  Let G be a reductive p-adic group and let H(G)^s be a Bernstein block of the
Hecke algebra of G. We consider two important topological completions of
H(G)^s: a direct summand S(G)^s of the Harish-Chandra--Schwartz algebra of G
and a two-sided ideal C*_r (G)^s of the reduced C*-algebra of G. These are
useful for the study of all tempered smooth G-representations.
  We suppose that H(G)^s is Morita equivalent to an affine Hecke algebra H(R,q)
-- as is known in many cases. The latter algebra also has a Schwartz completion
S(R,q) and a C*-completion C*_r (R,q), both defined in terms of the underlying
root datum R and the parameters q.
  We prove that, under some mild conditions, a Morita equivalence between
H(G)^s and H(R,q) extends to Morita equivalences between S(G)^s and S(R,q), and
between C*_r (G)^s and C*_r (R,q). We also check that our conditions are
fulfilled in all known cases of such Morita equivalences between Hecke
algebras. This is applied to compute the topological K-theory of the reduced
C*-algebra of a classical p-adic group.
