### A Characterization of a Class of Locally Compact Abelian Groups via Korovkin Theory.

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In the current work, a new notion of $n$-weak amenability of Banach algebras using homomorphisms, namely $(\varphi ,\psi )$-$n$-weak amenability is introduced. Among many other things, some relations between $(\varphi ,\psi )$-$n$-weak amenability of a Banach algebra $\mathcal{A}$ and ${M}_{m}\left(\mathcal{A}\right)$, the Banach algebra of $m\times m$ matrices with entries from $\mathcal{A}$, are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra ${L}^{1}\left(G\right)$ is ($\varphi ,\psi $)-$n$-weakly amenable for any...

We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe explicitly...

A properly infinite C*-algebra has no non-zero traces. We construct properly infinite Banach *-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C*-algebras, in the sense that they can be hermitian or *-semisimple.

Amenability and the Arens product are studied. Using the Arens product, derivations from A are extended to derivations from A**. This is used to show directly that A** amenable implies A amenable.

Several results are given about the amenability of certain algebras defined by locally compact groups. The algebras include the C*-algebras and von Neumann algebras determined by the representation theory of the group, the Fourier algebra A(G), and various subalgebras of these.

There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation π of G such that the image π(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image $\pi \left({L}^{1}\left(G\right)\right)$ of the ${L}^{1}$-group algebra does not containany nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a “generalized Heisenberg group”.

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 $A={\ell}^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras ${\ell}^{p}\left(\omega \right)$.