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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.
Let A be a Banach algebra, with second dual space A′′. We propose to study the space A′′ as a Banach algebra. There are two Banach algebra products on A′′, denoted by 2 and 3 . The Banach algebra A is Arens regular if the two products 2 and 3 coincide on A′′. In fact, A′′ has two topological centres denoted by Z (1) t (A ′′) and Z (2) t (A ′′) with A ⊂ Z t… (More)
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.
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 = ` p for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras ` (ω).
Let S be a semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second… (More)
Let G be a locally compact abelian group, and let p 2 1; 1). We show that the Segal algebra S p (G) is always weakly amenable, but that it is amenable only if G is discrete.
In , Dawson and the second author 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.… (More)
for each f, g ∈ L (G). For the theory of this Banach algebra, see , , , and [2, §3.3], for example. There are many standard left (and right) Banach L(G)-modules. Here we determine when these modules have certain well-known homological properties; we shall summarize some known results, and establish various new ones. In fact, we are seeking to… (More)
Let S be a (discrete) semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant.… (More)