Contractive homomor- phisms from Fourier algebras: not new Pham Le Hung Contractive homomorphisms from Fourier Contractive homomor- algebras: not new phisms L 1 ( H ) → M ( G ) Homomorphisms A ( G ) → B ( H ) Pham Le Hung Contractive homomor- phisms Victoria University of Wellington, New Zealand A ( G ) → B ( H ) Fields Institute, 15 April 2014
Contractive homomor- The Main Problem phisms from Fourier algebras: not new Pham Le Hung Contractive homomor- phisms Let G and H be locally compact groups. L 1 ( H ) → M ( G ) Problem Homomorphisms A ( G ) → B ( H ) Suppose that θ is a homomorphism from the Fourier Contractive algebras A ( G ) into the Fourier-Stieltjes algebra B ( H ) . homomor- phisms Describe θ . A ( G ) → B ( H )
Contractive homomor- Outline phisms from Fourier algebras: not new Pham Le Hung Contractive homomor- Contractive homomorphisms L 1 ( H ) → M ( G ) 1 phisms L 1 ( H ) → M ( G ) Homomorphisms A ( G ) → B ( H ) 2 Homomorphisms A ( G ) → B ( H ) Contractive homomor- phisms A ( G ) → B ( H ) Contractive homomorphisms A ( G ) → B ( H ) 3
Contractive homomor- (Contractive) homomorphisms phisms from Fourier L 1 ( G ) → M ( H ) algebras: not new Pham Le Hung Suppose that φ : G → M ( H ) is a uniformly bounded, Contractive homomor- weak ∗ -continuous homomorphism. phisms L 1 ( H ) → Define M ( G ) Homomorphisms φ t ( f )( t ) := � φ ( t ) , f � ( f ∈ C 0 ( H ) , t ∈ G ) . A ( G ) → B ( H ) Contractive Then φ t : C 0 ( H ) → C b ( G ) is a bounded linear map. homomor- phisms Define φ tt : M ( G ) → M ( H ) as follows. A ( G ) → B ( H ) � φ tt ( µ ) , f φ t ( f ) d µ . � � := G Then φ tt is a bounded homomorphism.
Contractive homomor- A contractive homomorphism phisms from Fourier L 1 ( G ) → M ( H ) algebras: not new Pham Le Hung Contractive Suppose that φ : G → H is a continuous group homomor- homomorphism. phisms L 1 ( H ) → Define M ( G ) φ t ( f ) := f ◦ φ ( f ∈ C 0 ( H )) . Homomorphisms A ( G ) → B ( H ) Then φ t : C 0 ( H ) → C b ( G ) is a contractive linear map. Contractive homomor- Define φ tt : M ( G ) → M ( H ) as follows. phisms A ( G ) → B ( H ) � φ tt ( µ ) , f φ t ( f ) d µ . � � := G Then φ tt is a contractive homomorphism.
Contractive homomor- A more complicated one phisms from Fourier algebras: not new Pham Le Suppose that K is a compact supgroup of H that commutes Hung with φ ( G ) i.e. Contractive K φ ( G ) ⊆ φ ( G ) K . homomor- phisms L 1 ( H ) → Then we can define φ tt K : M ( G ) → M ( H ) as follows. M ( G ) Homomorphisms φ tt K ( µ ) := φ tt ( µ ) ∗ m K ; A ( G ) → B ( H ) Contractive homomor- where m K is the normalized Haar measure on K . phisms A ( G ) → B ( H ) Then φ tt K is a contractive homomorphism. If in addition we have a “nice” character ρ : K → T , we could modify φ tt K ,ρ ( µ ) := φ tt ( µ ) ∗ ( ρ m K ) .
Contractive homomor- A further complication phisms from Fourier algebras: not new Pham Le Hung Contractive homomor- If L is a normal supgroup of G such that phisms L 1 ( H ) → M ( G ) φ ( s ) ∗ ( ρ m K ) = ρ m K ( s ∈ L ) , Homomorphisms A ( G ) → B ( H ) i.e. ψ ( s ) = ψ ( 1 G ) for all s ∈ L , where ψ := φ tt K ,ρ . Contractive homomor- phisms Consider ¯ ψ : G / L → M ( H ) . A ( G ) → B ( H ) ψ tt : M ( G / L ) → M ( H ) . Define ¯
Contractive homomor- A more concrete example phisms from Fourier algebras: not Take Ω 0 ⊆ Ω be closed subgroups of T × H with new Pham Le • Ω 0 is compact and normal in Ω , and Hung • π H : Ω 0 → H is injective. Contractive Set K := π H (Ω 0 ) and set ρ := π T ◦ ( π H | Ω 0 ) − 1 . homomor- phisms L 1 ( H ) → Then M ( G ) 1 π H : Ω → H is a homomorphism, Homomorphisms A ( G ) → B ( H ) 2 K is a compact subgroup of H , Contractive homomor- 3 π H (Ω) commutes with K , and phisms A ( G ) → B ( H ) 4 ρ is a “nice” character on K . Thus we have a contractive homomorphism Φ : M (Ω) → M ( H ) as above. Moreover, • Φ(Ω 0 ) = Φ( 1 Ω ) . Hence, a contractive homomorphism ˜ Φ : M (Ω / Ω 0 ) → M ( H ) .
Contractive homomor- Greenleaf’s theorem phisms from Fourier algebras: not new Pham Le Hung Contractive Let G and H be locally compact groups. Every contractive homomor- phisms homomorphism L 1 ( G ) → M ( H ) has the form L 1 ( H ) → M ( G ) Φ ◦ φ tt ˜ Homomorphisms A ( G ) → B ( H ) Contractive where homomor- phisms 1 ˜ A ( G ) → B ( H ) Φ : M (Ω / Ω 0 ) → M ( H ) as above, and 2 φ : G → Ω / Ω 0 is a continuous epimorphism.
Contractive homomor- Homomorphisms from A ( G ) . phisms from Fourier algebras: not new Pham Le • Suppose that θ : A ( G ) → B ( H ) is a homomorphism. Hung • For each t ∈ H , either θ ( f )( t ) = 0 ∀ f ∈ A ( G ) . Contractive homomor- phisms • Or, f �→ θ ( f )( t ) is a character of A ( G ) . L 1 ( H ) → M ( G ) • So that ∃ τ ( t ) ∈ G , θ ( f )( t ) = f ( τ ( t )) for all f ∈ A ( G ) . Homomorphisms A ( G ) → B ( H ) • Thus, ∃ an open subset Ω of H and a continuous map Contractive homomor- τ : Ω → G such that phisms A ( G ) → B ( H ) � f ( τ ( t )) if t ∈ Ω θ ( f )( t ) = if t ∈ H \ Ω , ( ∀ f ∈ A ( G )) . 0 • As a consequence, θ : A ( G ) → B ( H ) is automatically bounded.
Contractive homomor- Homomorphisms from A ( G ) phisms from Fourier algebras: not (cont.) new Pham Le Hung Contractive • Conversely, given a map τ : Ω → G , where Ω ⊆ H . homomor- phisms Define L 1 ( H ) → M ( G ) � f ( τ ( t )) if t ∈ Ω Homomorphisms θ τ ( f )( t ) = if t ∈ H \ Ω , ( ∀ f ∈ A ( G )) . A ( G ) → B ( H ) 0 Contractive homomor- phisms • Then θ τ : A ( G ) → ℓ ∞ ( H ) is a homomorphism. A ( G ) → B ( H ) • Where ℓ ∞ ( H ) is the algebra of bounded functions on H . Question For which τ , does θ τ ( A ( G )) ⊆ B ( H ) ?
Contractive homomor- A reduction lemma phisms from Fourier algebras: not new Pham Le Hung Let θ : A ( G ) → B ( H ) be a homomorphism. Contractive homomor- Then θ is induced by some continuous map τ : Ω → G . phisms L 1 ( H ) → The formula M ( G ) � f ( τ ( t )) Homomorphisms if t ∈ Ω A ( G ) → B ( H ) ϕ ( f )( t ) = Contractive 0 if t ∈ H \ Ω homomor- phisms A ( G ) → B ( H ) makes sense even if f ∈ B ( G d ) . In fact, ϕ is a homomorphism from A ( G d ) into B ( H d ) with � ϕ � ≤ � θ � .
Contractive homomor- Proof of the reduction phisms from Fourier It suffices to show that for u = � m i = 1 α i δ a i and v = � m algebras: not i = 1 β i δ b i new i = 1 | α i | 2 = � m i = 1 | β i | 2 = 1 we have in c 00 ( G 0 ) with � m Pham Le Hung � n � Contractive � � � γ k ϕ ( u ∗ ˇ v )( x k ) � ≤ � θ � (1) homomor- � � phisms � � L 1 ( H ) → � k = 1 M ( G ) Homomorphisms for every finite systems ( x k ) ⊆ H and ( γ k ) ⊂ C with A ( G ) → B ( H ) � � n k = 1 γ k ω H d ( x k ) � ≤ 1. Contractive homomor- The left hand side of (1) is phisms A ( G ) → B ( H ) � � � n � � � � � � � γ k ϕ ( u ∗ ˇ � γ k ( u ∗ ˇ � v )( x k ) � = v )( τ ( x k )) � � � � � � � � � k = 1 x k ∈ Ω � � � � m � � � � � � = γ k α i β j δ a i b − 1 ( τ ( x k )) � � j � � x k ∈ Ω i , j = 1 � �
Contractive Take a measurable set V to be chosen. homomor- phisms from Consider f = � m i = 1 α i χ a i V and g = � m i = 1 β i χ b i V in L 2 ( G ) Fourier algebras: not both of L 2 -norm � | V | . new So, f ∗ ˇ g ∈ A ( G ) with norm at most | V | . Therefore, Pham Le Hung � θ ( f ∗ ˇ g ) � ≤ � θ � | V | . Contractive homomor- phisms L 1 ( H ) → Thus M ( G ) � � Homomorphisms � n � � � A ( G ) → B ( H ) � � � � γ k θ ( f ∗ ˇ � γ k ( f ∗ ˇ � � θ � | V | ≥ g )( x k ) � = g )( τ ( x k )) � � Contractive � � � � homomor- � � � k = 1 x k ∈ Ω � � phisms A ( G ) → B ( H ) � � m � � � � � � � � = γ k α i β j � a i V ∩ τ ( x k ) b j V � � � � � x k ∈ Ω i , j = 1 � � � � m � � � � � � = γ k α i β j δ a i b − 1 ( τ ( x k )) · | V | . � � j � � x k ∈ Ω i , j = 1 � �
Contractive homomor- A reduction question phisms from Fourier algebras: not new Pham Le Hung Let θ : A ( G ) → B ( H ) be a homomorphism. Contractive Then θ is induced by some continuous map τ : Ω → G . homomor- phisms The formula L 1 ( H ) → M ( G ) � f ( τ ( t )) if t ∈ Ω Homomorphisms ϕ ( f )( t ) = A ( G ) → B ( H ) 0 if t ∈ H \ Ω Contractive homomor- phisms makes sense even if f ∈ B ( G d ) . A ( G ) → B ( H ) Is ϕ is a homomorphism from B ( G d ) into B ( H d ) with � ϕ � ≤ � θ � ?
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