  We extend to the case of a threshold ideal our result with J. Bourgain about
the essential centre of the commutant mod a diagonalization ideal for a n-tuple
of commuting Hermitian operators . We also compute the $K_0$-group of the
commutant mod trace-class of a unitary operator with spectrum equal to its
essential spectrum. We present the problem of computing the $K_1$-group for a
commutant mod trace-class in its simplest case.
