No abstract

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra BðEÞ of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c 0 and c p for 1ppoN: We add a new member to this family by showing that there are exactly four closed ideals in BðEÞ for the Banach space E :¼ ð"c n 2 Þ c0 ; that is, E is the c 0 -direct sum of the finite-dimensional Hilbert spaces c 1 2 ; c 2 2 ; y; c n 2 ; y . r 2004 Elsevier Inc. All rights reserved. MSC: primary 47L10; 46H10; secondary 47L20; 46B45

For a Banach space X, let B(X) denote the Banach algebra of all continuous linear operators on X. First, we study the lattice of closed ideals in B(Jp), where 1 < p < ∞ and Jp is the pth James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in B(Jp). Applications of this result include the following. (ii) For each natural number n and each n-tuple (m 1 , . . . , mn) in {k 2 | k ∈ N} ∪ {∞}, there is a Banach space X such that B(X) has exactly n maximal ideals, and these maximal ideals have codimensions m 1 , . . . , mn in B(X), respectively; the Banach space X is a finite direct sum of James spaces and p-spaces.Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces X such that all the maximal ideals in B(X) can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in B(X) for each hereditarily indecomposable Banach space X, and we prove that there are 2 2 ℵ 0 distinct maximal ideals in B(G), where G is the Banach space constructed by Gowers to solve Banach's hyperplane problem.

In 2012, Dales and Polyakov introduced the concepts of multi-norms and dual multi-norms based on a Banach space. Particular examples are the lattice multi-norm p}¨} L n q and the dual lattice multi-norm p}¨} DL n q based on a Banach lattice. Here we extend these notions to cover 'p-multi-norms' for 1 ď p ď 8, where 8-multi-norms and 1-multi-norms correspond to multinorms and dual multi-norms, respectively. We shall prove two representation theorems. First we modify a theorem of Pisier to show that an arbitrary multi-normed space can be represented as ppY n , }¨} L n q : n P Nq, where Y is a closed subspace of a Banach lattice; we then give a version for certain p-multi-norms. Second, we obtain a dual version of this result, showing that an arbitrary dual multi-normed space can be represented as pppX{Y q n , }¨} DL n q : n P Nq, where Y is a closed subspace of a Banach lattice X; again we give a version for certain p-multi-norms. We shall discuss several examples of p-multi-norms, including the weak p-summing norm and its dual and the canonical lattice p-multi-norm based on a Banach lattice. We shall determine the Banach spaces E such that the p-sum power-norm based on E is a p-multi-norm. This relies on a famous theorem of Kwapień; we shall present a simplified proof of this result. We shall relate p-multi-normed spaces to certain tensor products. Our representation theorems depend on the notion of 'strong' p-multi-norms, and we shall define these and discuss when p-multi-norms and strong p-multi-norms pass to subspaces, quotients, and duals; we shall also consider whether these multi-norms are preserved when we interpolate between couples of p-multi-normed spaces. We shall discuss multi-bounded operators between p-multi-normed spaces, and identify the classes of these spaces in some cases, in particular for spaces of operators between Banach lattices taken with their canonical lattice p-multi-norms. Acknowledgements. The authors are grateful to the London Mathematical Society for the award of Scheme 2 grant 21202 that allowed Troitsky to come to Lancaster in May 2013; to the Fields Institute in Toronto, for invitations to Dales, Laustsen, and Troitsky to participate in the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras in March and April, 2014; to the Lorentz Center in Leiden for invitations to Dales, Laustsen, and Troitsky to participate in a meeting on Ordered Banach Algebras in July, 2014. Oikhberg acknowledges with thanks the support of the Simons Foundation Travel Grant 210060, and Troitsky acknowledges with thanks the support of an NSERC grant.

Abstract.A properly infinite C * -algebra has no non-zero traces. We construct properly infinite Banach * -algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C * -algebras, in the sense that they can be hermitian or * -semisimple.