  It is shown that for a certain class of the Kato functions the Trotter-Kato
product formulae converge in Dixmier ideal C 1,$\infty$ in topology, which is
defined by the $\times$ 1,$\infty$-norm. Moreover, the rate of convergence in
this topology inherits the error-bound estimate for the corresponding
operator-norm convergence. 1 since [24], [14]. Note that a subtle point of this
program is the question about the rate of convergence in the corresponding
topology. Since the limit of the Trotter-Kato product formula is a strongly
continuous semigroup, for the von Neumann-Schatten ideals this topology is the
trace-norm $\times$ 1 on the trace-class ideal C 1 (H). In this case the limit
is a Gibbs semigroup [25]. For self-adjoint Gibbs semigroups the rate of
convergence was estimated for the first time in [7] and [9]. The authors
considered the case of the Gibbs-Schr{\"o}dinger semigroups. They scrutinised
in these papers a dependence of the rate of convergence for the (exponential)
Trotter formula on the smoothness of the potential in the Schr{\"o}dinger
generator. The first abstract result in this direction was due to [19]. In this
paper a general scheme of lifting the operator-norm rate convergence for the
Trotter-Kato product formulae was proposed and advocated for estimation the
rate of the trace-norm
