Non--commutative Integration Calculus

We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0ccr{eps}{u_1}cdotsccr{eps}{u_n}$ with $eps$ a grading operator on a Hilbert space $cH$ and $u_i$ bounded operators on $cH$ which naturally contains the compactly supported de Rham forms on $R^d$ (i.e. $eps$ is the sign of the free Dirac operator on $R^d$ and $cH$ a $L^2$--space on $R^d$). We present an elementary proof that the integral of $d$--forms $int_{R^d}trac{X_0dd X_1cdots dd X_d}$ for $X_iinMap(R^d;gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $Gamma X_0ccr{eps}{X_1}cdotsccr{eps}{X_d}$ where $Gamma=1$ for $d$ odd and $Gamma=gamma_{d+1}$ (`$gamma_5$--matrix) a spin matrix anticommuting with $eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace.