Topological amenability and Köthe co-echelon algebras

We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). Using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii–Sheinberg’s criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.

Orthogonal stability of the generalized quadratic functional equations in the sense of Rätz

Let (X, ⊥) be an orthogonality module in the sense of Rätz over a unital Banach algebra A and Y be a real Banach module over A. In this paper, we apply the alternative fixed point theorem for proving the Hyers-Ulam stability of the orthogonally generalized k-quadratic functional equation of the formaf(kx + y) + af(kx - y) = f(ax + ay) + f(ax - ay) + \left( {2{k^2} - 2} \right)f(ax)for some |k| > 1, for all a ɛ A1 := {u ɛ A||u|| = 1} and for all x, y ɛ X with x⊥y, where f maps from X to Y.

Operator Valued Measures as Multipliers of L1(I,X) with order Convolution

Let I = (0;∞) with the usual topology and product as max multiplication. Then I becomes a locally compact topo- logical semigroup. Let X be a Banach Space. Let L1(I;X) be the Banach space of X-valued measurable functions f such that ,we define It turns out that ƒ ∗ g ∈ L1(I;X) and L1(I;X) becomes an L1(I)-Banach module. A bounded linear operator T on L1(I;X) is called a multiplier of L1(I;X) if T(f ∗ g) = f ∗ Tg for all f ∈ L1(I) and g ∈ L1(I;X). We characterize the multipliers of L1(I;X) in terms of operator valued measures with point-wise finite variation and give an easy proof of some results of Tewari[12].

On derivations and elementary operators on C*-algebras

Let A be a unital C*-algebra with the canonical (H) C*-bundle $\mathfrak{A}$ over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of $\mathfrak{A}$ is a prime C*-algebra. We also consider separable C*-algebras A for which $\mathfrak{A}$ is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of $\mathfrak{A}$ have uniformly finite dimensions, and each restriction bundle of $\mathfrak{A}$ over a set where its fibres are of constant dimension is of finite type as a vector bundle.