Halpern has defined a center valued essential spectrum, ΣI(A), and numerical range, Wʓ(A), for operators A in a von Neumann algebra ɸ. By restricting our attention to algebras ɸ which act on a separable Hilbert space, we can use a direct integral decomposition of ɸ to obtain simple characterizations of these qualities, and this in turn enables us to prove analogues of some classical results.
since the essential spectrum is defined relative to a central ideal, we first show that, under the separability assumption, every ideal, modulo the center, is an ideal generated by finite projections. This leads to the following decomposition theorem:
Theorem: Z = ʃΛ ⊕ c(λ)dµ ∈ ΣI(A) if and only if c(λ) ∈ σe(A(λ)) µ-a.e., where A = ʃΛ ⊕ A(λ)dµ and σe is a suitable spectrum in the algebra ɸ(λ).
Using mainly measure-theoretic arguments, we obtain a similar decomposition result for the norm closure of the central numerical range:
Theorem: Z = ʃΛ ⊕ c(λ)dµ ∈ Wʓ(A) if and only if c(λ) ∈ W(A(λ)) µ-a.e.
By means of these theorems, questions about ΣI(A) and W (A) in ɸ can be reduced to the factors ɸ(λ). As examples, we show that spectral mapping holds for ΣI, namely f(ΣI(A)) = ΣI(f(A)), and that a generalization of the power inequality holds for Wʓ(A).
Dropping the separability assumption, we show that central ideals can be defined in purely algebraic terms, and that the following perturbation result holds:
Thereom: ΣI(A + X) = ΣI(A) for all A ∈ɸ if and only if X ∈ I.
