The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are shown to be preserved under lifting to the universal covering as a result of their local feature. In contrast, regularised traces, such as $\zeta$-regularised traces, are shown to differ from their lifted counterpart by an ordinary trace of an operator with smooth kernel. The latter vanishes for regularised traces of differential operators; the $\Z_2$-graded generalisation of this statement yields back Atiyah's $L^2$-index theorem. In contrast, the eta invariant of an essentially self-adjoint invertible differential operator differs from the $L^2$-eta invariant of its lift by an ordinary trace of some operator with smooth kernel.
More generally, we prove that due to their local feature, defect formula that express discrepancies of regularised traces for integer order classical pseudodifferential on closed manifolds, lift to their universal covering.