894 research outputs found
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 for each .
Our theory is analogous to, but distinct from, an existing theory of
`operator spaces'; it is designed to relate to general spaces for , and in particular to -spaces, rather than to -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 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 of bounded operators acting on an
infinite-dimensional Banach pace :
(Q1) Does always contain a maximal left ideal which is not
finitely generated? (Q2) Is every finitely-generated, maximal left ideal of
necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for
some non-zero ?
Since the two-sided ideal 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
contains ; (iii) the answer to Question (Q2) is positive for
many, but not all, Banach spaces.Comment: to appear in Studia Mathematic
Biogenesis of Poxviruses: Inactivation of Host DNA Polymerase by a Component of the Invading Inoculum Particle
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 a completely
contractive Banach algebra , which can be viewed as a
deformed Fourier algebra of . 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 makes
sense as long as is of Kac type. Finally we analyse as an explicit
example the algebras , , 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
Pointwise approximate identities in Banach function algebras
In this memoir, we shall study Banach function algebras that have bounded pointwise approximate identities, and especially those that have contractive pointwise approximate identities. ABanach function algebra A is (pointwise) contractive if A and every non-zero, maximal modular ideal in A have contractive (pointwise) approximate identities. Let A be a Banach function algebra with character space Phi_A. We shall show that the existence of a contractive pointwise approximate identity in A depends closely on whether ||varphi|| =1 for each varphi in Phi_A$. The linear span of Phi_A in the dual space A' is denoted by L(A), and this is used to define the BSE norm on A; the algebra A has a BSE norm if this norm is equivalent to the given norm. We shall then introduce and study in some detail the quotient Banach function algebra {mathcal Q}(A)= A''/L(A)^\perp; we shall give various examples, especially uniform algebras and those involving algebras that are standard in abstract harmonic analysis, including Segal algebras with respect to the group algebra of a locally compact group. We shall characterize the Banach function algebrasfor which overline{L(A)}= \ell^{1}(\Phi_A), and then classify contractive and pointwise contractive algebras in the class of unital Banach function algebras that have a BSE norm; they are uniform algebras with specific properties. We shall also give examples of such algebras that do not have a BSE norm. Finally we shall discuss when some classical Banach function algebras of harmonic analysis have non-trivial reflexive closed ideals, and make some remarks on weakly compact homo\-morphisms between Banach function algebra
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