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.
`∞ is a G-space.
1. Definition and Examples
Definition: A Banach space X is called a Grothendieck space (G-space) if
∗ ∗ ∗ w ∗ ∗ w xn ∈ X , xn −→ 0 ⇔ xn −→ 0.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces `∞ is a G-space.
1. Definition and Examples
Definition: A Banach space X is called a Grothendieck space (G-space) if
∗ ∗ ∗ w ∗ ∗ w xn ∈ 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 ∈ X , xn −→ 0 ⇔ xn −→ 0.
Examples: All reflexive Banach spaces are G-spaces.
`∞ 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 )`∞(Γ), 2 6 p 6 ∞, Γ is countable (R¨abiger,1985). P ϕ ϕ ( ⊕L )`∞(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp,∞, 1 < p < ∞ (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 )`∞(Γ), 2 6 p 6 ∞, Γ is countable (R¨abiger,1985). P ϕ ϕ ( ⊕L )`∞(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp,∞, 1 < p < ∞ (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 )`∞(Γ), 2 6 p 6 ∞, Γ is countable (R¨abiger,1985). P ϕ ϕ ( ⊕L )`∞(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp,∞, 1 < p < ∞ (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 ϕ ϕ ( ⊕L )`∞(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp,∞, 1 < p < ∞ (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 )`∞(Γ), 2 6 p 6 ∞, 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 ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp,∞, 1 < p < ∞ (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 )`∞(Γ), 2 6 p 6 ∞, Γ is countable (R¨abiger,1985). P ϕ ϕ ( ⊕L )`∞(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 < p < ∞ (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 )`∞(Γ), 2 6 p 6 ∞, Γ is countable (R¨abiger,1985). P ϕ ϕ ( ⊕L )`∞(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ 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 )`∞(Γ), 2 6 p 6 ∞, Γ is countable (R¨abiger,1985). P ϕ ϕ ( ⊕L )`∞(Γ), L is an Orlicz function space with the weakly sequentially complete dual (Leung, 1988). ∗ ∗ `∞⊗ˆ π`p, 2 < p < ∞ and `∞⊗ˆ πT , T is the original Tsirelson space (Gonz´alezand Guti´errez,1995). the weak Lp-space Lp,∞, 1 < p < ∞ (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⊗ˆ πF is a Banach lattice. `2⊗ˆ 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 ∈ E, yk ∈ F , u = xk ⊗ yk . k=1 k=1
Let E⊗ˆ 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⊗ˆ 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 ∈ E, yk ∈ F , u = xk ⊗ yk . k=1 k=1
Let E⊗ˆ πF denote the completion of E⊗F with respect to k · kπ, called the Grothendieck projective tensor product.
E, F are Banach lattices ; E⊗ˆ 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 ∈ E, yk ∈ F , u = xk ⊗ yk . k=1 k=1
Let E⊗ˆ πF denote the completion of E⊗F with respect to k · kπ, called the Grothendieck projective tensor product.
E, F are Banach lattices ; E⊗ˆ πF is a Banach lattice. `2⊗ˆ 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 ∈ N, xk ∈ E , yk ∈ F . k=1 The positive projective tensor norm on E⊗¯ F is defined by
n n n X + + X o kuk|π| = inf kxk k·kyk k : xk ∈ E , yk ∈ F , |u| 6 xk ⊗yk . k=1 k=1
Let E⊗ˆ |π|F denote the completion of E⊗¯ F with respect to k · k|π|, 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⊗ˆ π`2 6= `2⊗ˆ |π|`2.
Theorem (Cartwright and Lotz, 1975): Let E and F be Banach lattices. If L(E, F ) and Lr (E, F ) are isometrically isomorphic, then either E is an AL-space or F is an AM-space.
∗ ∗ ∗ r ∗ (E⊗ˆ πF ) = L(E, F ), (E⊗ˆ |π|F ) = L (E, F ).
If E and F are Banach lattices, then E⊗ˆ πF is isometrically isomorphic to E⊗ˆ |π|F if and only if either E or F is isometrically isomorphic to an AL-space.
2. Fremlin Tensor Product
E, F are Banach lattices ; E⊗ˆ πF = E⊗ˆ |π|F .
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Theorem (Cartwright and Lotz, 1975): Let E and F be Banach lattices. If L(E, F ) and Lr (E, F ) are isometrically isomorphic, then either E is an AL-space or F is an AM-space.
∗ ∗ ∗ r ∗ (E⊗ˆ πF ) = L(E, F ), (E⊗ˆ |π|F ) = L (E, F ).
If E and F are Banach lattices, then E⊗ˆ πF is isometrically isomorphic to E⊗ˆ |π|F if and only if either E or F is isometrically isomorphic to an AL-space.
2. Fremlin Tensor Product
E, F are Banach lattices ; E⊗ˆ πF = E⊗ˆ |π|F . `2⊗ˆ π`2 6= `2⊗ˆ |π|`2.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces ∗ ∗ ∗ r ∗ (E⊗ˆ πF ) = L(E, F ), (E⊗ˆ |π|F ) = L (E, F ).
If E and F are Banach lattices, then E⊗ˆ πF is isometrically isomorphic to E⊗ˆ |π|F if and only if either E or F is isometrically isomorphic to an AL-space.
2. Fremlin Tensor Product
E, F are Banach lattices ; E⊗ˆ πF = E⊗ˆ |π|F . `2⊗ˆ π`2 6= `2⊗ˆ |π|`2.
Theorem (Cartwright and Lotz, 1975): Let E and F be Banach lattices. If L(E, F ) and Lr (E, F ) are isometrically isomorphic, then either E is an AL-space or F is an AM-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces If E and F are Banach lattices, then E⊗ˆ πF is isometrically isomorphic to E⊗ˆ |π|F if and only if either E or F is isometrically isomorphic to an AL-space.
2. Fremlin Tensor Product
E, F are Banach lattices ; E⊗ˆ πF = E⊗ˆ |π|F . `2⊗ˆ π`2 6= `2⊗ˆ |π|`2.
Theorem (Cartwright and Lotz, 1975): Let E and F be Banach lattices. If L(E, F ) and Lr (E, F ) are isometrically isomorphic, then either E is an AL-space or F is an AM-space.
∗ ∗ ∗ r ∗ (E⊗ˆ πF ) = L(E, F ), (E⊗ˆ |π|F ) = L (E, F ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 2. Fremlin Tensor Product
E, F are Banach lattices ; E⊗ˆ πF = E⊗ˆ |π|F . `2⊗ˆ π`2 6= `2⊗ˆ |π|`2.
Theorem (Cartwright and Lotz, 1975): Let E and F be Banach lattices. If L(E, F ) and Lr (E, F ) are isometrically isomorphic, then either E is an AL-space or F is an AM-space.
∗ ∗ ∗ r ∗ (E⊗ˆ πF ) = L(E, F ), (E⊗ˆ |π|F ) = L (E, F ).
If E and F are Banach lattices, then E⊗ˆ πF is isometrically isomorphic to E⊗ˆ |π|F if and only if either E or F is isometrically isomorphic to an AL-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 3. Grothendieck Tensor Product being a G-space
3. Grothendieck Projective Tensor Product being a Grothendieck space
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces `∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ `p = `q, 1 < q < 2. L(`∞, `q) = K(`∞, `q).
3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces ∗ `p = `q, 1 < q < 2. L(`∞, `q) = K(`∞, `q).
3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces L(`∞, `q) = K(`∞, `q).
3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ `p = `q, 1 < q < 2.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ `p = `q, 1 < q < 2. L(`∞, `q) = K(`∞, `q).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces ∗ ∗ `∞⊗ˆ πT is a G-space, T is the original Tsirelson space. ∗ Let T be the dual of T . Then L(`∞, T ) = K(`∞, T ).
3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces ∗ Let T be the dual of T . Then L(`∞, T ) = K(`∞, T ).
3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ ∗ `∞⊗ˆ πT is a G-space, T is the original Tsirelson space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ ∗ `∞⊗ˆ πT is a G-space, T is the original Tsirelson space. ∗ Let T be the dual of T . Then L(`∞, T ) = K(`∞, T ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Question: For what Banach lattices E and F , the Fremlin projective tensor product E⊗ˆ |π|F can be a G-space?
3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ ∗ `∞⊗ˆ πT is a G-space, T is the original Tsirelson space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 3. Grothendieck Tensor Product being a G-space
Theorem (Gonz´alezand Guti´errez,1995): Let E and F be Banach spaces. If E is a G-space, F is ∗ ∗ reflexive, and L(E, F ) = K(E, F ), then E⊗ˆ πF is a G-space.
`∞⊗ˆ π`p is a G-space for 2 < p < ∞. ∗ ∗ `∞⊗ˆ πT is a G-space, T is the original Tsirelson space.
Question: For what Banach lattices E and F , the Fremlin projective tensor product E⊗ˆ |π|F can be a G-space?
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 4. Fremlin Tensor Product being a G-space
4. Fremlin Projective Tensor Product being a Grothendieck Space
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 4. Fremlin Tensor Product being a G-space
Let λ be a Banach sequence lattice, X be a Banach lattice. Define n o N ∗ ∗ ∗+ λε(X ) = x¯ = (xi )i ∈ X : x (|xi |) i ∈ λ, ∀ x ∈ X
and n ∗ ∗ o x¯ = sup x (|xi |) : x ∈ B ∗+ . λε(X ) i λ X
Then λε(X ) is a Banach lattice. Let n o λε,0(X ) = x¯ ∈ λε(X ) : lim (0,..., 0, xn, xn+1,... ) = 0 . n λε(X )
Then λε,0(X ) is an ideal of λε(X ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 4. Fremlin Tensor Product being a G-space
Let λ0 be the K¨othe dual of λ. Define
n ∞ o N X ∗ ∗ 0 ∗ + λπ(X ) = x¯ = (xi )i ∈ X : xi (|xi |) < +∞, ∀(xi )i ∈ λε(X ) i=1 and ∞ n X ∗ ∗ o x¯ = sup x (|xi |):(x )i ∈ B 0 ∗ + . λπ(X ) i i λε(X ) i=1
Then λπ(X ) is a Banach lattice. Let n o λπ,0(X ) = x¯ ∈ λπ(X ) : lim (0,..., 0, xn, xn+1,... ) = 0 . n λπ(X )
Then λπ,0(X ) is an ideal of λπ(X ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Lemma 1: 0 (n) (0) Let λ be σ-order continuous and letx ¯ , x¯ ∈ λε,0(X ). Then (n) (0) limn x¯ =x ¯ weakly in λε,0(X ) if and only if (n) (0) (n) limn xi = xi weakly in X and supn kx¯ kλε(X ) < ∞.
Lemma 2: ∗(n) ∗(0) ∗ Let λ be σ-order continuous and letx ¯ , x¯ ∈ λπ,0(X ) . Then ∗(n) ∗(0) ∗ ∗ limn x¯ =x ¯ weak in λπ,0(X ) if and only if ∗(n) ∗(0) ∗ ∗ ∗(n) lim x = x weak in X and sup kx¯ k 0 ∗ < ∞. n i i n λε(X )
4. Fremlin Tensor Product being a G-space
Theorem (Bu and Wong, 2012): ∗ 0 ∗ ∗ 0 ∗ λε,0(X ) = λπ(X ) and λπ,0(X ) = λε(X ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Lemma 2: ∗(n) ∗(0) ∗ Let λ be σ-order continuous and letx ¯ , x¯ ∈ λπ,0(X ) . Then ∗(n) ∗(0) ∗ ∗ limn x¯ =x ¯ weak in λπ,0(X ) if and only if ∗(n) ∗(0) ∗ ∗ ∗(n) lim x = x weak in X and sup kx¯ k 0 ∗ < ∞. n i i n λε(X )
4. Fremlin Tensor Product being a G-space
Theorem (Bu and Wong, 2012): ∗ 0 ∗ ∗ 0 ∗ λε,0(X ) = λπ(X ) and λπ,0(X ) = λε(X ).
Lemma 1: 0 (n) (0) Let λ be σ-order continuous and letx ¯ , x¯ ∈ λε,0(X ). Then (n) (0) limn x¯ =x ¯ weakly in λε,0(X ) if and only if (n) (0) (n) limn xi = xi weakly in X and supn kx¯ kλε(X ) < ∞.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 4. Fremlin Tensor Product being a G-space
Theorem (Bu and Wong, 2012): ∗ 0 ∗ ∗ 0 ∗ λε,0(X ) = λπ(X ) and λπ,0(X ) = λε(X ).
Lemma 1: 0 (n) (0) Let λ be σ-order continuous and letx ¯ , x¯ ∈ λε,0(X ). Then (n) (0) limn x¯ =x ¯ weakly in λε,0(X ) if and only if (n) (0) (n) limn xi = xi weakly in X and supn kx¯ kλε(X ) < ∞.
Lemma 2: ∗(n) ∗(0) ∗ Let λ be σ-order continuous and letx ¯ , x¯ ∈ λπ,0(X ) . Then ∗(n) ∗(0) ∗ ∗ limn x¯ =x ¯ weak in λπ,0(X ) if and only if ∗(n) ∗(0) ∗ ∗ ∗(n) lim x = x weak in X and sup kx¯ k 0 ∗ < ∞. n i i n λε(X )
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Theorem (Bu and Wong, 2012):
If λ is σ-order continuous then λπ(X ) = λπ,0(X ). 0 0 ∗ 0 ∗ If λ is σ-order continuous then λε(X ) = λε,0(X ) if and only ∗ if every positive linear operator from λ0 to X is compact.
Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
4. Fremlin Tensor Product being a G-space
Lemma 3:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a 0 ∗ 0 ∗ G-space if and only if X is a G-space and λε(X ) = λε,0(X ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 0 0 ∗ 0 ∗ If λ is σ-order continuous then λε(X ) = λε,0(X ) if and only ∗ if every positive linear operator from λ0 to X is compact.
Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
4. Fremlin Tensor Product being a G-space
Lemma 3:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a 0 ∗ 0 ∗ G-space if and only if X is a G-space and λε(X ) = λε,0(X ).
Theorem (Bu and Wong, 2012):
If λ is σ-order continuous then λπ(X ) = λπ,0(X ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
4. Fremlin Tensor Product being a G-space
Lemma 3:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a 0 ∗ 0 ∗ G-space if and only if X is a G-space and λε(X ) = λε,0(X ).
Theorem (Bu and Wong, 2012):
If λ is σ-order continuous then λπ(X ) = λπ,0(X ). 0 0 ∗ 0 ∗ If λ is σ-order continuous then λε(X ) = λε,0(X ) if and only ∗ if every positive linear operator from λ0 to X is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 4. Fremlin Tensor Product being a G-space
Lemma 3:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a 0 ∗ 0 ∗ G-space if and only if X is a G-space and λε(X ) = λε,0(X ).
Theorem (Bu and Wong, 2012):
If λ is σ-order continuous then λπ(X ) = λπ,0(X ). 0 0 ∗ 0 ∗ If λ is σ-order continuous then λε(X ) = λε,0(X ) if and only ∗ if every positive linear operator from λ0 to X is compact.
Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Theorem (Bu and Buskes, 2009):
If λ is σ-order continuous then λ⊗ˆ |π|X is lattice isometric to λπ,0(X ).
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
4. Fremlin Tensor Product being a G-space
Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
4. Fremlin Tensor Product being a G-space
Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Theorem (Bu and Buskes, 2009):
If λ is σ-order continuous then λ⊗ˆ |π|X is lattice isometric to λπ,0(X ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 4. Fremlin Tensor Product being a G-space
Lemma 4:
Let λ be a reflexive Banach sequence lattice. Then λπ,0(X ) is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Theorem (Bu and Buskes, 2009):
If λ is σ-order continuous then λ⊗ˆ |π|X is lattice isometric to λπ,0(X ).
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 5. New Examples of Non-reflexive G-spaces
5. New Examples of Non-reflexive Grothendieck Spaces
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Theorem (Gonz´alezand Guti´errez,1995): Let T ∗ be the original Tsirelson space and T be the dual of T ∗. ∗ ∗ ∗ ∗ Then L(`∞, T ) = K(`∞, T ). Thus L(T , `∞) = K(T , `∞).
New Example 1: ∗ The Fremlin projective tensor product `∞⊗ˆ |π|T is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces ∗ ∗ ∗ ∗ Thus L(T , `∞) = K(T , `∞).
New Example 1: ∗ The Fremlin projective tensor product `∞⊗ˆ |π|T is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Theorem (Gonz´alezand Guti´errez,1995): Let T ∗ be the original Tsirelson space and T be the dual of T ∗. Then L(`∞, T ) = K(`∞, T ).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces New Example 1: ∗ The Fremlin projective tensor product `∞⊗ˆ |π|T is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Theorem (Gonz´alezand Guti´errez,1995): Let T ∗ be the original Tsirelson space and T be the dual of T ∗. ∗ ∗ ∗ ∗ Then L(`∞, T ) = K(`∞, T ). Thus L(T , `∞) = K(T , `∞).
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Theorem (Gonz´alezand Guti´errez,1995): Let T ∗ be the original Tsirelson space and T be the dual of T ∗. ∗ ∗ ∗ ∗ Then L(`∞, T ) = K(`∞, T ). Thus L(T , `∞) = K(T , `∞).
New Example 1: ∗ The Fremlin projective tensor product `∞⊗ˆ |π|T is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Fact: Let 1 < q < ∞ and X be an AM-space with an order unit. Then every positive linear operator from X to `q is compact.
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Fact: Let 1 < q < ∞ and X be an AM-space with an order unit. Then every positive linear operator from X to `q is compact.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 5. New Examples of Non-reflexive G-spaces
Theorem 1: Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. Then λ⊗ˆ |π|X is a G-space if and only if X is a G-space and every positive linear operator from λ to X ∗ is compact.
Fact: Let 1 < q < ∞ and X be an AM-space with an order unit. Then every positive linear operator from X to `q is compact.
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Fact: If K is a compact stonean space, a compact σ-stonean space, or a F -space, then C(K) is a G-space.
New Example 2: Let 1 < p < ∞ and K be a compact stonean space, a compact σ-stonean space, or a F -space. Then the Fremlin projective tensor product `p⊗ˆ |π|C(K) is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces New Example 2: Let 1 < p < ∞ and K be a compact stonean space, a compact σ-stonean space, or a F -space. Then the Fremlin projective tensor product `p⊗ˆ |π|C(K) is a G-space.
5. New Examples of Non-reflexive G-spaces
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
Fact: If K is a compact stonean space, a compact σ-stonean space, or a F -space, then C(K) is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 5. New Examples of Non-reflexive G-spaces
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
Fact: If K is a compact stonean space, a compact σ-stonean space, or a F -space, then C(K) is a G-space.
New Example 2: Let 1 < p < ∞ and K be a compact stonean space, a compact σ-stonean space, or a F -space. Then the Fremlin projective tensor product `p⊗ˆ |π|C(K) is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces New Example 3:
The Fremlin projective tensor product `p⊗ˆ |π|`∞ is a G-space for 1 < p < ∞.
Ole Example (Gonz´alezand Guti´errez,1995):
The Grothendieck projective tensor product `p⊗ˆ π`∞ is a G-space if and only if 2 < p < ∞.
5. New Examples of Non-reflexive G-spaces
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Ole Example (Gonz´alezand Guti´errez,1995):
The Grothendieck projective tensor product `p⊗ˆ π`∞ is a G-space if and only if 2 < p < ∞.
5. New Examples of Non-reflexive G-spaces
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
New Example 3:
The Fremlin projective tensor product `p⊗ˆ |π|`∞ is a G-space for 1 < p < ∞.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces 5. New Examples of Non-reflexive G-spaces
Theorem 2: Let 1 < p < ∞ and X be both an AM-space with an order unit and a G-space. Then `p⊗ˆ |π|X is a G-space.
New Example 3:
The Fremlin projective tensor product `p⊗ˆ |π|`∞ is a G-space for 1 < p < ∞.
Ole Example (Gonz´alezand Guti´errez,1995):
The Grothendieck projective tensor product `p⊗ˆ π`∞ is a G-space if and only if 2 < p < ∞.
Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces Thanks
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Qingying Bu Department of Mathematics University of Mississippi Oxford, MS 38677, USA New Examples of Non-reflexive Grothendieck Spaces