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Multi-normed spaces

100%
Dales H.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We modify the very well known theory of normed spaces (E,||·||) within functional analysis by considering a sequence (||·||ₙ: n ∈ ℕ) of norms, where ||·||ₙ is defined on the product space Eⁿ for each n ∈ 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 ∈ [1,∞], and in particular to L¹-spaces, rather than to L²-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' ((Eⁿ,||·||ₙ): n ∈ ℕ), where (E,||·||) 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, maximum, and (p,q)-multi-norms 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 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. Applications of this theory will be presented elsewhere.
2
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Weighted convolution algebras on subsemigroups of the real line

65%
Dales H., Dedania H.
Instytut Matematyczny Polskiej Akademii Nauk
EN
In this memoir, we shall consider weighted convolution algebras on discrete groups and semigroups, concentrating on the group (ℚ,+) of rational numbers, the semigroup $(ℚ^{+•},+)$ of strictly positive rational numbers, and analogous semigroups in the real line ℝ. In particular, we shall discuss when these algebras are Arens regular, when they are strongly Arens irregular, and when they are neither, giving a variety of examples. We introduce the notion of 'weakly diagonally bounded' weights, weakening the known concept of 'diagonally bounded' weights, and thus obtaining more examples. We shall also construct an example of a weighted convolution algebra on ℕ that is neither Arens regular nor strongly Arens irregular, and an example of a weight ω on $ℚ^{+•}$ such that $lim inf_{s→ 0+}ω(s) =0$.
3
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Approximate amenability of semigroup algebras and Segal algebras

65%
Dales H., Loy R.
Instytut Matematyczny Polskiej Akademii Nauk
EN
In recent years, there have been several studies of various 'approximate' versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004. The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability. There are a number of open questions on these relationships; these will be considered. In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work; we shall concentrate on the case of Banach function algebras. In Chapter 2, we shall discuss these properties for the semigroup algebra ℓ¹(S) of a semigroup S. In the case where S has only finitely many idempotents, ℓ¹(S) is approximately amenable if and only if it is amenable. In Chapter 3, we shall consider the class of weighted semigroup algebras of the form $ℓ¹(ℕ_{∧},ω)$, where ω: ℤ → [1,∞) is an arbitrary function. We shall determine necessary and sufficient conditions on ω for these Banach sequence algebras to have each of the various approximate amenability properties that interest us. In this way we shall illuminate the implications between these properties. In Chapter 4, we shall discuss Segal algebras on 𝕋 and on ℝ. It is a conjecture that every proper Segal algebra on 𝕋 fails to be approximately amenable; we shall establish this conjecture for a wide class of Segal algebras.
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Generators of maximal left ideals in Banach algebras

60%
Dales H., Żelazko W.
Studia Mathematica
|
2012
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tom 212
|
nr 2
173-193
EN
In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces 'closed ideals' by 'maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples. We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional.
5
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Approximate identities in Banach function algebras

60%
Dales H., Ülger A.
Studia Mathematica
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2015
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tom 226
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nr 2
155-187
EN
In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these algebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a contractive Banach function algebra which is not equivalent to a uniform algebra. We shall also obtain results about Banach sequence algebras and Banach function algebras that are ideals in their second duals.
6
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Second duals of measure algebras

54%
Dales H., Lau A., Strauss D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L¹(G) and the measure algebra M(G) on G, concentrating on their second dual algebras. As a preliminary we shall study the second dual C₀(Ω)'' of the C*-algebra C₀(Ω) for a locally compact space Ω, recognizing this space as C(Ω̃), where Ω̃ is the hyper-Stonean envelope of Ω. We shall study the C*-algebra $B^{b}(Ω)$ of bounded Borel functions on Ω, and we shall determine the exact cardinality of a variety of subsets of Ω̃ that are associated with $B^{b}(Ω)$. We shall identify the second duals of the measure algebra (M(G),∗) and the group algebra (L¹(G),∗) as the Banach algebras (M(G̃),□ ) and (M(Φ),□ ), respectively, where □ denotes the first Arens product and G̃ and Φ are certain compact spaces, and we shall then describe many of the properties of these two algebras. In particular, we shall show that the hyper-Stonean envelope G̃ determines the locally compact group G. We shall also show that (G̃,□ ) is a semigroup if and only if G is discrete, and we shall discuss in considerable detail the product of point masses in M(G̃). Some important special cases will be considered. We shall show that the spectrum of the C*-algebra $L^{∞}(G)$ is determining for the left topological centre of L¹(G)'', and we shall discuss the topological centre of the algebra (M(G)'',□ ).
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Equivalence of multi-norms

54%
Dales H., Daws M., Pham H., Ramsden P.
Instytut Matematyczny Polskiej Akademii Nauk
EN
The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of 'equivalence' of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces $L^{r}(Ω)$ are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show that the standard [t]-multi-norm defined on $L^{r}(Ω)$ is not equivalent to a (p,q)-multi-norm in most cases, leaving some cases open. We discuss the equivalence of the Hilbert space multi-norm, the (p,q)-multi-norm, and the maximum multi-norm based on a Hilbert space. We calculate the value of some constants that arise. Several results depend on the classical theory of (q,p)-summing operators.
8
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Equivalences involving (p,q)-multi-norms

49%
Blasco O., Dales H., Pham H.
Studia Mathematica
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2014
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tom 225
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nr 1
29-59
EN
We consider (p,q)-multi-norms and standard t-multi-norms based on Banach spaces of the form $L^{r}(Ω)$, and resolve some question about the mutual equivalence of two such multi-norms. We introduce a new multi-norm, called the [p,q]-concave multi-norm, and relate it to the standard t-multi-norm.
9
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A properly infinite Banach *-algebra with a non-zero, bounded trace

49%
Dales H., Laustsen N., Read C.
Studia Mathematica
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2003
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tom 155
|
nr 2
107-129
EN
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.
10
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Approximate amenability for Banach sequence algebras

49%
Dales H., Loy R., Zhang Y.
Studia Mathematica
|
2006
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tom 177
|
nr 1
81-96
EN
We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p}(ω)$.
11
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Maximal left ideals of the Banach algebra of bounded operators on a Banach space

43%
Dales H., Kania T., Kochanek T., Koszmider P., Laustsen N.
Studia Mathematica
|
2013
|
tom 218
|
nr 3
245-286
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