John Pym

On -amenability of Banach algebras

Generalizing the notion of left amenability for so-called F-algebras [12], we study the concept of -amenability of a Banach algebra A, where is a homomorphism from A to . We establish several characterizations of -amenability as well as some hereditary properties. In addition, some illuminating examples are given.

Haar measure for compact right topological groups

Compact right topological groups arise in topological dynamics and in other settings. Following H. Furstenbergs seminal work on distal flows, R. Ellis and I. Namioka have shown that the compact right topological groups of dynamical type always admit a probability measure invariant under the continuous left translations; however, this invariance property is insufficient to identify a unique probability measure (in contrast to the case of compact topological groups). In the present paper, we amplify on the proofs of Ellis and Namioka to show that a right invariant probability measure on the compact right topological group G exists provided G admits an appropriate system of normal subgroups, that it is uniquely determined and that it is also invariant under the continuous left translations. Using Namiokas work, we show that G has such a system of subgroups if its topological centre contains a countable dense subset, or if it is a closed subgroup of such a group.

LOCALLY COMPACT GROUPS, INVARIANT MEANS AND THE CENTRES OF COMPACTIFICATIONS

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L ∞ ( G ), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.

Subsemigroups of Stone-Čech compactifications

The Stone–Cech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([ 23 ], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [ 3 ] showed that β G was a semigroup for each discrete group G , and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[ 6 ] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [ 22 ] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

Subsemigroups of βN

Abstract The Stone-Cech compactification β N of the additive semigroup (β N , +) has a natural— and much studied—semigroup structure. We contribute some new pathological results to our knowledge of this semigroup. We find 2 c disjoint compact semigroups in β N whose free product is also a subsemigroup; each of these compact semigroups itself has a complex structure. In particular, the free semigroup product of 2 c idempotents can be found in β N . We also discuss three different types of subsemigroups of β N , each of which is, in some sense, singly generated by a right cancellable element of β N ⧹ N .

Elements of finite order in Stone-Čech compactifications

Let S be a free semigroup (on any set of generators). When S is given the discrete topology, its Stone-Cech compactification has a natural semigroup structure. We give two results about elements p of finite order in βS . The first is that any continuous homomorphism of βS into any compact group must send p to the identity. The second shows that natural extensions, to elements of finite order, of relationships between idempotents and sequences with distinct finite sums, do not hold.

The action of a semisimple Lie group on its maximal compact subgroup

In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if G = KAN is an Iwasawa decomposition of the group G, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer M of AN in K. Complete descriptions are given for the enveloping semigroups in the cases G = SL(2, C) and G = SL(2, R).

κ-topologies for right topological semigroups

Given a cardinal κ and a right topological semigroup S with topology τ, we consider the new topology obtained by declaring any intersection of at most κ members of τ to be open. Under appropriate hypotheses, we show that this process turns S into a topological semigroup. We also show that under these hypotheses the points of any subsemigroup T with card T≤κ can be replaced by (new) open sets that algebraically behave like T. Examples are given to demonstrate the nontriviality of these results

Convolution and the second dual of a Banach algebra

We shall point out a connexion between convolution (as defined in ( 5 ); see also ( 7 )) and the Arens multiplication on the second dual of a Banach algebra ( 1, 2 ). This enables us to demonstrate in an easy way the existence of a commutative Banach algebra whose second dual is not commutative. Examples can also be found in ( 3, 9 ) and elsewhere.

Multiplications in Additive Compactifications of N and Z

Abstract There is a natural action (n,x)↦nx of ( N ,·) on any semigroup (S,+). When S is compact, there is always an extension to a map β N ×S→S . When S has additional properties, there are extensions to other familiar semigroup compactifications of N , for example, wap N , ap N and sap N ≅ b Z (the Bohr compactification of Z ). Special cases of these extensions yield multiplications ∗ on these compactifications. The properties of multiplication, and its relationships with the natural addition, in each compactification are discussed. In particular, ( b Z ,+,∗) is a ring and its properties can often be pulled back to the other structures. The final section is devoted to the enveloping semigroups (which are in fact rings) of the actions of Z on compact groups. There turn out to be few possibilities: for example, if the group is not totally disconnected, then the enveloping ring for the action n↦nx is just ( b Z ,+,∗) .