Affine cellular algebras and asymptotic algebras

The theory of affine cellular algebras A is extended to incorporate their asymptotic algebras A ˆ , clarifying unexpected differences between classical and affine situations and comparing with Lusztig's asymptotic Hecke algebras. The main new results are about a double centraliser property between A and A ˆ , about constructing A ˆ from cell modules of A , about existence of an embedding A → A ˆ and about a faithful functor from torsionless A ˆ -modules to A -modules as well as about the embedding being weakly spectrum preserving (in the sense of Baum and Nistor) and about non-zero endomorphisms of cell modules being injective, while there are no non-zero homomorphisms between non-isomorphic cell modules.