Differences of idempotents in C*-algebras and the quantum Hall effect. II. Unbounded idempotents

Abstract

Let a von Neumann algebra M of operators act on a Hilbert space H, I be the unit of M, τ be a faithful semifinite normal trace on M. Let (M, τ) be the ∗-algebra of all τ-measurable operators and L1(M, τ) be the Banach space of all τ-integrable operators, P, Q ∈ S(M, τ) be idempotents. If P - Q ∈ L1(M, τ) then τ(P - Q) ∈ R. In particular, if A = A3 ∈ L1(M, τ), then τ(A) ∈ R. If P - Q ∈ L1(M, τ) and P Q ∈ M, then for all n ∈ N we have (P - Q)2n+1 ∈ L1(M, τ) and τ((P - Q)2n+1) = τ(P - Q) ∈ R. If A ∈ L2(M, τ) and U ∈ M is an isometry, then ||UA - A||22 ≤ 2||(I - U)AA∗||1.