Approximation of the spectral action functional in the case of $\tau$-compact resolvents

We establish estimates and representations for the remainders of Taylor approximations of the spectral action functional $V\mapsto \tau (f(H_0+V))$ on bounded self-adjoint perturbations, where $H_0$ is a self-adjoint operator with $\tau$-compact resolvent in a semifinite von Neumann algebra and $f$ belongs to a broad set of compactly supported functions including $n$-times differentiable functions with bounded $n$-th derivative. Our results significantly extend analogous results in \cite{SkAnJOT}, where $f$ was assumed to be compactly supported and $(n+1)$-times continuously differentiable. If, in addition, the resolvent of $H_0$ belongs to the noncommutative $L^n$-space, stronger estimates are derived and extended to noncompactly supported functions with suitable decay at infinity.