  This article is dedicated to the proof of the following theorem. Let G be a
finite group, p be a prime number, and e be a p-block of G. Assume that the
centraliser C_G(P) of an e-subpair (P,e_P) "strongly" controls the fusion of
the block e, and that a defect group of e is either abelian or (for odd p) has
a non-cyclic center. Then there exists a stable equivalence of Morita type
between the block algebras OGe and OC_G(P)e_P, where O is a complete discrete
valuation ring of residual characteristic p. This stable equivalence is
constructed by gluing together a family of local Morita equivalences, which are
induced by bimodules with fusion-stable endo-permutation sources.
  Brou\'e had previously obtained a similar result for principal blocks, in
relation with the search for a modular proof of the odd Z*p-theorem. Thus our
theorem points towards a block-theoretic analogue of the Z*p-theorem, which we
state in terms of fusion control and Morita equivalences.

Representation Theory