Affine cellular algebras and asymptotic algebras

The theory of affine cellular algebras $A$ is extended to incorporate their asymptotic algebras $\widehat{A}$, 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 $\widehat{A}$, about constructing $\widehat{A}$ from cell modules of $A$, about existence of an embedding $A\to \widehat{A}$ and about a faithful functor from torsionless $\widehat{A}$-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.