Characterizations of Riesz sets

Let $G$ be a compact abelian group, $M(G)$ its measure algebra and $L^{1}(G)$ its group algebra. For a subset $E$ of the dual group $\widehat{G}$, let $M_{E}(G)=\{\mu\in M(G):\widehat{\mu}=0$ on $\widehat{G} \backslash E\}$ and $L_{E}^{1}(G)=\{a\in L^{1}(G):\widehat{a}=0$ on $\widehat{G}\backslash E\}$. The set $E$ is said to be a Riesz set if $M_{E}(G)=L_{E}^{1}(G)$. In this paper we present several characterizations of the Riesz sets in terms of Arens multiplication and in terms of the properties of the Gelfand transform $\Gamma :L_{E}^{1}(G)\rightarrow c_{0}(E)$.

Unital Banach algebras and their subalgebras

Let A be a Banach algebra with a bounded approximate identity. We prove that if A is not unital, then there is a nonunital subalgebra B of A with a sequential bounded approximate identity. It follows that A must be unital if A is weakly sequentially complete and B** under the first Arens multiplication has a unique right identity for every subalgebra B of A with a sequential bounded approximate identity. As a consequence, we prove a result of Ülger that if A is both weakly sequentially complete and Arens regular, then A must be unital.

Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the functionis also in LUC (G). Hence we may define a productfor n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.

On James' Quasi-Reflexive Banach Space as a Banach Algebra

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.

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.

The Measure Algebra as an Operator Algebra

In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G).Let △ denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of △ in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative B*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space.