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1. Introduction and Statement of the Main Result THE ALGEBRAS OF BOUNDED OPERATORS ON THE TSIRELSON AND BAERNSTEIN SPACES ARE NOT GROTHENDIECK SPACES KEVIN BEANLAND, TOMASZ KANIA, AND NIELS JAKOB LAUSTSEN Abstract. We present two new examples of reexive Banach spaces X for which the associated Banach algebra B(X) of bounded operators on X is not a Grothendieck space, th namely X = T (the Tsirelson space) and X = Bp (the p Baernstein space) for 1 < p < 1. 1. Introduction and statement of the main result A Grothendieck space is a Banach space X for which every weak*-convergent sequence in the dual space X∗ converges weakly. The name originates from a result of Grothen- dieck, who showed that `1, and more generally every injective Banach space, has this property. Plainly, every reexive Banach space X is a Grothendieck space because the weak and weak* topologies on X∗ coincide. By the HahnBanach theorem, the class of Grothendieck spaces is closed under quotients, and hence in particular under passing to complemented subspaces. Substantial eorts have been devoted to the study of Grothendieck spaces over the years, especially in the case of C(K)-spaces. Notable achievements include the constructions by Talagrand [16] (assuming the Continuum Hypothesis) and Haydon [10] of compact Haus- dor spaces KT and KH , respectively, such that the Banach spaces C(KT ) and C(KH ) are Grothendieck, but C(KT ) has no quotient isomorphic to `1, while C(KH ) has the weaker property that it contains no subspace isomorphic to `1. However, KH has the signicant advantage over KT that it exists within ZFC. Haydon, Levy and Odell [11, Corollary 3F] have subsequently shown that Talagrand's result cannot be obtained within ZFC itself because under the assumption of Martin's Axiom and the negation of the Continuum Hypo- thesis, every non-reexive Grothendieck space has a quotient isomorphic to `1. Brech [4] has pursued these ideas even further using forcing to construct a model of ZFC in which there is a compact Hausdor space KB such that C(KB) is a Grothendieck space and has density strictly smaller than the continuum, so in particular no quotient of C(KB) is 1 isomorphic to `1. In another direction, Bourgain [3] has shown that H is a Grothendieck space. Nevertheless, a general structure theory of Grothendieck spaces is yet to materialize, and many fundamental questions about the nature of this class remain open. Diestel [8, 2010 Mathematics Subject Classication. 47L10, 46B03 (primary); 46B10, 46B45 (secondary). Key words and phrases. Banach space, Grothendieck space, Tsirelson space, Baernstein space, bounded operator, diagonal operator. 1 2 K. BEANLAND, T. KANIA, AND N. J. LAUSTSEN 3] produced an expository list of such questions in 1973. It is remarkable how few of these questions that have been resolved in the meantime. We shall make a small contribution towards the resolution of the seemingly very dicult problem of describing the Banach spaces X for which the associated Banach algebra B(X) of bounded operators on X is a Grothendieck space by proving the following result. th Theorem 1.1. Let X be either the Tsirelson space T or the p Baernstein space Bp, where 1 < p < 1. Then the Banach algebra B(X) is not a Grothendieck space. For details of the spaces T and Bp, we refer to Section 2. We remark that these spaces are not the rst examples of reexive Banach spaces X for which B(X) is known not to be a Grothendieck space due to the following result of the second-named author [13]. Theorem 1.2 (Kania). Let X = L `n , where 1 < p < 1 and either q = 1 or n2N q `p q = 1. Then the Banach algebra B(X) is not a Grothendieck space. This paper is organized as follows: in Section 2, we prove Theorem 1.1, followed by a discussion of the key dierence between the two cases (see Proposition 2.7 and the paragraph preceding it for details), before we conclude with a short section listing some related open problems. 2. The proof of Theorem 1.1 The proof that B(T ) is not a Grothendieck space relies on abstracting the strategy used to prove Theorem 1.2. The following notion will play a key role in this approach. Let E be a Banach space with a normalized, -unconditional basis 1 (where ` -unconditional' 1 (en)n=1 1 means that Pn Pn for each and all choices of j=1 αjβjej 6 max16j6n jαjj · j=1 βjej n 2 N scalars .) The -direct sum of a sequence 1 of Banach spaces α1; : : : ; αn; β1; : : : ; βn E (Xn)n=1 is given by 1 M and the series X converges in Xn = (xn): xn 2 Xn (n 2 N) kxnken E : E n2N n=1 This is a Banach space with respect to the coordinate-wise dened operations and the norm 1 X k(xn)k = kxnken : n=1 Analogously, we write L X for the Banach space of uniformly bounded sequences n2N n ` with for each 1, equipped with the coordinate-wise dened operations (xn) xn 2 Xn n 2 N and the norm . k(xn)k = supn kxnk The main property of the E-direct sum that we shall require is that every uniformly bounded sequence 1 of operators, where for each , induces (Un)n=1 Un 2 B(Xn) n 2 N a `diagonal operator' diag(U ) on L X by the denition n n2N n E 1 1 diag(Un)(xn)n=1 = (Unxn)n=1; THE ALGEBRAS B(T ) AND B(Bp) ARE NOT GROTHENDIECK SPACES 3 and . k diag(Un)k = supn kUnk We shall also use the following result of W. B. Johnson [12]. Theorem 2.1 (Johnson). The Banach space L `n contains a complemented copy n2N 1 `1 of ` . Hence a Banach space that contains a complemented copy of L `n is not 1 n2N 1 ` a Grothendieck space. 1 Lemma 2.2. Let X = L X , where E is a Banach space with a normalized, 1-un- n2N n E conditional basis and (Xn) is a sequence of Banach spaces. Then B(X) contains a com- L plemented subspace which is isometrically isomorphic to Xn . n2N `1 Proof. We may suppose that is non-zero for each . Let , and take Xn n 2 N n 2 N wn 2 Xn and ∗ such that . For , the rank-one operator fn 2 Xn kwnk = kfnk = 1 = hwn; fni xn 2 Xn given by xn ⊗ fn : y 7! hy; fnixn;Xn ! Xn; has the same norm as xn, so the map M ∆: (xn) 7! diag(xn ⊗ fn); Xn ! B(X); (2.1) `1 n2N is an isometry. It is clearly linear, and therefore the image of ∆ is a subspace of B(X) which L is isometrically isomorphic to Xn . This subspace is complemented in B(X) n2N `1 because ∆ has a bounded, linear left inverse, namely the map given by 1 M U 7! (QnUJnwn)n=1; B(X) ! Xn ; `1 n2N th where Jn : Xn ! X and Qn : X ! Xn denote the n coordinate embedding and projection, respectively. Corollary 2.3. Let X be a Banach space which contains a complemented subspace that is isomorphic to L `mn for some unbounded sequence (m ) of natural numbers and n2N 1 E n some Banach space E with a normalized, 1-unconditional basis. Then B(X) is not a Gro- thendieck space. Proof. Let Y = L `mn . Lemma 2.2 implies that (Y ) contains a complemented n2N 1 E B copy of L `mn , which is isomorphic to L `n by Peªczy«ski's decomposition n2N 1 `1 n2N 1 `1 method, and therefore B(Y ) is not a Grothendieck space by Theorem 2.1. The assump- tion means that we can nd bounded operators U : X ! Y and V : Y ! X such that UV = IY . This implies that B(X) contains a complemented copy of B(Y ) because the operator R 7! URV; B(X) ! B(Y ); is a left inverse of S 7! V SU; B(Y ) ! B(X). Therefore B(X) is not a Grothendieck space. Following Figiel and Johnson [9], we use the term `the Tsirelson space' and the symbol T to denote the dual of the reexive Banach space originally constructed by Tsirelson [17] with the property that it does not contain any of the classical sequence spaces c0 and `p for 1 6 p < 1. We refer to [6] for an attractive introduction to the Tsirelson space, including 4 K. BEANLAND, T. KANIA, AND N. J. LAUSTSEN background information, its formal denition and a comprehensive account of what was known about it up until the late 1980's. The following notion plays a key role in the study of the Tsirelson space (and in the denition of the Baernstein spaces, to be given below). Denition 2.4. A non-empty, nite subset M of N is (Schreier-)admissible if jMj 6 min M, where jMj denotes the cardinality of M. Proof of Theorem 1.1 for X = T . The unit vector basis is a normalized, 1-unconditional basis for . We shall denote it by 1 throughout this proof. Take natural numbers T (tn)n=1 , and set 1 = m1 6 k1 < m2 6 k2 < m3 6 k3 < ··· and Mn = [mn; mn+1) \ N Fn = spanftj : j 2 Mng (n 2 N); so that (Fn) is an unconditional nite-dimensional Schauder decomposition of T . For , [5, Corollary 7(i)] shows that the series P1 converges in if and xn 2 Fn (n 2 N) n=1 xn T only if the series P1 converges in , and when they both converge, the norms n=1 kxnktkn T of their sums are related by 1 1 1 1 X X X kxnktkn xn 18 kxnktkn : 3 6 6 n=1 n=1 n=1 Consequently T is 54-isomorphic to the direct sum L F , where E denotes the n2N n E closed linear span of .
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