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Aspects of the theory of derivations

Gerard Murphy (1994)

Banach Center Publications

We survey some old and new results in the theory of derivations on Banach algebras. Although our overview is broad ranging, our principal interest is in recent results concerning conditions on a derivation implying that its range is contained in the radical of the algebra.

Banach algebra techniques in the theory of arithmetic functions

Lutz G. Lucht (2008)

Acta Mathematica Universitatis Ostraviensis

For infinite discrete additive semigroups X [ 0 , ) we study normed algebras of arithmetic functions g : X endowed with the linear operations and the convolution. In particular, we investigate the problem of scaling the mean deviation of related multiplicative functions for X = log . This involves an extension of Banach algebras of arithmetic functions by introducing weight functions and proving a weighted inversion theorem of Wiener type in the frame of Gelfand’s theory of commutative Banach algebras.

Properties of derivations on some convolution algebras

Thomas Pedersen (2014)

Open Mathematics

For all convolution algebras L 1[0, 1); L loc1 and A(ω) = ∩n L 1(ωn), the derivations are of the form D μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D μ to a natural dual space is weak-star continuous.

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