Possible extensions of the noncommutative integral

Abstract

In this paper we will discuss the problem of extending a trace σ defined on a dense von Neumann subalgebraMof a topological *-algebra A to some subspaces of A. In particular, we will prove that extensions of the trace σ that go beyond the space L1(σ ) really exist and we will explicitly construct one of these extensions. We will continue the analysis undertaken in Bongiorno et al. (Rocky Mt. J. Math. 40(6):1745–1777, 2010) on the general problem of extending positive linear functionals on a *-algebra