauberian regular semisimple commutative Banach
algebra.
Remark
A(K G /K ) is yet to be understood completely.
Problem for my thesis
We want to study spectral synthesis problem for this Banach
algebra A(K G /K ).

Preliminaries
Spectral synthesis
References
P. Eymard, L'algèbre de Fourier d'un groupe localement
compact, Bull. Soc. Math. France. 92 (1964), 181236.
B. Forrest, Fourier analysis on coset spaces, Rocky Mountain J.
Math. 28 (1998), 173-190.
E. Hewitt and K.A. Ross, Abstract harmonic analysis, II,
Springer-Verlag, Berlin-Heidelberg -New York, 1969.
C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier
23 (1973), 91-123.
E. Kaniuth A course in commutative Banach algebras
,Graduate texts in mathematics, Springer, 2009.
E. Kaniuth and A. T. Lau Spectral synthesis for A(G ) and
subspaces of VN(G ), Proc. Amer. Math. Soc. 129(2001),
3253-3263.

Preliminaries
Spectral synthesis
References
P. Malliavin, Impossibilitsè de la synthsèse spectrale sur les
groupes abèliens non compacts, Inst. Hautes Èt. Sci. Publ.
Math. 2 (1959), 61-68.
V. Muruganandam, The Fourier algebra of a Hypergroup- I,
Journal of Australian Mathematical Society, 82 (2007), 59-83.
V. Muruganandam, The Fourier algebra of a Hypergroup- II
Spherical hypergroups, Mathematische Nachrichten, 281,(11),
(2008), 1590-1603.
W. Rudin, Fourier analysis on groups, Interscience, New York -
London, 1962.
M. Takesaki and N. Tatsuuma, Duality and subgroups. II, J.
Funct. Anal. 11 (1972), 184-190.

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