On Positive Injective Tensor Products Being Grothendieck Spaces
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DOI: 10.1007/s13226-020-0461-1
ON POSITIVE INJECTIVE TENSOR PRODUCTS BEING GROTHENDIECK SPACES Shaoyong Zhang∗ , Zhaohui Gu∗∗ and Yongjin Li∗∗∗ ∗ Department
of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China
∗∗ School
of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, P. R. China
∗∗∗ Department
of Mathematics, Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, P. R. China
e-mails: [email protected]; [email protected]; [email protected] (Received 9 March 2019; accepted 1 July 2019) Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. In this paper, we show ˇ |ε| X is a Grothendieck space if and only if X is a that the positive injective tensor product λ⊗ Grothendieck space and every positive linear operator from λ∗ to X ∗∗ is compact. Key words : Banach lattice; injective tensor product; Grothendieck spaces. 2010 Mathematics Subject Classification : 46B20, 46B28.
1. I NTRODUCTION A Banach space Z is called a Grothendieck space if every separably valued bounded linear operator on Z is weakly compact, or equivalently, if every weak∗ convergent sequence in Z ∗ is weakly convergent (see, e.g., [3, 6]). All reflexive Banach spaces are Grothendieck spaces. There are a few examples of non-reflexive Grothendieck spaces. For instance, if K is a stonean space then C(K), in particular, `∞ , is a Grothendieck space (see, e.g., [6]). Recently, Li and Bu [7] characterized the positive projective ˆ |π| X that is a Grothendieck space, where λ is a Banach sequence lattice and X is a tensor product λ⊗ ˇ |ε| X that is Banach lattice. In this paper, we will characterize the positive injective tensor product λ⊗ a Grothendieck space.
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SHAOYONG ZHANG, ZHAOHUI GU AND YONGJIN LI 2. P RELIMINARIES
For a Banach space Z, Z ∗ will denote its topological dual and BZ will denote its closed unit ball. For a vector lattice X, the X-valued sequence space X N is a vector lattice with the following order x ¯ ≥ 0 ⇐⇒ xi ≥ 0 ∀i ∈ N,
x ¯ = (xi )i ∈ X N ,
and with the following lattice operations x ¯ ∧ y¯ = (xi ∧ yi )i ,
x ¯ ∨ y¯ = (xi ∨ yi )i ,
x ¯ = (xi )i , y¯ = (yi )i ∈ X N .
X + will denote the positive cone of X. For each n ∈ N and each x ¯ = (xi )i ∈ X N , let x ¯(≤ n) = (x1 , · · · , xn , 0, 0, · · · ),
x ¯(≥ n) = (0, · · · , 0, xn , xn+1 , · · · ).
For Banach lattices X and Y , Lr (X, Y ) will denote the space of regular linear operators from X to Y with the usual regular operator norm k · kr , and Kr (X, Y ) will denote the linear span of compact positive operators from X to Y . It follows from [9, section 1.3] that if Y is Dedekind complete then (Lr (X, Y ), k · kr ) is a Banach lattice. Recall that a Banach lattice X is called σ-order continuous if 0 ≤ xn ↓ 0 in X then xn → 0 in X; and is called a σ-Levi space if 0 ≤ xn ↑ and supn kxn k < ∞ then supn xn exists in X. Banach Lattice-valued Sequence Spaces Let λ be a solid sequence space, that is, a subspace of RN such that if |ai | ≤ |bi | for all i ∈ N and (
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