Essential central spectrum and range for elements of a von Neumann algebra

Abstract

A closed two-sided ideal ^f in a von Neumann algebra is defined to be a central ideal if ^ A%Pi is in ^ for every set {Pi} of orthogonal projections in the center %* of J ^ and every bounded subset {A*} of KJ'. Central ideals are characterized in terms of the existence of continuous fields and their form is completely determined. If ^ is a central ideal of Jzf and A e J ^ then Ao e %* is said to be in the essential central spectrum of A if Ao — A is not invertible in Sϊf modulo the smallest closed ideal containing ^ and ζ for every maximal ideal ζ of %*. It is shown that the essential central spectrum is a nonvoid, strongly closed subset of %? and that it satisfies many of the relations of the essential spectrum of operators on Hubert space. Let j y ~ be the space of all bounded ^-module homomorphisms of J ^ into -S. The essential central numerical range of i e with respect to ^ is defined to be Sέ^(A)={φ(A) φe ~, II Φ II ^ 1, 0(1) = IV, Φ(^) = (0)}. Here P ^ is the orthogonal complement of the largest central projection in *Jζ The essential central numerical range is shown to be a weakly closed, bounded, ^-convex subset of %£. It possesses many of the properties of the essential numerical range but in a form more suited to the fact that A is in Ssf rather than a bounded operator. It is shown that if Sf is properly infinite and ^ is the ideal of finite elements (resp. the strong radical) of J ^ then «-%S(A) is the intersection of JΓ with the weak (resp. uniform) closure of the convex hull of {UAU~ U unitary in