Let \({\mathcal M}\) denote a semifinite von Neumann algebra of operators on a Hilbert space with a distinguished faithful semifinite normal trace \(\tau\). It was shown by E. Nelson that the class \(\widetilde {\mathcal M}\) of \(\tau\)-measurable operators in a topological *-algebra where the topology under consideration is the topology of convergence in measure.
A \(\tau\)-measurable operator \(T\) is called \(\tau\)-compact if its generalised singular function decreases to zero. This is equivalent to say that \(\tau(E_{(t,\infty)})<\infty\) for all \(t>0\); where \(E_{(\cdot)}\) denotes the spectral measure of \(| T|\). These operators were considered by Fack and Kosaki. It can easily be seen that the class \(\widetilde{\mathcal K}\) of \(\tau\)-compact operators is a two-sided ideal which is closed in the topology of convergence in measure and contains the non-commutative \(L^ p\)-spaces. Part of the paper is to give a representation for \(\widetilde{\mathcal M}/ \widetilde{\mathcal K}\) as a \(C^*\)-algebra.