PROPERTY (u) IN JH ˜⊗ǫJH

It is shown that the tensor product JH⊗̃ǫJH fails Pe lczńyski’s property (u). The proof uses a result of Kwapień and Pe lczńyski on the main triangle projection in matrix spaces. The Banach space JH constructed by Hagler [1] has a number of interesting properties. For instance, it is known that JH contains no isomorph of l, and has property (S): every normalized weakly null sequence has a subsequence equivalent to the c0-basis. This easily implies that JH is c0-saturated, i.e., every infinite dimensional closed subspace contains an isomorph of c0. In answer to a question raised originally in [1], Knaust and Odell [2] showed that every Banach space which has property (S) also has Pe lczyński’s property (u). In [4], the author showed that the Banach space JH⊗̃ǫJH is c0-saturated. It is thus natural to ask whether JH⊗̃ǫJH has also the related properties (S) and/or (u). In this note, we show that JH⊗̃ǫJH fails property (u) (and hence property (S) as well). Our proof makes use of a result, due to Kwapień and Pe lczyński, that the main triangle projection is unbounded in certain matrix spaces. We use standard Banach space notation as may be found in [5]. Recall that a series ∑ xn in a Banach space E is called weakly unconditionally Cauchy (wuC) if there is a constant K < ∞ such that ‖ ∑k n=1 ǫnxn‖ ≤ K for all choices of signs ǫn = ±1 and all k ∈ N. A Banach space E has property (u) if whenever (xn) is a weakly Cauchy sequence in E, there is a wuC series ∑ yk in E such that xn− ∑n k=1 yk → 0 weakly as n → ∞. If E and F are Banach spaces, and L(E , F ) is the space of all bounded linear operators from E ′ into F endowed with the operator norm, then the tensor product E⊗̃ǫF is the closed subspace of L(E , F ) generated by the weak*-weakly continuous operators of finite rank. In particular, for any x ∈ E, and y ∈ F , one obtains an element x⊗ y ∈ E⊗̃ǫF defined by (x⊗ y)x ′ = 〈x, x〉y for all x ∈ E . Let us also recall the definition of the space JH , as well as fix some terms and notation. Let T = ∪∞n=0{0, 1} n be the dyadic tree. The elements of T are called nodes. If φ is a node of the form (ǫi) n i=1, we 1991 Mathematics Subject Classification. 46B20, 46B28. 1