Ideals of operators on $(\oplus \ell ^\infty (n))_{\ell ^1}$

Abstract

Abstract. The unique maximal ideal in the Banach algebra L(E), E = (⊕ℓ ∞ (n)) ℓ 1 , is identified. The proof relies on techniques developed by Laustsen, Loy and Read [7] and a dichotomy result for operators mapping into L 1 due to Laustsen, Odell, Schlumprecht and Zsák [8].Given a Banach space E, it is a natural problem to try to understand the ideal structure of the Banach algebra L(E). In the classical period, the only Banach spaces E for which the lattice of closed ideals in L(E) are fully understood are Hilbert space [2,4,11] and the sequence spaces c 0 and ℓ p , 1 ≤ p < ∞ [3]. In the past decade, starting with [7], there has been a resurgence of interest in the problem and new results have been obtained, see, e.g., [5,6,8,9,10,12,13]. One should also mention the recent breakthrough example AH by Argyros and Haydon [1], where L(AH) is well understood because of a very different reason. In this note, we make a small contribution to the program by identifying the unique maximal ideal in the algebra L(E), where E = (⊕ℓ ∞ (n)) ℓ 1 . Our method is slightly more general in the sense that it works also for the space E = (⊕ℓ 2 (n)) ℓ 1 , which gives an alternate proof of the main result of [9]. Let E n be finite dimensional Banach spaces and let E = (⊕E n ) ℓ 1 . Given a subset I of N, denote by E I the subspace (⊕ n∈I E n ) ℓ 1 . The natural embedding of E I into E is denoted by J I and the natural projection from E onto E I is denoted by P I . If I = {n}, then we write for short J n and P n respectively. Let T be a bounded linear operator on E, then T has a natural matrix representation (T mn ), where T mn = P m T J n : E n → E m . Denote the nth column in this representation by T n . Thus T n = T J n :Let X, Y, Z be Banach spaces. If T : X → Y is a bounded linear operator and ε > 0, defineFor Z = c 0 , respectively ℓ 1 , we write Fac In the sequel, T will always denote a bounded linear operator on E.