and their amenability notions - fields institutehypergroups and their amenability notions mahmood...
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
Hypergroupsand their amenability notions
Mahmood AlaghmandanFields institute
May 30, 2014
May 26, 2014
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
CONTENTS
Hypergroups
Amenable hypergroups
Leptin’s conditions
Amenability of Hypergroup algebra
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HYPERGROUPSA locally compact space H is a hypergroup if∃∗ : M(H)×M(H)→M(H) called convolution:
I ∀x, y ∈ H, δx ∗ δy is a positive measure with compactsupport and ‖δx ∗ δy‖M(H) = 1.
I (x, y) 7→ δx ∗ δy is a continuous map from H ×H into M(H)equipped with the weak∗ topology.
I (x, y)→ supp(δx ∗ δy) is a continuous mapping from H ×Hinto K(H) equipped with the Michael topology.
I ∃e ∈ H, δe is the identity of M(H).
I ∃ a homeomorphism x→ x of H called involution such that(δx ∗ δy) = δy ∗ δx.
I e ∈ supp(δx ∗ δy) if and only if y = x.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HYPERGROUPSA locally compact space H is a hypergroup if∃∗ : M(H)×M(H)→M(H) called convolution:
I ∀x, y ∈ H, δx ∗ δy is a positive measure with compactsupport and ‖δx ∗ δy‖M(H) = 1.
I (x, y) 7→ δx ∗ δy is a continuous map from H ×H into M(H)equipped with the weak∗ topology.
I (x, y)→ supp(δx ∗ δy) is a continuous mapping from H ×Hinto K(H) equipped with the Michael topology.
I ∃e ∈ H, δe is the identity of M(H).
I ∃ a homeomorphism x→ x of H called involution such that(δx ∗ δy) = δy ∗ δx.
I e ∈ supp(δx ∗ δy) if and only if y = x.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HYPERGROUPSA locally compact space H is a hypergroup if∃∗ : M(H)×M(H)→M(H) called convolution:
I ∀x, y ∈ H, δx ∗ δy is a positive measure with compactsupport and ‖δx ∗ δy‖M(H) = 1.
I (x, y) 7→ δx ∗ δy is a continuous map from H ×H into M(H)equipped with the weak∗ topology.
I (x, y)→ supp(δx ∗ δy) is a continuous mapping from H ×Hinto K(H) equipped with the Michael topology.
I ∃e ∈ H, δe is the identity of M(H).
I ∃ a homeomorphism x→ x of H called involution such that(δx ∗ δy) = δy ∗ δx.
I e ∈ supp(δx ∗ δy) if and only if y = x.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HYPERGROUPSA locally compact space H is a hypergroup if∃∗ : M(H)×M(H)→M(H) called convolution:
I ∀x, y ∈ H, δx ∗ δy is a positive measure with compactsupport and ‖δx ∗ δy‖M(H) = 1.
I (x, y) 7→ δx ∗ δy is a continuous map from H ×H into M(H)equipped with the weak∗ topology.
I (x, y)→ supp(δx ∗ δy) is a continuous mapping from H ×Hinto K(H) equipped with the Michael topology.
I ∃e ∈ H, δe is the identity of M(H).
I ∃ a homeomorphism x→ x of H called involution such that(δx ∗ δy) = δy ∗ δx.
I e ∈ supp(δx ∗ δy) if and only if y = x.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HAAR MEASURELet f ∈ Cc(H),
Lxf (y) = δx ∗ δy(f ) =: f (δx ∗ δy).
A positive non-zero Borel measure h is called a Haar measure if
h(Lxf ) = h(f ), ∀f ∈ Cc(H), x ∈ H.
For a commutative and/or compact and/or discretehypergroup, the existence of a Haar measure can be proven.
For A,B ⊆ H, A ∗ B ⊆ H where
A ∗ B :=⋃
x∈A, y∈B
supp(δx ∗ δy).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HAAR MEASURELet f ∈ Cc(H),
Lxf (y) = δx ∗ δy(f ) =: f (δx ∗ δy).
A positive non-zero Borel measure h is called a Haar measure if
h(Lxf ) = h(f ), ∀f ∈ Cc(H), x ∈ H.
For a commutative and/or compact and/or discretehypergroup, the existence of a Haar measure can be proven.
For A,B ⊆ H, A ∗ B ⊆ H where
A ∗ B :=⋃
x∈A, y∈B
supp(δx ∗ δy).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
HYPERGROUP ALGEBRA
For every f , g ∈ L1(H, h),
f ∗ g =
∫H
f (y) Lyg dh(y), f ∗(y) = f (y).
L1(H)(= L1(H, h)) forms a ∗-algebra called hypergroup algebra.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
FOURIER SPACE OF HYPERGROUPS
[Muruganandam, 07] defined Fourier Stieltjes space ofhypergroups, similar to group case, and consequently Fourierspace of H, A(H).
A(H)∗ = VN(H) = λ(L1(H))′′ ⊆ B(L2(H)).
Proposition. [A.’14]For a hypergroup H,
A(H) := {f ∗ g : f , g ∈ L2(H)}.
And ‖u‖A(H) = inf{‖f‖2‖g‖2} for all f , g ∈ L2(H) s.t. u = f ∗ g.
The hypergroup H is called regular Fourier hypergroup if A(H)is a Banach algebra with respect to pointwise multiplication.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
FOURIER SPACE OF HYPERGROUPS
[Muruganandam, 07] defined Fourier Stieltjes space ofhypergroups, similar to group case, and consequently Fourierspace of H, A(H).
A(H)∗ = VN(H) = λ(L1(H))′′ ⊆ B(L2(H)).
Proposition. [A.’14]For a hypergroup H,
A(H) := {f ∗ g : f , g ∈ L2(H)}.
And ‖u‖A(H) = inf{‖f‖2‖g‖2} for all f , g ∈ L2(H) s.t. u = f ∗ g.
The hypergroup H is called regular Fourier hypergroup if A(H)is a Banach algebra with respect to pointwise multiplication.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
COMMUTATIVE HYPERGROUPS
Let H be a commutative hypergroup,
H := {α ∈ Cb(H) : α(δx∗δy) = α(x)α(y), α(x) = α(x), andα 6= 0}.
H is the Gelfand spectrum of L1(H). H is called the dual of H.
H is not necessarily a hypergroup any more!
Fourier-Stieltjes transform and Fourier transform defined:
F : M(H)→ Cb(H) where F(µ)(α) :=
∫Hα(x)dµ(x).
F : L1(H)→ C0(H) where F(f )(α) :=
∫H
f (x)α(x)dh(x)
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
COMMUTATIVE HYPERGROUPS
Let H be a commutative hypergroup,
H := {α ∈ Cb(H) : α(δx∗δy) = α(x)α(y), α(x) = α(x), andα 6= 0}.
H is the Gelfand spectrum of L1(H). H is called the dual of H.
H is not necessarily a hypergroup any more!
Fourier-Stieltjes transform and Fourier transform defined:
F : M(H)→ Cb(H) where F(µ)(α) :=
∫Hα(x)dµ(x).
F : L1(H)→ C0(H) where F(f )(α) :=
∫H
f (x)α(x)dh(x)
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
PLANCHEREL MEASURE
Theorem.Let H be a commutative hypergroup. Then there exists anon-negative measure π on H, called Plancherel measure of Hsuch that ∫
H|f (x)|2dx =
∫H|f (α)|2dπ(α)
for all f ∈ L1(H) ∩ L2(H).
Note that for an arbitrary hypergroup H (unlike group case) thesupport of the Plancherel measure,
supp(π) 6= H.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
PLANCHEREL MEASURE
Theorem.Let H be a commutative hypergroup. Then there exists anon-negative measure π on H, called Plancherel measure of Hsuch that ∫
H|f (x)|2dx =
∫H|f (α)|2dπ(α)
for all f ∈ L1(H) ∩ L2(H).
Note that for an arbitrary hypergroup H (unlike group case) thesupport of the Plancherel measure,
supp(π) 6= H.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 0.LOCALLY COMPACT GROUPS
Every locally compact group G, it is a regular Fourierhypergroup.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 1.REPRESENTATIONS OF COMPACT GROUPS
Let G be a compact (quantum) group and G the set of allirreducible unitary (co-)representations of G.
For each π1, π2 ∈ G, π1 ⊗ π2 ∼= σ1 ⊕ · · · ⊕ σn for σ1, · · · , σn ∈ G.
Define a convolution on `1(G) and make G into a commutativediscrete hypergroup which is called the fusion algebra of G.
`1(G) is isometrically isomorphic toZA(G) = {f ∈ A(G) : f (yxy−1) = f (y) for all x, y ∈ G}.
[A. ’13]:G is a regular Fourier hypergroup and A(G) ∼= ZL1(G).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 1.REPRESENTATIONS OF COMPACT GROUPS
Let G be a compact (quantum) group and G the set of allirreducible unitary (co-)representations of G.
For each π1, π2 ∈ G, π1 ⊗ π2 ∼= σ1 ⊕ · · · ⊕ σn for σ1, · · · , σn ∈ G.
Define a convolution on `1(G) and make G into a commutativediscrete hypergroup which is called the fusion algebra of G.
`1(G) is isometrically isomorphic toZA(G) = {f ∈ A(G) : f (yxy−1) = f (y) for all x, y ∈ G}.
[A. ’13]:G is a regular Fourier hypergroup and A(G) ∼= ZL1(G).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 2.CONJUGACY CLASS OF [FC]
BGROUPS
The space of all orbits in a locally compact group G for somerelatively compact subgroup B of automorphisms of Gincluding inner ones denoted by ConjB(G).
ConjB(G) forms a commutative hypergroup.L1(ConjB(G)) is isometrically isomorphic toZBL1(G) = {f ∈ L1(G) : f ◦ β = f for all β ∈ B}.
ConjB(G) is a regular Fourier hypergroup. (Muruganandam’07)
When B is the set of all inner automorphisms, we use Conj(G).Then A(Conj(G)) ∼= ZA(G).
I Conj(G) is a compact hypergroup if G is compact.I Conj(G) is a discrete hypergroup if G is discrete.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 2.CONJUGACY CLASS OF [FC]
BGROUPS
The space of all orbits in a locally compact group G for somerelatively compact subgroup B of automorphisms of Gincluding inner ones denoted by ConjB(G).
ConjB(G) forms a commutative hypergroup.L1(ConjB(G)) is isometrically isomorphic toZBL1(G) = {f ∈ L1(G) : f ◦ β = f for all β ∈ B}.
ConjB(G) is a regular Fourier hypergroup. (Muruganandam’07)
When B is the set of all inner automorphisms, we use Conj(G).Then A(Conj(G)) ∼= ZA(G).
I Conj(G) is a compact hypergroup if G is compact.I Conj(G) is a discrete hypergroup if G is discrete.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 2.CONJUGACY CLASS OF [FC]
BGROUPS
The space of all orbits in a locally compact group G for somerelatively compact subgroup B of automorphisms of Gincluding inner ones denoted by ConjB(G).
ConjB(G) forms a commutative hypergroup.L1(ConjB(G)) is isometrically isomorphic toZBL1(G) = {f ∈ L1(G) : f ◦ β = f for all β ∈ B}.
ConjB(G) is a regular Fourier hypergroup. (Muruganandam’07)
When B is the set of all inner automorphisms, we use Conj(G).Then A(Conj(G)) ∼= ZA(G).
I Conj(G) is a compact hypergroup if G is compact.I Conj(G) is a discrete hypergroup if G is discrete.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 3.DOUBLE COSET HYPERGROUPS
Let G be a locally compact group and K be a compact subgroupof G.
G//K := {KxK : x ∈ G}.
forms a hypergroup.
L1(G//K) ∼= {f ∈ L1(G) : f is constant on double cosets of K}.
[Muruganandam ’08]:G//K is a regular Fourier hypergroup and
A(G//K) ∼= {f ∈ A(G) : f is constant on double cosets of K}.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 3.DOUBLE COSET HYPERGROUPS
Let G be a locally compact group and K be a compact subgroupof G.
G//K := {KxK : x ∈ G}.
forms a hypergroup.
L1(G//K) ∼= {f ∈ L1(G) : f is constant on double cosets of K}.
[Muruganandam ’08]:G//K is a regular Fourier hypergroup and
A(G//K) ∼= {f ∈ A(G) : f is constant on double cosets of K}.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 4.POLYNOMIAL HYPERGROUPS
Let N0 = N ∪ {0}. Let (an)n∈N0 and (cn)n∈N0 be sequences ofnon-zero real numbers and (bn)n∈N0 be a sequence of realnumbers with the property
a0 + b0 = 1an + bn + cn = 1, n ≥ 1.
If (Rn)n∈N0 is a sequence of polynomials defined by
R0(x) = 1,R1(x) = 1
a0(x− b0),
R1(x)Rn(x) = anRn+1(x) + bnRn(x) + cnRn−1(x), n ≥ 1,
Then,
Rn(x)Rm(x) =
n+m∑k=|n−m|
g(n,m; k)Rk(x)
where g(n,m; k) ∈ R+ for all |n−m| ≤ k ≤ n + m.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 4.POLYNOMIAL HYPERGROUPS
Let N0 = N ∪ {0}. Let (an)n∈N0 and (cn)n∈N0 be sequences ofnon-zero real numbers and (bn)n∈N0 be a sequence of realnumbers with the property
a0 + b0 = 1an + bn + cn = 1, n ≥ 1.
If (Rn)n∈N0 is a sequence of polynomials defined by
R0(x) = 1,R1(x) = 1
a0(x− b0),
R1(x)Rn(x) = anRn+1(x) + bnRn(x) + cnRn−1(x), n ≥ 1,
Then,
Rn(x)Rm(x) =
n+m∑k=|n−m|
g(n,m; k)Rk(x)
where g(n,m; k) ∈ R+ for all |n−m| ≤ k ≤ n + m.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLE 4.POLYNOMIAL HYPERGROUPS
Define ∗ on `1(N0) such that
δn ∗ δm =
n+m∑k=|n−m|
g(n,m; k)δk
and n = n.
Then (N0, ∗, ) is a discrete commutative hypergroup with theunit element 0 which is called the polynomial hypergroup on N0induced by (Rn)n∈N0 .
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
CONTENTS
Hypergroups
Amenable hypergroups
Leptin’s conditions
Amenability of Hypergroup algebra
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
LEFT INVARIANT MEAN[SKANTHARAJAH ’92]:
A linear functional m ∈ L∞(H)∗ is called a mean if it has norm1 and is non-negative, i.e. f ≥ 0 a.e. implies m(f ) ≥ 0.
m is called left invariant mean if m(Lxf ) = m(f ).
A hypergroup H is called amenable if it has a left invariantmean.
Theorem.Every commutative and/or compact hypergroup is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
LEFT INVARIANT MEAN[SKANTHARAJAH ’92]:
A linear functional m ∈ L∞(H)∗ is called a mean if it has norm1 and is non-negative, i.e. f ≥ 0 a.e. implies m(f ) ≥ 0.
m is called left invariant mean if m(Lxf ) = m(f ).
A hypergroup H is called amenable if it has a left invariantmean.
Theorem.Every commutative and/or compact hypergroup is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
LEFT INVARIANT MEAN[SKANTHARAJAH ’92]:
A linear functional m ∈ L∞(H)∗ is called a mean if it has norm1 and is non-negative, i.e. f ≥ 0 a.e. implies m(f ) ≥ 0.
m is called left invariant mean if m(Lxf ) = m(f ).
A hypergroup H is called amenable if it has a left invariantmean.
Theorem.Every commutative and/or compact hypergroup is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
REITER’S CONDITIONS[SKANTHARAJAH ’92]:
H satisfies (Pr), 1 ≤ r <∞, if whenever ε > 0 and a compact setE ⊆ H are given, then there exists f ∈ Lr(H), f ≥ 0, ‖f‖r = 1such that
‖Lxf − f‖r < ε (x ∈ E).
[Skantharajah ’92]:
Amenablity⇔ (P1)⇐ (P2)⇔ (Pr) 1<r<∞⇔ 1 ∈ supp(π).
When H is commutative.Note that (P1) ; (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
REITER’S CONDITIONS[SKANTHARAJAH ’92]:
H satisfies (Pr), 1 ≤ r <∞, if whenever ε > 0 and a compact setE ⊆ H are given, then there exists f ∈ Lr(H), f ≥ 0, ‖f‖r = 1such that
‖Lxf − f‖r < ε (x ∈ E).
[Skantharajah ’92]:
Amenablity⇔ (P1)⇐ (P2)⇔ (Pr) 1<r<∞
⇔ 1 ∈ supp(π).
When H is commutative.Note that (P1) ; (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
REITER’S CONDITIONS[SKANTHARAJAH ’92]:
H satisfies (Pr), 1 ≤ r <∞, if whenever ε > 0 and a compact setE ⊆ H are given, then there exists f ∈ Lr(H), f ≥ 0, ‖f‖r = 1such that
‖Lxf − f‖r < ε (x ∈ E).
[Skantharajah ’92]:
Amenablity⇔ (P1)⇐ (P2)⇔ (Pr) 1<r<∞⇔ 1 ∈ supp(π).
When H is commutative.
Note that (P1) ; (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
REITER’S CONDITIONS[SKANTHARAJAH ’92]:
H satisfies (Pr), 1 ≤ r <∞, if whenever ε > 0 and a compact setE ⊆ H are given, then there exists f ∈ Lr(H), f ≥ 0, ‖f‖r = 1such that
‖Lxf − f‖r < ε (x ∈ E).
[Skantharajah ’92]:
Amenablity⇔ (P1)⇐ (P2)⇔ (Pr) 1<r<∞⇔ 1 ∈ supp(π).
When H is commutative.Note that (P1) ; (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
CONTENTS
Hypergroups
Amenable hypergroups
Leptin’s conditions
Amenability of Hypergroup algebra
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
LEPTIN CONDITION
[Singh ’96]:
(L) H satisfies the Leptin condition if for every compact subsetK of H and ε > 0, ∃V measurable in H such that0 < h(V) <∞ and
h(K ∗ V)
h(V)< 1 + ε.
Theorem. [Singh 96]Let H be a hypergroup satisfying (L). Then it does (Pr) for1 ≤ r <∞.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
LEPTIN CONDITION
[Singh ’96]:
(L) H satisfies the Leptin condition if for every compact subsetK of H and ε > 0, ∃V measurable in H such that0 < h(V) <∞ and
h(K ∗ V)
h(V)< 1 + ε.
Theorem. [Singh 96]Let H be a hypergroup satisfying (L). Then it does (Pr) for1 ≤ r <∞.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
LEPTIN HYPERGROUPS
Hypergroups satisfying (L) condition:
I Amenable locally compact groups. (Leptin ’68)
I Some simple polynomial hypergroups. (Singh ’96)
I Every compact hypergroup.
I SU(2). (A. ’13)
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
MODIFIED LEPTIN CONDITION
[A. ’14]:(LD) H satisfies the D-Leptin condition for some D ≥ 1 if for
every compact subset K of H and ε > 0, ∃V measurable inH such that 0 < h(V) <∞ and
h(K ∗ V)
h(V)< D + ε.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLES OF (LD) HYPERGROUPS.[A. ]:
I Let G be an FD group. Then Conj(G) satisfies the D-Leptincondition for D = |G′|.
I SU(3) satisfies 38-Leptin condition.I By [Banica- Vergnioux ’09]: dual of connected simply
connected compact real Lie group satisfies some D-Leptincondition:
classic computation BV algorithm
SU(2) 1 15SU(3) 38 = 6561 18240SU(4) ? ≥ 18 ∗ 1014
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLES OF (LD) HYPERGROUPS.[A. ]:
I Let G be an FD group. Then Conj(G) satisfies the D-Leptincondition for D = |G′|.
I SU(3) satisfies 38-Leptin condition.
I By [Banica- Vergnioux ’09]: dual of connected simplyconnected compact real Lie group satisfies some D-Leptincondition:
classic computation BV algorithm
SU(2) 1 15SU(3) 38 = 6561 18240SU(4) ? ≥ 18 ∗ 1014
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLES OF (LD) HYPERGROUPS.[A. ]:
I Let G be an FD group. Then Conj(G) satisfies the D-Leptincondition for D = |G′|.
I SU(3) satisfies 38-Leptin condition.I By [Banica- Vergnioux ’09]: dual of connected simply
connected compact real Lie group satisfies some D-Leptincondition:
classic computation BV algorithm
SU(2) 1 15SU(3) 38 = 6561 18240SU(4) ? ≥ 18 ∗ 1014
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
EXAMPLES OF (LD) HYPERGROUPS.[A. ]:
I Let G be an FD group. Then Conj(G) satisfies the D-Leptincondition for D = |G′|.
I SU(3) satisfies 38-Leptin condition.I By [Banica- Vergnioux ’09]: dual of connected simply
connected compact real Lie group satisfies some D-Leptincondition:
classic computation BV algorithm
SU(2) 1 15SU(3) 38 = 6561 18240SU(4) ? ≥ 18 ∗ 1014
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AN APPLICATION OF LEPTIN CONDITION
Theorem. [Choi-Ghahramani ’12]Every proper Segal algebra of Td is not approximatelyamenable.
Theorem. [A. ’14]Let G be a compact group such that G satisfies D-Leptincondition. Every proper Segal algebra of G is notapproximately amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AN APPLICATION OF LEPTIN CONDITION
Theorem. [Choi-Ghahramani ’12]Every proper Segal algebra of Td is not approximatelyamenable.
Theorem. [A. ’14]Let G be a compact group such that G satisfies D-Leptincondition. Every proper Segal algebra of G is notapproximately amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
D-LEPTIN AND (P2)?
Question.Let H be a hypergroup satisfying (LD). Does it satisfy (P2)?
If H is a locally compact group: Yes!
1 (LD) implies that A(H) has a D-bounded approximateidentity.
2 Leptin’s Theorem: A(H) has a bounded approximateidentity if and only H is amenable
3 H is amenable if and only if (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
D-LEPTIN AND (P2)?
Question.Let H be a hypergroup satisfying (LD). Does it satisfy (P2)?
If H is a locally compact group: Yes!
1 (LD) implies that A(H) has a D-bounded approximateidentity.
2 Leptin’s Theorem: A(H) has a bounded approximateidentity if and only H is amenable
3 H is amenable if and only if (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
D-LEPTIN AND (P2)?
Question.Let H be a hypergroup satisfying (LD). Does it satisfy (P2)?
If H is a locally compact group: Yes!
1 (LD) implies that A(H) has a D-bounded approximateidentity.
2 Leptin’s Theorem: A(H) has a bounded approximateidentity if and only H is amenable
3 H is amenable if and only if (P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
(LD)⇒ (P2) FOR HYPERGROUPS
Proposition. [A. ’14]Let H be a regular Fourier hypergroup. If H satisfies (LD). ThenA(H) has a D-bounded approximate identity.
Leptin’s Theorem for Hypergroups. [A. ’14]If H is a regular Fourier hypergroup. Then A(H) has a boundedapproximate identity if and only if H satisfies (P2). Then thereis a 1-bounded approximate identity for A(H).
(LD)⇒ (D− b.a.i⇔)(P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
(LD)⇒ (P2) FOR HYPERGROUPS
Proposition. [A. ’14]Let H be a regular Fourier hypergroup. If H satisfies (LD). ThenA(H) has a D-bounded approximate identity.
Leptin’s Theorem for Hypergroups. [A. ’14]If H is a regular Fourier hypergroup. Then A(H) has a boundedapproximate identity if and only if H satisfies (P2).
Then thereis a 1-bounded approximate identity for A(H).
(LD)⇒ (D− b.a.i⇔)(P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
(LD)⇒ (P2) FOR HYPERGROUPS
Proposition. [A. ’14]Let H be a regular Fourier hypergroup. If H satisfies (LD). ThenA(H) has a D-bounded approximate identity.
Leptin’s Theorem for Hypergroups. [A. ’14]If H is a regular Fourier hypergroup. Then A(H) has a boundedapproximate identity if and only if H satisfies (P2). Then thereis a 1-bounded approximate identity for A(H).
(LD)⇒ (D− b.a.i⇔)(P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
(LD)⇒ (P2) FOR HYPERGROUPS
Proposition. [A. ’14]Let H be a regular Fourier hypergroup. If H satisfies (LD). ThenA(H) has a D-bounded approximate identity.
Leptin’s Theorem for Hypergroups. [A. ’14]If H is a regular Fourier hypergroup. Then A(H) has a boundedapproximate identity if and only if H satisfies (P2). Then thereis a 1-bounded approximate identity for A(H).
(LD)⇒ (D− b.a.i⇔)(P2).
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
APPLICATION OF LEPTIN’S THEOREM.
Corollary. [A. ’14]Let G be a locally compact group. Then G//K satisfies (P2) forevery compact subgroup K if and only if G is amenable .
Proof.I G//K is a regular Fourier hypergroup and A(G//K) is
f ∈ A(G) which are constant on double cosets of K.(Murugunandam ’08)
(i) If G is amenable if and only if A(G) has a boundedapproximate identity. (Leptin ’68)
(ii) A(G) has a bounded approximate identity if and only ifA(G//K) has a bounded approximate identity.
(iii) By Leptin’s Theorem, G//K satisfies (P2) if and only ifA(G//K) has a b.a.i.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
APPLICATION OF LEPTIN’S THEOREM.
Corollary. [A. ’14]Let G be a locally compact group. Then G//K satisfies (P2) forevery compact subgroup K if and only if G is amenable .
Proof.I G//K is a regular Fourier hypergroup and A(G//K) is
f ∈ A(G) which are constant on double cosets of K.(Murugunandam ’08)
(i) If G is amenable if and only if A(G) has a boundedapproximate identity. (Leptin ’68)
(ii) A(G) has a bounded approximate identity if and only ifA(G//K) has a bounded approximate identity.
(iii) By Leptin’s Theorem, G//K satisfies (P2) if and only ifA(G//K) has a b.a.i.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
APPLICATION OF LEPTIN’S THEOREM.
Corollary. [A. ’14]Let G be a locally compact group. Then G//K satisfies (P2) forevery compact subgroup K if and only if G is amenable .
Proof.I G//K is a regular Fourier hypergroup and A(G//K) is
f ∈ A(G) which are constant on double cosets of K.(Murugunandam ’08)
(i) If G is amenable if and only if A(G) has a boundedapproximate identity. (Leptin ’68)
(ii) A(G) has a bounded approximate identity if and only ifA(G//K) has a bounded approximate identity.
(iii) By Leptin’s Theorem, G//K satisfies (P2) if and only ifA(G//K) has a b.a.i.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
APPLICATION OF LEPTIN’S THEOREM.
Corollary. [A. ’14]Let G be a locally compact group. Then G//K satisfies (P2) forevery compact subgroup K if and only if G is amenable .
Proof.I G//K is a regular Fourier hypergroup and A(G//K) is
f ∈ A(G) which are constant on double cosets of K.(Murugunandam ’08)
(i) If G is amenable if and only if A(G) has a boundedapproximate identity. (Leptin ’68)
(ii) A(G) has a bounded approximate identity if and only ifA(G//K) has a bounded approximate identity.
(iii) By Leptin’s Theorem, G//K satisfies (P2) if and only ifA(G//K) has a b.a.i.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
APPLICATION OF LEPTIN’S THEOREM.
Corollary. [A. ’14]Let G be a locally compact group. Then G//K satisfies (P2) forevery compact subgroup K if and only if G is amenable .
Proof.I G//K is a regular Fourier hypergroup and A(G//K) is
f ∈ A(G) which are constant on double cosets of K.(Murugunandam ’08)
(i) If G is amenable if and only if A(G) has a boundedapproximate identity. (Leptin ’68)
(ii) A(G) has a bounded approximate identity if and only ifA(G//K) has a bounded approximate identity.
(iii) By Leptin’s Theorem, G//K satisfies (P2) if and only ifA(G//K) has a b.a.i.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
Let H be a regular Fourier hypergroup.(m) H is amenable.
(LD) H satisfies the D-Leptin condition for some D ≥ 1.(BD) A(H) has a D-bounded approximate identity for some
D ≥ 1.(P2) H satisfies (P2).(P1) H satisfies Reiter condition.
(AM) L1(H) is an amenable Banach algebra.
(L1)
��
+3 (P2) :+3 (P1) (AM);
ks
(LD) +3 (BD) ks +3 (B1)��
KS
(m)��
KS
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
CONTENTS
Hypergroups
Amenable hypergroups
Leptin’s conditions
Amenability of Hypergroup algebra
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE HYPERGROUP ALGEBRAS
Theorem. [Johnson ’72]G is amenable⇔ L1(G) is amenable.
Theorem. [Skantharajah ’92]H is amenable⇐ L1(H) is amenable.
Example. [Azimifard-Samei-Spronk ’09]L1(Conj(SU(2)))(= ZL1(SU(2))) is not amenable. ButConj(SU(2)) is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE HYPERGROUP ALGEBRAS
Theorem. [Johnson ’72]G is amenable⇔ L1(G) is amenable.
Theorem. [Skantharajah ’92]H is amenable⇐ L1(H) is amenable.
Example. [Azimifard-Samei-Spronk ’09]L1(Conj(SU(2)))(= ZL1(SU(2))) is not amenable. ButConj(SU(2)) is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE HYPERGROUP ALGEBRAS
Theorem. [Johnson ’72]G is amenable⇔ L1(G) is amenable.
Theorem. [Skantharajah ’92]H is amenable⇐ L1(H) is amenable.
Example. [Azimifard-Samei-Spronk ’09]L1(Conj(SU(2)))(= ZL1(SU(2))) is not amenable. ButConj(SU(2)) is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE POLYNOMIAL HYPERGROUP ALGEBRAS
Chebyshev polynomial hypergroup on N0 = {0, 1, 2, 3, . . .}:
δm ∗ δn =12δ|n−m| +
12δn+m.
[Lasser ’07]: `1(N0) is amenable.
Proof.
I `1(N0) is isometrically Banach algebra isomorphic toZ±1A(T) = {f + f : f ∈ A(T)}.
I Z±1A(T) is a subalgebra of an amenable Banach algebra(A(T)) invariant under a finite subgroup of Aut(A(T)).[Kepert ’94]: Z±1A(T) is amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE POLYNOMIAL HYPERGROUP ALGEBRAS
Chebyshev polynomial hypergroup on N0 = {0, 1, 2, 3, . . .}:
δm ∗ δn =12δ|n−m| +
12δn+m.
[Lasser ’07]: `1(N0) is amenable.
Proof.
I `1(N0) is isometrically Banach algebra isomorphic toZ±1A(T) = {f + f : f ∈ A(T)}.
I Z±1A(T) is a subalgebra of an amenable Banach algebra(A(T)) invariant under a finite subgroup of Aut(A(T)).[Kepert ’94]: Z±1A(T) is amenable.
![Page 65: and their amenability notions - Fields InstituteHypergroups and their amenability notions Mahmood Alaghmandan Fields institute May 30, 2014 May 26, 2014. HypergroupsAmenable hypergroupsLeptin’s](https://reader033.vdocuments.us/reader033/viewer/2022060322/5f0d66787e708231d43a2aaf/html5/thumbnails/65.jpg)
Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE POLYNOMIAL HYPERGROUP ALGEBRAS
Chebyshev polynomial hypergroup on N0 = {0, 1, 2, 3, . . .}:
δm ∗ δn =12δ|n−m| +
12δn+m.
[Lasser ’07]: `1(N0) is amenable.
Proof.
I `1(N0) is isometrically Banach algebra isomorphic toZ±1A(T) = {f + f : f ∈ A(T)}.
I Z±1A(T) is a subalgebra of an amenable Banach algebra(A(T)) invariant under a finite subgroup of Aut(A(T)).[Kepert ’94]: Z±1A(T) is amenable.
![Page 66: and their amenability notions - Fields InstituteHypergroups and their amenability notions Mahmood Alaghmandan Fields institute May 30, 2014 May 26, 2014. HypergroupsAmenable hypergroupsLeptin’s](https://reader033.vdocuments.us/reader033/viewer/2022060322/5f0d66787e708231d43a2aaf/html5/thumbnails/66.jpg)
Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABLE POLYNOMIAL HYPERGROUP ALGEBRAS
Chebyshev polynomial hypergroup on N0 = {0, 1, 2, 3, . . .}:
δm ∗ δn =12δ|n−m| +
12δn+m.
[Lasser ’07]: `1(N0) is amenable.
Proof.
I `1(N0) is isometrically Banach algebra isomorphic toZ±1A(T) = {f + f : f ∈ A(T)}.
I Z±1A(T) is a subalgebra of an amenable Banach algebra(A(T)) invariant under a finite subgroup of Aut(A(T)).[Kepert ’94]: Z±1A(T) is amenable.
![Page 67: and their amenability notions - Fields InstituteHypergroups and their amenability notions Mahmood Alaghmandan Fields institute May 30, 2014 May 26, 2014. HypergroupsAmenable hypergroupsLeptin’s](https://reader033.vdocuments.us/reader033/viewer/2022060322/5f0d66787e708231d43a2aaf/html5/thumbnails/67.jpg)
Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF DISCRETE HYPERGROUP ALGEBRAS
Theorem. [Lasser ’07]Let N0 be a polynomial hypergroup and for each N > 0,{x ∈ N0 : h(x) ≤ N} is finite. Then `1(N0) is not amenable.
Example. `1(SU(2))(= ZA(SU(2))) is not amenable.
Theorem. [A. ’14]Let H be a commutative discrete hypergroup satisfying (P2)and for each N > 0, {x ∈ H : h(x) ≤ N} is finite. Then `1(H) isnot amenable.
Example. For every tall compact group G, `1(G)(= ZA(G)) isnot amenable.
Example. For every FC group such that |C| → ∞,`1(Conj(G))(= Z`1(G)) is not amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF DISCRETE HYPERGROUP ALGEBRAS
Theorem. [Lasser ’07]Let N0 be a polynomial hypergroup and for each N > 0,{x ∈ N0 : h(x) ≤ N} is finite. Then `1(N0) is not amenable.
Example. `1(SU(2))(= ZA(SU(2))) is not amenable.
Theorem. [A. ’14]Let H be a commutative discrete hypergroup satisfying (P2)and for each N > 0, {x ∈ H : h(x) ≤ N} is finite. Then `1(H) isnot amenable.
Example. For every tall compact group G, `1(G)(= ZA(G)) isnot amenable.
Example. For every FC group such that |C| → ∞,`1(Conj(G))(= Z`1(G)) is not amenable.
![Page 69: and their amenability notions - Fields InstituteHypergroups and their amenability notions Mahmood Alaghmandan Fields institute May 30, 2014 May 26, 2014. HypergroupsAmenable hypergroupsLeptin’s](https://reader033.vdocuments.us/reader033/viewer/2022060322/5f0d66787e708231d43a2aaf/html5/thumbnails/69.jpg)
Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF DISCRETE HYPERGROUP ALGEBRAS
Theorem. [Lasser ’07]Let N0 be a polynomial hypergroup and for each N > 0,{x ∈ N0 : h(x) ≤ N} is finite. Then `1(N0) is not amenable.
Example. `1(SU(2))(= ZA(SU(2))) is not amenable.
Theorem. [A. ’14]Let H be a commutative discrete hypergroup satisfying (P2)and for each N > 0, {x ∈ H : h(x) ≤ N} is finite. Then `1(H) isnot amenable.
Example. For every tall compact group G, `1(G)(= ZA(G)) isnot amenable.
Example. For every FC group such that |C| → ∞,`1(Conj(G))(= Z`1(G)) is not amenable.
![Page 70: and their amenability notions - Fields InstituteHypergroups and their amenability notions Mahmood Alaghmandan Fields institute May 30, 2014 May 26, 2014. HypergroupsAmenable hypergroupsLeptin’s](https://reader033.vdocuments.us/reader033/viewer/2022060322/5f0d66787e708231d43a2aaf/html5/thumbnails/70.jpg)
Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF DISCRETE HYPERGROUP ALGEBRAS
Theorem. [Lasser ’07]Let N0 be a polynomial hypergroup and for each N > 0,{x ∈ N0 : h(x) ≤ N} is finite. Then `1(N0) is not amenable.
Example. `1(SU(2))(= ZA(SU(2))) is not amenable.
Theorem. [A. ’14]Let H be a commutative discrete hypergroup satisfying (P2)and for each N > 0, {x ∈ H : h(x) ≤ N} is finite. Then `1(H) isnot amenable.
Example. For every tall compact group G, `1(G)(= ZA(G)) isnot amenable.
Example. For every FC group such that |C| → ∞,`1(Conj(G))(= Z`1(G)) is not amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF `1(H)
Conjecture.`1(H) is amenable if and only if supx∈H h(x) <∞.
Theorem. [A.-Spronk]Let G be a compact group which has an open commutativesubgroup. Then `1(G)(= ZA(G)) is amenable.
Theorem. [Azimifard-Samei- Spronk ’09]Let G be an FD group; then `1(Conj(G))(= Z`1(G))) is amenable.
Theorem. [A.-Choi-Samei ’13]Let G be an RDPF group. Then `1(Conj(G))(= Z`1(G))) isamenable if and only if G is FD.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF `1(H)
Conjecture.`1(H) is amenable if and only if supx∈H h(x) <∞.
Theorem. [A.-Spronk]Let G be a compact group which has an open commutativesubgroup. Then `1(G)(= ZA(G)) is amenable.
Theorem. [Azimifard-Samei- Spronk ’09]Let G be an FD group; then `1(Conj(G))(= Z`1(G))) is amenable.
Theorem. [A.-Choi-Samei ’13]Let G be an RDPF group. Then `1(Conj(G))(= Z`1(G))) isamenable if and only if G is FD.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF `1(H)
Conjecture.`1(H) is amenable if and only if supx∈H h(x) <∞.
Theorem. [A.-Spronk]Let G be a compact group which has an open commutativesubgroup. Then `1(G)(= ZA(G)) is amenable.
Theorem. [Azimifard-Samei- Spronk ’09]Let G be an FD group; then `1(Conj(G))(= Z`1(G))) is amenable.
Theorem. [A.-Choi-Samei ’13]Let G be an RDPF group. Then `1(Conj(G))(= Z`1(G))) isamenable if and only if G is FD.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
AMENABILITY OF `1(H)
Conjecture.`1(H) is amenable if and only if supx∈H h(x) <∞.
Theorem. [A.-Spronk]Let G be a compact group which has an open commutativesubgroup. Then `1(G)(= ZA(G)) is amenable.
Theorem. [Azimifard-Samei- Spronk ’09]Let G be an FD group; then `1(Conj(G))(= Z`1(G))) is amenable.
Theorem. [A.-Choi-Samei ’13]Let G be an RDPF group. Then `1(Conj(G))(= Z`1(G))) isamenable if and only if G is FD.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
COMPACT HYPERGROUPS
A compact hypergroup satisfies Leptin, so does (P2).
For a compact hypergroup H, L1(H) is weakly amenable.
Conjecture.If G is a compact group. Then L1(Conj(G))(= ZL1(G)) isamenable if and only if G has an open abelian group.
Theorem. [Azimifard-Samei- Spronk ’09]If G is a non-abelian connected compact group, thenL1(Conj(G))(= ZL1(G)) is not amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
COMPACT HYPERGROUPS
A compact hypergroup satisfies Leptin, so does (P2).
For a compact hypergroup H, L1(H) is weakly amenable.
Conjecture.If G is a compact group. Then L1(Conj(G))(= ZL1(G)) isamenable if and only if G has an open abelian group.
Theorem. [Azimifard-Samei- Spronk ’09]If G is a non-abelian connected compact group, thenL1(Conj(G))(= ZL1(G)) is not amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
COMPACT HYPERGROUPS
A compact hypergroup satisfies Leptin, so does (P2).
For a compact hypergroup H, L1(H) is weakly amenable.
Conjecture.If G is a compact group. Then L1(Conj(G))(= ZL1(G)) isamenable if and only if G has an open abelian group.
Theorem. [Azimifard-Samei- Spronk ’09]If G is a non-abelian connected compact group, thenL1(Conj(G))(= ZL1(G)) is not amenable.
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Hypergroups Amenable hypergroups Leptin’s conditions Amenability of Hypergroup algebra
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