Vol. 43, No. 2, 1972

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

Herbert Paul Halpern

Vol. 43 (1972), No. 2, 349–380

A closed two-sided ideal in a von Neumann algebra 𝒜 is defined to be a central ideal if AiPi is in for every set {Pi} of orthogonal projections in the center 𝒵 of 𝒜r and every bounded subset {Ai} of . Central ideals are characterized in terms of the existence of continuous fields and their form is completely determined.

If 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 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 is defined to be 𝒦(A) = {ϕ(A)|ϕ ∈𝒜,ϕ1(1) = P() = (0)}. Here P 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 is the ideal of finite elements (resp. the strong radical) of 𝒜 then 𝒦(A) is the intersection of 𝒵 with the weak (resp. uniform) closure of the convex hull of {UAU1|U unitary in 𝒜}.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 15 April 1971
Published: 1 November 1972
Herbert Paul Halpern