We initiate the study of certain linear operators from a Banach algebra A into a Banach A-bimodule X, which we call approximately local derivations. We show that when A is a C*-algebra, a Banach algebra generated by idempotents, a semisimple annihilator Banach algebra, or the group algebra of a SIN or a totally disconnected group, bounded approximately local derivations from A into X are derivations. We also prove that the same result holds if p ∈ (1, infinity) and A is the Figa-Talamanca-Herz algebra Ap( G) of a locally compact group G whose principle component is abelian. Later on, we extend this idea to the space of n-cocycles and we show that, for some of the above algebras, bounded approximately local n-cocycles from A(n) into X are n-cocycles. Finally, we consider the quantization of these results and apply them to the Figa-Talamanca-Herz algebra Ap(G) of a locally compact group G for p ∈ (1, infinity). We show that Ap(G), equipped with an appropriate operator space structure, is operator weakly amenable. We also show that completely bounded approximately local n-cocycles from Ap( G)(n) into any quantized Ap(G)-bimodule are n-cocycles.
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