A closed two-sided ideal ℐ in a
von Neumann algebra 𝒜 is defined to be a central ideal if ∑AiPi is in ℐ for every set
{Pi} of orthogonal projections in the center 𝒵 of 𝒜r and every bounded subset {Ai}
of ℐ. Central ideals are characterized in terms of the existence of continuous fields
and their form is completely determined.
If ℐ is a central ideal of 𝒜 and A ∈ then A0∈𝒵 is said to be in the essential
central spectrum of A if A0− A is not invertible in 𝒜 modulo the smallest closed
ideal containing ℐ and ζ for every maximal ideal ζ of 𝒵. It is shown that the essential
central spectrum is a nonvoid, strongly closed subset of 𝒵 and that it satisfies many
of the relations of the essential spectrum of operators on Hilbert space. Let 𝒜∼
be the space of all bounded Z-module homomorphisms of 𝒜 into 𝒵. The
essential central numerical range of A ∈𝒜 with respect to ℐ is defined to be
𝒦ℐ(A) = {ϕ(A)|ϕ ∈𝒜∼,∥ϕ∥≦ 1,ϕ(1) = Pℐ,ϕ(ℐ) = (0)}. Here Pℐ is the
orthogonal complement of the largest central projection in The essential
central numerical range is shown to be a weakly closed, bounded, 𝒵-convex
subset of 𝒵. It possesses many of the properties of the essential numerical
range but in a form more suited to the fact that A is in 𝒜 rather than a
bounded operator. It is shown that if 𝒜 is properly infinite and ℐ is the
ideal of finite elements (resp. the strong radical) of 𝒜 then 𝒦ℐ(A) is the
intersection of 𝒵 with the weak (resp. uniform) closure of the convex hull of
{UAU−1|U unitary in 𝒜}.