New Examples of Non-reflexive Grothendieck Spaces Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA This is a joint work with Yongjin Li Positivity IX{University of Alberta, July 17-22, 2017 Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 1. Definition and Examples 1. Definition and Examples Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Examples: All reflexive Banach spaces are G-spaces. `1 is a G-space. 1. Definition and Examples Definition: A Banach space X is called a Grothendieck space (G-space) if ∗ ∗ ∗ w ∗ ∗ w xn 2 X ; xn −! 0 , xn −! 0: Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces `1 is a G-space. 1. Definition and Examples Definition: A Banach space X is called a Grothendieck space (G-space) if ∗ ∗ ∗ w ∗ ∗ w xn 2 X ; xn −! 0 , xn −! 0: Examples: All reflexive Banach spaces are G-spaces. Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 1. Definition and Examples Definition: A Banach space X is called a Grothendieck space (G-space) if ∗ ∗ ∗ w ∗ ∗ w xn 2 X ; xn −! 0 , xn −! 0: Examples: All reflexive Banach spaces are G-spaces. `1 is a G-space. Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces C(K), K is a compact σ-stonean space (Ando, 1961). C(K), K is a F -space (Seever, 1968). P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces C(K), K is a F -space (Seever, 1968). P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). C(K), K is a compact σ-stonean space (Ando, 1961). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). C(K), K is a compact σ-stonean space (Ando, 1961). C(K), K is a F -space (Seever, 1968). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). C(K), K is a compact σ-stonean space (Ando, 1961). C(K), K is a F -space (Seever, 1968). P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). C(K), K is a compact σ-stonean space (Ando, 1961). C(K), K is a F -space (Seever, 1968). P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). C(K), K is a compact σ-stonean space (Ando, 1961). C(K), K is a F -space (Seever, 1968). P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 1. Definition and Examples Examples of Non-reflexive G-spaces: C(K), K is a compact stonean space (Grothendieck, 1953). C(K), K is a compact σ-stonean space (Ando, 1961). C(K), K is a F -space (Seever, 1968). P p ( ⊕L )`1(Γ), 2 6 p 6 1, Γ is countable (R¨abiger,1985). P ' ' ( ⊕L )`1(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `1⊗^ π`p, 2 < p < 1 and `1⊗^ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp;1, 1 < p < 1 (Lotz, 2010). Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 2. Two Projective Tensor Products 2. Grothendieck and Fremlin Projective Tensor Products Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces E; F are Banach lattices ; E⊗^ πF is a Banach lattice. `2⊗^ π`2 is not a Banach lattice. 2. Grothendieck Tensor Product Definition: Let E and F be Banach spaces. The projective tensor norm on E⊗F is defined by n n n X X o kukπ = inf kxk k · kyk k : xk 2 E; yk 2 F ; u = xk ⊗ yk : k=1 k=1 Let E⊗^ πF denote the completion of E⊗F with respect to k · kπ, called the Grothendieck projective tensor product. Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces `2⊗^ π`2 is not a Banach lattice. 2. Grothendieck Tensor Product Definition: Let E and F be Banach spaces. The projective tensor norm on E⊗F is defined by n n n X X o kukπ = inf kxk k · kyk k : xk 2 E; yk 2 F ; u = xk ⊗ yk : k=1 k=1 Let E⊗^ πF denote the completion of E⊗F with respect to k · kπ, called the Grothendieck projective tensor product. E; F are Banach lattices ; E⊗^ πF is a Banach lattice. Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 2. Grothendieck Tensor Product Definition: Let E and F be Banach spaces. The projective tensor norm on E⊗F is defined by n n n X X o kukπ = inf kxk k · kyk k : xk 2 E; yk 2 F ; u = xk ⊗ yk : k=1 k=1 Let E⊗^ πF denote the completion of E⊗F with respect to k · kπ, called the Grothendieck projective tensor product. E; F are Banach lattices ; E⊗^ πF is a Banach lattice. `2⊗^ π`2 is not a Banach lattice. Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 2. Fremlin Tensor Product Definition (Fremlin, 1974): Let E and F be Banach lattices, E⊗¯ F be the Risez space tensor product of E and F with the positive cone n n X + +o Cp = xk ⊗ yk : n 2 N; xk 2 E ; yk 2 F : k=1 The positive projective tensor norm on E⊗¯ F is defined by n n n X + + X o kukjπj = inf kxk k·kyk k : xk 2 E ; yk 2 F ; juj 6 xk ⊗yk : k=1 k=1 Let E⊗^ jπjF denote the completion of E⊗¯ F with respect to k · kjπj, called the Fremlin projective tensor product. Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces `2⊗^ π`2 6= `2⊗^ jπj`2.