Simultaneous lifting from irreducible representations of C*-algebras

Given a sequence (pi(n)) of irreducible representations of a liminal C*-algebra A, and a sequence (b(n)) of trace class operators with b(n) is an element of pi(n)(A), we investigate necessary conditions and sufficient conditions for the existence of a simultaneous lifting a is an element of A such that, for each n, the trace of sigma(a) is bounded for irreducible representations sigma in a neighbourhood of pi(n).