Multi-normed spaces

We modify the very well known theory of normed spaces (E, \norm) within functional analysis by considering a sequence (\norm_n : n\in\N) of norms, where \norm_n is defined on the product space $E^n$ for each $n\in\N$. Our theory is analogous to, but distinct from, an existing theory of `operator spaces'; it is designed to relate to general spaces $L^p$ for $p\in [1,\infty]$, and in particular to $L^1$-spaces, rather than to $L^2$-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a `multi-normed space' ((E^n, \norm_n) : n\in \N), where (E, \norm) is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum and maximum multi-norm based on a given space. Multi-norms measure `geometrical features' of normed spaces, in particular by considering their `rate of growth'. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to `multi-topological linear spaces' through `multi-null sequences', and to `multi-bounded' linear operators, which are exactly the `multi-continuous' operators. We define a new Banach space ${\mathcal M}(E,F)$ of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of `orthogonal decompositions' of a normed space with respect to a multi-norm, and apply this to construct a `multi-dual' space.Comment: Many update

Multi-norms

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

Continuity of homomorphisms and derivations from algebras of approximable and nuclear operators

1. Let be a Banach algebra. We say that homomorphisms from are continuous if every homomorphism from into a Banach algebra is automatically continuous, and that derivations from are continuous if every derivation from into a Banach -bimodule is automatically continuou

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

A Collection of Problems on Spectrally Bounded Operators

We discuss several open problems on spectrally bounded operators, some new, some old, adding in a few new insights.Comment: 15 pages,; submitte

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