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Let (X; d) be a compact metric space and let K be a nonempty compact subset of X. Let  2 (0; 1] and let Lip(X; K; d ) denote the Banach algebra of all continuous complex-valued functions f on X for which...

This paper continues the investigation of the first author begun in part one. The hereditary properties of n -homomorphism AMENABILITY for Banach algebras are investigated and the relations between n -homomorphism AMENABILITY of a Banach algebra and its ideals are found. Analogous to the character AMENABILITY, it is shown that the tensor product of two unital Banach algebras is n -homomorphism amenable if and only if each one is n -homomorphism amenable.

We introduce the notions of $\mathcal{T}_{M}$-AMENABILITY and $\phi$-$\mathcal{T}_{M}$-AMENABILITY. Then, we characterize $\phi$-$\mathcal{T}_{M}$-AMENABILITY  in terms of $WAP$-diagonals and $\phi$-invariant means. Some concrete cases are also discussed.

Let G be a locally compact group, and let w be a weight on G. We show that the weighted measure algebra M(G, w) is amenable if and only if G is a discrete, amenable group and sup{w (g) w (g−1) :gÎG} < ¥ , where w (g)³1 (gÎG).

notion of AMENABILITY for topological semigroups is introduced. A topological semigroup S is called Johnson amenable if for every Banach S -bimodule E, every bounded crossed homomorphism from S to E* is principal. In this paper it is shown that a discrete semigroup S is Johnson amenable if and only if L1 (S) is an amenable Banach algebra. Also, we show that if a topological semigroup S is Johnson amenable, then it is amenable, but the converse is not true.