Ramsey properties and ultrahomogeneity for Banach and operator spaces. J. Lopez-Abad Instituto de Ciencias Matem´ aticas,CSIC, Madrid joint work with D. Bartoˇ sov´ a, M. Lupini and B. Mbombo Transfinite methods in Banach spaces and algebras of operators. Bedlewo, July 22, 2016 J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 1 / 28
Outline 1 Ramsey properties of Grassmannians Grassmannians over a finite field Fields F = Q , R , C 2 Approximate Ramsey, Ultrahomogeneity and Topological dynamics J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 2 / 28
Main Results to discuss (I) The group of isometries of the Gurarij space G with its SOT is extremely amenable (II) The universal minimal flow of the group of affine homeomorphism of the Poulsen simplex P is the natural action of it in P . (III) Analogues for the non-commutative case. (IV) A version of the Graham-Leeb-Rothschild Theorem for Grassmannians over Q , R , C . (V) The Approximate Ramsey Property of the finite dimensional normed spaces. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 3 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field Given a vector space V over F , and k ∈ N , let Gr ( k , V ) be the collection of all subspaces of V of dimension exactly k . J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 4 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field Given a vector space V over F , and k ∈ N , let Gr ( k , V ) be the collection of all subspaces of V of dimension exactly k . Theorem (Graham-Leeb-Rothschild (1972)) For every k , m ∈ N and r ∈ N there exists n such that for every coloring c : Gr ( k , F n ) → { 1 , 2 , . . . , r } there exists V ∈ Gr ( m , F n ) such that c is constant on Gr ( k , V ) . J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 4 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The pigeonhole principle is a consequence of a factorization result. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 5 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The pigeonhole principle is a consequence of a factorization result. (i) Let emb n × k ( F ) (or emb ( F k , F n )) be the collection of n × k -matrices of rank k . J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 5 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The pigeonhole principle is a consequence of a factorization result. (i) Let emb n × k ( F ) (or emb ( F k , F n )) be the collection of n × k -matrices of rank k . (ii) Let GL k ( F ) be the group of invertible k × k -matrices. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 5 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The pigeonhole principle is a consequence of a factorization result. (i) Let emb n × k ( F ) (or emb ( F k , F n )) be the collection of n × k -matrices of rank k . (ii) Let GL k ( F ) be the group of invertible k × k -matrices. (iii) Given A ∈ emb n × k , let A = red ( A ) · τ ( A ) be the unique decomposition of A by the reduced column echelon form of A and a unique invertible matrix τ ( A ) ∈ GL k ( F ). J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 5 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The pigeonhole principle is a consequence of a factorization result. (i) Let emb n × k ( F ) (or emb ( F k , F n )) be the collection of n × k -matrices of rank k . (ii) Let GL k ( F ) be the group of invertible k × k -matrices. (iii) Given A ∈ emb n × k , let A = red ( A ) · τ ( A ) be the unique decomposition of A by the reduced column echelon form of A and a unique invertible matrix τ ( A ) ∈ GL k ( F ). (iv) Let E n × k := { A ∈ emb n × k : red ( A ) = A } . J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 5 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field Theorem (Ramsey degree of full rank matrices) For every k , m ∈ N and every r ∈ N there exists n such that for every coloring f : Emb n × k ( F ) → { 1 , . . . , r } there exists R ∈ E n × m and g : GL k ( F ) → { 1 , . . . , r } such that J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 6 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field Theorem (Ramsey degree of full rank matrices) For every k , m ∈ N and every r ∈ N there exists n such that for every coloring f : Emb n × k ( F ) → { 1 , . . . , r } there exists R ∈ E n × m and g : GL k ( F ) → { 1 , . . . , r } such that f R · Emb m × d ( F ) { 1 , . . . , r } � τ g GL k ( F ) J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 6 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field Theorem (Ramsey degree of full rank matrices) For every k , m ∈ N and every r ∈ N there exists n such that for every coloring f : Emb n × k ( F ) → { 1 , . . . , r } there exists R ∈ E n × m and g : GL k ( F ) → { 1 , . . . , r } such that f R · Emb m × d ( F ) { 1 , . . . , r } � τ g GL k ( F ) Since every subspace W ∈ Gr ( k , F n ) is the image of a matrix A ∈ E n × k , this result gives the Graham-Leeb-Rothschild Theorem. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 6 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The factorization result is a consequence of the Dual Ramsey Theorem. Definition Let P k ( n ) be the collection of all k-partitions of { 1 , . . . , n } ; that is, partitions with exactly k-many pieces. Given a partition Π ∈ P m ( n ) , let � Π � k the collection of all k-partitions formed by joining the pieces of Π . J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 7 / 28
Ramsey properties of Grassmannians Grassmannians over a finite field The factorization result is a consequence of the Dual Ramsey Theorem. Definition Let P k ( n ) be the collection of all k-partitions of { 1 , . . . , n } ; that is, partitions with exactly k-many pieces. Given a partition Π ∈ P m ( n ) , let � Π � k the collection of all k-partitions formed by joining the pieces of Π . Theorem (Dual Ramsey Theorem; Graham and Rothschild (1971)) For every k , m and every r there is n such that for every coloring c : P k ( n ) → { 1 , . . . , r } there is Π ∈ P m ( n ) such that c ↾ � Π � k is constant. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 7 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C (P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr ( k , F n ) a metric space. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 8 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C (P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr ( k , F n ) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 8 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C (P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr ( k , F n ) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. Goal: (G1) For every n find a good metric d n on Gr ( k , F n ). J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 8 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C (P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr ( k , F n ) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. Goal: (G1) For every n find a good metric d n on Gr ( k , F n ). (G2) Find a compact metric space ( K k , d ) and a Lipschitz map ν : Gr ( k , F n ) → K such that: J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 8 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C (P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr ( k , F n ) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. Goal: (G1) For every n find a good metric d n on Gr ( k , F n ). (G2) Find a compact metric space ( K k , d ) and a Lipschitz map ν : Gr ( k , F n ) → K such that: For every k , m , every ε > 0 , C > 0 and every compact metric ( L , ̺ ) there exists n such that for every Lipschitz coloring f : ( Gr ( k , F n ) , d n ) → ( L , ̺ ) with Lip ( f ) ≤ C there exists V ∈ Gr ( m , F n ) and a C -Lipschitz ¯ f : ( K , d ) → ( L , ̺ ) such that f Gr ( k , V ) L � ε ν ¯ f K k J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 8 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C Definition Given a norm M on F n , we define the gap (opening) distance Λ ( F n , M ) ( V , W ) between V , W ∈ Gr ( k , F n ) as the Hausdorff distance (with respect to M) between the unit balls of ( V , M ) and ( W , M ) . J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 9 / 28
Ramsey properties of Grassmannians Fields F = Q , R , C Definition Given a norm M on F n , we define the gap (opening) distance Λ ( F n , M ) ( V , W ) between V , W ∈ Gr ( k , F n ) as the Hausdorff distance (with respect to M) between the unit balls of ( V , M ) and ( W , M ) . Note that this is non-complete (Cauchy sequences of k -dimensional subspaces might converge to a < k -dimensional subspace). Its completion is Gr ( ≤ k , F n ) is now compact. J. Lopez-Abad (ICMAT) Ramsey properties Bedlewo 9 / 28
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