Integration of virtually continuous functions over bistochastic measures and the trace formula for nuclear operators (Q2797728)

In the present paper, the authors work with virtually continuous functions (a notion weaker than a continuous function on a product of two measure spaces) in order to give a description of the integral trace of a nuclear operator. A virtually continuous function must coincide with a \textit{properly} virtually continuous function \(f\) on a subset of \((X,\mu)\times (Y,\nu)\) of full measure. The function \(f\) must satisfy that for any \(\varepsilon>0\), there are subsets \(X'\subseteq X\), \(Y'\subseteq Y\) with \(\mu(X\setminus X'),\nu(Y\setminus Y')<\varepsilon\) and admissible semimetrics \(\rho_X\) and \(\rho_Y\) on \(X'\) and \(Y'\) resp., such that \(f\) is continuous on \((X', \rho_X)\times(Y',\rho_Y)\). The authors show that the kernel of a nuclear operator is a virtually continuous function, and deduce the main result of [\textit{M. Sh. Birman}, St. Petersbg. Math. J. 27, No. 2, 327--331 (2016); translation from Algebra Anal. 27, No. 2, 211--217 (2015; Zbl 1336.47023)] from this.