The Spectral Scale and the Numerical Range

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1 = (c+ c )/2 and b2 = (c− c)/2i. Also write B for the spectral scale of {b1, b2} relative to τ . In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information about the operator c. In this paper we expand that information base by showing that the numerical range of c is encoded in B also. We begin by proving that the k-numerical range of an arbitrary operator d in B(H) coincides with the numerical range of d when the von Neumann algebra generated by d contains no finite rank operators. Thus, the k-numerical range is not useful for most operators considered here. We next show that the boundary of the numerical range of c is exactly the set of radial complex slopes on B at the origin. Further, we show that points on this boundary that lie in the numerical range are visible as line segments in the boundary of B. Also, line segments on the boundary which lie in the numerical range show up as faces of dimension two in the boundary of B. Finally, when N is abelian, we prove that the point spectrum of c appears as complex slopes of 1-dimensional faces of B. AMS Subject Classification Numbers 47A12, 47C15 ∗The second author was partially supported by the National Science Foundation during the period of research that resulted in this paper.