  We reprove the surjectivity statement of Braverman-Kazhdan's spectral
description of Lusztig's asymptotic Hecke algebra $J$ in the context of
$p$-adic groups. The proof is based on Bezrukavnikov-Ostrik's description of
$J$ in terms of equivariant $K$-theory. As a porism, we prove that the action
of $J$ extends from the non-strictly positive unramified characters to the
complement of a finite union of divisors, and that the trace pairing between
the Ciubotaru-He rigid cocentre of an affine Hecke algebra with equal
parameters and the rigid quotient of its Grothendieck group is perfect whenever
the parameter $q$ is not a root of the Poincar\'{e} polynomial of the finite
Weyl group.
  Without recourse to $K$-theory, we prove a weak version of Xi's description
of $J$ in type $A$.
