On Extending the Trace as a Linear Functional, II

A positive bounded selfadjoint operator is in the trace class of von Neumann and Schatten ([4]) if the sum of its diagonal matrix elements with respect to some orthonormal basis is finite, and the trace is then defined to be this sum, which is independent of the basis. A bounded selfadjoint but not necessarily positive operator x is in the trace class if in the decomposition x = x+ – x−, with x+ and x− positive and x+x− = 0, both x+ and x−are in the trace class; the trace of x is then defined to be the difference of the finite traces of x+ and x−. The trace defined in this way is a linear functional on the trace class, and is unitarily invariant; if u is a unitary operator, the trace of uxu−1 is the same as the trace of x.