A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Interpolation sets and quotients of function spaces on a locally compact group II Jorge Galindo Instituto de Matem´ aticas y Aplicaciones de Castell´ on, Universitat Jaume I International Conference on Abstract Harmonic Analysis, Granada May 20-24, 2013 Based on Joint work with Mahmoud Filali (University of Oulu, Finland). ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: a general theorem Theorem 1 Let A ⊂ B ⊂ LUC ( G ) , two admissible subalgebras of LUC ( G ) . Let U ∈ N ( e ) be compact such that T is right U-uniformly discrete. If G contains a family of sets { T η : η < κ } with: 1 T η ∩ T η ′ = ∅ for every η < η ′ < κ . 2 T η fails to be an A -interpolation set for any η < κ . � T η is an aproximable B -interpolation set. 3 η<κ Then, there is a linear isometry Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A . Back to ENAR ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: a general theorem Theorem 1 Let A ⊂ B ⊂ LUC ( G ) , two admissible subalgebras of LUC ( G ) . Let U ∈ N ( e ) be compact such that T is right U-uniformly discrete. If G contains a family of sets { T η : η < κ } with: 1 T η ∩ T η ′ = ∅ for every η < η ′ < κ . 2 T η fails to be an A -interpolation set for any η < κ . � T η is an aproximable B -interpolation set. 3 η<κ Then, there is a linear isometry Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A . Back to ENAR ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: a general theorem Theorem 1 Let A ⊂ B ⊂ LUC ( G ) , two admissible subalgebras of LUC ( G ) . Let U ∈ N ( e ) be compact such that T is right U-uniformly discrete. If G contains a family of sets { T η : η < κ } with: 1 T η ∩ T η ′ = ∅ for every η < η ′ < κ . 2 T η fails to be an A -interpolation set for any η < κ . � T η is an aproximable B -interpolation set. 3 η<κ Then, there is a linear isometry Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A . Back to ENAR ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: a general theorem Theorem 1 Let A ⊂ B ⊂ LUC ( G ) , two admissible subalgebras of LUC ( G ) . Let U ∈ N ( e ) be compact such that T is right U-uniformly discrete. If G contains a family of sets { T η : η < κ } with: 1 T η ∩ T η ′ = ∅ for every η < η ′ < κ . 2 T η fails to be an A -interpolation set for any η < κ . � T η is an aproximable B -interpolation set. 3 η<κ Then, there is a linear isometry Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A . Back to ENAR ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: a general theorem Theorem 1 Let A ⊂ B ⊂ LUC ( G ) , two admissible subalgebras of LUC ( G ) . Let U ∈ N ( e ) be compact such that T is right U-uniformly discrete. If G contains a family of sets { T η : η < κ } with: 1 T η ∩ T η ′ = ∅ for every η < η ′ < κ . 2 T η fails to be an A -interpolation set for any η < κ . � T η is an aproximable B -interpolation set. 3 η<κ Then, there is a linear isometry Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A . Back to ENAR ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: a general theorem Theorem 1 Let A ⊂ B ⊂ LUC ( G ) , two admissible subalgebras of LUC ( G ) . Let U ∈ N ( e ) be compact such that T is right U-uniformly discrete. If G contains a family of sets { T η : η < κ } with: 1 T η ∩ T η ′ = ∅ for every η < η ′ < κ . 2 T η fails to be an A -interpolation set for any η < κ . � T η is an aproximable B -interpolation set. 3 η<κ Then, there is a linear isometry Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A Ψ: ℓ ∞ ( κ ) → B / A . Back to ENAR ../IMAC/imac.pdf Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: WAP 0 ( G ) C 0 ( G ) We need a family { T η : η < κ } of pairwise disjoint sets such that: None of the T η ’s is a C 0 ( G )-interpolation set. � T = T η is a uniformly discrete approximable WAP 0 ( G )-interpolation η<κ set. Useful data: A C 0 -interpolation set must be relatively compact. If T is right U 2 -uniformly discrete and both UTs ∩ UT and sUT ∩ UT are relatively compact for all s ∈ G \ K with K compact ( UT is a t-set), then T is an approximable WAP 0 ( G )-interpolation set. Let G be SIN. Then construct T with | T | = κ ( G ) ( κ ( G )= compact covering number of G ) such that UT is a t-set and T is right � U 2 -uniformly discrete. Any partition T = T η will do. η<κ Theorem 2 (Chou for κ = ω ) ../IMAC/imac.pdf If G is a locally compact SIN group, then there is a linear isometry ψ : ℓ ∞ ( κ ( G )) → WAP 0 ( G ) . Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: WAP 0 ( G ) C 0 ( G ) We need a family { T η : η < κ } of pairwise disjoint sets such that: None of the T η ’s is a C 0 ( G )-interpolation set. � T = T η is a uniformly discrete approximable WAP 0 ( G )-interpolation η<κ set. Useful data: A C 0 -interpolation set must be relatively compact. If T is right U 2 -uniformly discrete and both UTs ∩ UT and sUT ∩ UT are relatively compact for all s ∈ G \ K with K compact ( UT is a t-set), then T is an approximable WAP 0 ( G )-interpolation set. Let G be SIN. Then construct T with | T | = κ ( G ) ( κ ( G )= compact covering number of G ) such that UT is a t-set and T is right � U 2 -uniformly discrete. Any partition T = T η will do. η<κ Theorem 2 (Chou for κ = ω ) ../IMAC/imac.pdf If G is a locally compact SIN group, then there is a linear isometry ψ : ℓ ∞ ( κ ( G )) → WAP 0 ( G ) . Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: WAP 0 ( G ) C 0 ( G ) We need a family { T η : η < κ } of pairwise disjoint sets such that: None of the T η ’s is a C 0 ( G )-interpolation set. � T = T η is a uniformly discrete approximable WAP 0 ( G )-interpolation η<κ set. Useful data: A C 0 -interpolation set must be relatively compact. If T is right U 2 -uniformly discrete and both UTs ∩ UT and sUT ∩ UT are relatively compact for all s ∈ G \ K with K compact ( UT is a t-set), then T is an approximable WAP 0 ( G )-interpolation set. Let G be SIN. Then construct T with | T | = κ ( G ) ( κ ( G )= compact covering number of G ) such that UT is a t-set and T is right � U 2 -uniformly discrete. Any partition T = T η will do. η<κ Theorem 2 (Chou for κ = ω ) ../IMAC/imac.pdf If G is a locally compact SIN group, then there is a linear isometry ψ : ℓ ∞ ( κ ( G )) → WAP 0 ( G ) . Jorge Galindo Interpolation sets and quotients of group function spaces II
A general theorem Some quotients One application: strong Arens irregularity of L 1( G ) Linearly isometric copies of ℓ ∞ ( κ ) in quotients: WAP 0 ( G ) C 0 ( G ) We need a family { T η : η < κ } of pairwise disjoint sets such that: None of the T η ’s is a C 0 ( G )-interpolation set. � T = T η is a uniformly discrete approximable WAP 0 ( G )-interpolation η<κ set. Useful data: A C 0 -interpolation set must be relatively compact. If T is right U 2 -uniformly discrete and both UTs ∩ UT and sUT ∩ UT are relatively compact for all s ∈ G \ K with K compact ( UT is a t-set), then T is an approximable WAP 0 ( G )-interpolation set. Let G be SIN. Then construct T with | T | = κ ( G ) ( κ ( G )= compact covering number of G ) such that UT is a t-set and T is right � U 2 -uniformly discrete. Any partition T = T η will do. η<κ Theorem 2 (Chou for κ = ω ) ../IMAC/imac.pdf If G is a locally compact SIN group, then there is a linear isometry ψ : ℓ ∞ ( κ ( G )) → WAP 0 ( G ) . Jorge Galindo Interpolation sets and quotients of group function spaces II
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