Multi-norms

We give a survey of the theory of multi-norms, based on a talk given in Tartu on 5 September 2013

Maximal left ideals of the Banach algebra of bounded operators on a Banach space

We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (Q1) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (Q2) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (Q2) is positive for many, but not all, Banach spaces.Comment: to appear in Studia Mathematic

Integration over the quantum diagonal subgroup and associated Fourier-like algebras

By analogy with the classical construction due to Forrest, Samei and Spronk we associate to every compact quantum group $\mathbb{G}$ a completely contractive Banach algebra $A_\Delta(\mathbb{G})$, which can be viewed as a deformed Fourier algebra of $\mathbb{G}$. To motivate the construction we first analyse in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and So{\l}tan, the corresponding integration represented by a certain idempotent state on $C(\mathbb{G})$ makes sense as long as $\mathbb{G}$ is of Kac type. Finally we analyse as an explicit example the algebras $A_\Delta(O_N^+)$, $N\ge 2$, associated to Wang's free orthogonal groups, and show that they are not operator weakly amenable.Comment: Minor updates; Remark 5.7 has been added; 31 page

Maximal left ideals in Banach algebras

Let A be a Banach algebra. Then frequently each maximal left ideal in A is closed, but there are easy examples that show that a maximal left ideal can be dense and of codimension 1 in A. It has been conjectured that these are the only two possibilities: each maximal left ideal in a Banach algebra A is either closed or of codimension 1 (or both). We shall show that this is the case for many Banach algebras that satisfy some extra condition, but we shall also show that the conjecture is not always true by constructing, for each n is an element of N, examples of Banach algebras that have a dense maximal left ideal of codimension n. In particular, we shall exhibit a semi-simple Banach algebra with this property. We shall show that the questions concerning maximal left ideals in a Banach algebra A that we are considering are related to automatic continuity questions: When are A-module homomorphisms from A into simple Banach left A-modules automatically continuous

Amenability of algebras of approximable operators

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space.Comment: 20 pages, to appear in Israel Journal of Mathematic