Differences of Idempotents In C*-Algebras and the Quantum Hall Effect

Let ϕ be a trace on the unital C*-algebra A and Mϕ be the ideal of the definition of the trace ϕ. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P − Q ∈ Mϕ, then ϕ((P − Q)2n+1) = ϕ(P − Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ Mϕ with ϕ(U-A) ∈ R. Then I-A, I−U ∈ Mϕ and ϕ(I−U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U − V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.