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Let S be a semigroup. We shall consider the centres of the semigroup $$(\beta \,S, \,\Box \,)$$ ( β S , □ ) and of the algebra $$(M(\beta \,S), \,\Box \,)$$ ( M ( β S ) , □ ) , where $$M(\beta \,S)$$ M ( β S ) is the bidual of the semigroup algebra $$(\ell ^{\,1}(S),\,\star \,)$$ ( ℓ 1 ( S ) , ⋆ ) , and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of $$S^*$$ S ∗ and of $$M(S^*)$$ M ( S ∗ ) that are ‘determining for the left topological centre’ (DLTC sets) of $$\beta \,S$$ β S and $$M(\beta \,S)$$ M ( β S ) . It is known that, when the semigroup S is cancellative, $$\ell ^{\,1}(S)$$ ℓ 1 ( S ) is strongly Arens irregular and that there is a DLTC set consisting of two points of $$S^*$$ S ∗ . In contrast, there is little that has been published about the Arens regularity of $$\ell ^{\,1}(S)$$ ℓ 1 ( S ) when S is not cancellative. Totally ordered, abelian semigroups, with the map $$(s,t)\rightarrow s \wedge t$$ ( s , t ) → s ∧ t as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of $$\beta \,S$$ β S and of $$M(\beta \,S)$$ M ( β S ) for all such semigroups, and give several examples, showing that the minimum cardinality of DTC sets may be arbitrarily large, and, in particular, we shall give an example of a countable, totally ordered, abelian semigroup S with this operation for which there is no countable DTC set for $$\beta S$$ β S or for $$M(\beta S)$$ M ( β S ) . There was no previously-known example of an abelian semigroup S for which $$\beta \,S$$ β S or $$M(\beta S)$$ M ( β S ) did not have a finite DTC set.

Let (Formula presented.) be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras (Formula presented.) such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.). For example, we shall prove that, in the case where (Formula presented.) is an infinite-dimensional, unital Banach function algebra, (Formula presented.) is a maximal abelian subalgebra in infinitely-many closed subalgebras of (Formula presented.) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples (Formula presented.) that are local algebras. We shall also give examples of uniform algebras of the form (Formula presented.), where (Formula presented.) is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras (Formula presented.) such that each (Formula presented.) contains (Formula presented.) as a closed subalgebra and is such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.).

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?

Let A be a (non-unital) commutative Banach algebra. We consider when A has a variety of factorization properties: we list the (ob-vious) implications between these properties, and then consider whether any of these implications can be reversed in various classes of commu-tative Banach algebras. We summarize the known counterexamples to these possible reverse implications, and add further counterexamples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products.

Let K be a compact space, and denote by (K) over tilde its hyper-Stonean envelope. We discuss the class of spaces K with the property that (K) over tilde is homeomorphic to (I) over tilde, the hyper-Stonean envelope of the closed unit interval d. Certainly each uncountable, compact, metrizable space K belongs to this class. We describe several further classes of compact spaces K for which (K) over tilde = (I) over tilde. In fact, (K) over tilde = (I) over tilde if and only if the Banach spaces M(K) and M(I) of measures on K and If are isometrically isomorphic.

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 Lr (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 Lr (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.

Let G be a locally compact group, and take p∈(1, ∞). We prove that the Banach left L 1(G)-module L p(G) is injective (if and) only if the group G is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces.

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 `1(S) of a semigroup S. In the case where S has only finitely many idempotents, `1(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 `1(N^; !), where ! : Z ! [1; 1) is an arbitrary function. We shall determine necessary and sufficient conditions on ! 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.

We shall now define, in \(\S\) 5.1, the compact spaces of most interest to us, the ‘hyper-Stonean spaces’, and characterize them in terms of the existence of category measures. For a locally compact space K and μ ∈ P(K), we shall discuss in \(\S\) 5.2 the commutative C∗-algebra L ∞ (K, μ), which has been identified with the space C(Φ μ ). In particular, we shall describe Φ m , where m is Lebesgue measure on \(\mathbb{I}\): in this case, Φ m is called \(\mathbb{H}\), the hyper-Stonean space of the unit interval. We shall give a topological characterization of \(\mathbb{H}\) in \(\S\)5.3.

We shall now give some background in the theory of normed and Banach spaces, including the key definitions of dual and bidual spaces and of an isomorphism and an isometric isomorphism between two normed spaces. In particular, we shall show how certain bidual spaces can be embedded in other Banach spaces. In \(\S\)2.3, we shall also recall some basic results and theorems concerning Banach lattices. We shall define complemented subspaces of a Banach space in \(\S\)2.4, and also we shall discuss, in \(\S\)2.5, the projective and injective objects in the category of Banach spaces and bounded operators. We shall conclude the chapter by discussing dentability and the Krein–Milman property for Banach spaces in \(\S\)2.6.

The main aim of this chapter is to determine when a space of the form C(K) for a compact space K is a dual space or a bidual space,either isometrically or isomorphically. However, we shall first discuss when two spaces C(K) and C(L) are isomorphic and when they are isometrically isomorphic. Some results come from rather elementary considerations, but some require more sophisticated background.

This chapter will first give the basic background that we shall require concerning Banach algebras, C∗-algebras, and von Neumann algebras. In particular, in \(\S\)3.1, we shall discuss the bidual of a Banach algebra, taken with its Arens products. In \(\S\)3.3, we shall exhibit the Baire classes as examples of commutative C∗-algebras. We shall conclude the chapter in \(\S\)3.4 with a few remarks on the generalizations of some of our discussions concerning the commutative C∗-algebras C 0(K) to general (non-commutative) C∗-algebras; as we said, these generalizations will not be used within our main text.

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that $ \mathrm{tsr}(A) \geq \mathrm{tsr}(C(\Phi_A))$ whenever $ A$ is approximately regular.

We study the structure of derivations D : AlgA C [X] → A, where A is a function algebra defined on a compact subset of C such that A contains C [X]. We show that, under certain conditions on the algebra A, all such derivations are zero. We give examples to show the necessity of these conditions.

Let $G$ be a locally compact group, and let $L^1 (G)$ be the Banach algebra which is the group algebra of $G$. We consider a variety of Banach left $L^1 (G)$ -modules over $L^1 (G)$ , and seek to determine conditions on $G$ that determine when these modules are either projective or injective or flat in the category. The answers typically involve $G$ being compact or discrete or amenable. For example, in the case where $G$ is discrete and $1 < p < \infty$, we find that the module $\ell^p (G)$ is injective whenever $G$ is amenable, and that, if it is amenable, then $G$ is ?pseudo-amenable?, a property very close to that of amenability.

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.

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 al- gebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a con- tractive 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.

In this paper we shall prove that the measure algebra M(G) of a locally compact group G is amenable as a Banach algebra if and only if G is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that M(G) is not amenable in the case where the group G is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

NORMED ALGEBRAS OF DIFFERENTIABLE FUNCTIONS ON COMPACT PLANE SETS

Let A be a Banach algebra, and let D: A → A* be a continuous derivation, where A* is the topological dual space of A. The paper discusses the situation when the second transpose D**: A** → (A**)* is also a derivation in the case where A** has the first Arens product.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map Formula is continuous, and a linear map D:A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous. The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

Let G G be a locally compact abelian group, and let p ∈ [ 1 , ∞ ) p \in [1,\infty ) . We show that the Segal algebra S p ( G ) S_p(G) is always weakly amenable, but that it is amenable only if G G is discrete.

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

In the 1970s, a question of Kaplansky about discontinuous homomorphisms from certain commutative Banach algebras was resolved. Let A be the commutative C*-algebra C(Ω), where Ω is an infinite compact space. Then, if the continuum hypothesis (CH) be assumed, there is a discontinuous homomorphism from C(Ω) into a Banach algebra [2, 7]. In fact, let A be a commutative Banach algebra. Then (with (CH)) there is a discontinuous homomorphism from A into a Banach algebra whenever the character space ΦA of A is infinite [3, Theorem 3] and also whenever there is a non-maximal, prime ideal P in A such that ∣A/P∣=2ℵ0 [4, 8]. (It is an open question whether or not every infinite-dimensional, commutative Banach algebra A satisfies this latter condition.)

Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.

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., to appear in Studia Mathematica

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.

Let A be a Banach algebra. Then there are two natural products on the second dual A′′ of A arising from left and right translations by elements of A; they are called the Arens products; we denote these products by and ♦, respectively. For definitions and discussions of these products, see [2], [4], and [5], for example. We briefly recall the definitions. For a ∈ A, λ ∈ A′, and Φ ∈ A′′, define λ · a and a · λ in A′ by

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 continuous.

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 updates

The theory of translation-invariant operators on various spaces of functions (or measures or distributions) is a well-trodden field. The problem is to decide, first, whether or not a linear operator between two function spaces on, say, ℝ or ℝ+ which commutes with one or many translations on the two spaces is necessarily continuous, and, second, to give a canonical form for all such continuous operators. In some cases each such operator is zero. The second problem is essentially the ‘multiplier problem’, and it has been extensively discussed; see [7], for example.

We investigate the completeness and completions of the normed algebras (D(1)(X),∥•∥) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D(1)(X),∥•∥) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D(1)(X),∥•∥) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. In an earlier paper of Bland and Feinstein, the notion of an F-derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D ((1))/F corresponding to the normed algebras (D(1)(X),∥•∥). In the present paper, we obtain stronger results concerning the questions when (D(1)(X),∥•∥) and D ((1))/F (X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘F-regular'. An example of Bishop shows that the completion of (D(1)(X),∥•∥) need not be semisimple. We show that the completion of (D(1)(X),∥•∥) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X. We prove that the character space of D(1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D(1)(X),∥•∥) is complete. In particular, characters on the normed algebras (D(1)(X),∥•∥) are automatically continuous.

In this paper we study the question whether certain non-semisimple, commutative Banach algebras have a Wedderburn decomposition U = B ⊕ rad U , where B is a subalgebra of U . Let A(G) be the Fourier algebra of a locally compact abelian group G, and let E be a closed subset of G. Let J(E) be the smallest closed ideal in A(G) whose hull is E. We prove that, when E is a set of non-synthesis, the quotient algebra A(G) J(E) never has a Wedderburn decomposition even in the purely algebraic sense. This result is extended to cover certain Beurling algebras Ax(Rk) and Ax(Tk).

In this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers–Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.