Trace and Differences of Idempotents in C*-Algebras

Abstract

Let φ be atrace on aunital C*-algebra \(\mathcal{A}\), let \(\mathfrak{M}_\varphi\) be the ideal of definition of the trace φ, and let \(P,Q\in\mathcal{A}\) be idempotents such that QP = P. If \(Q\in\mathfrak{M}_\varphi\) then \(P\in\mathfrak{M}_\varphi\) and 0 ≤ φ(P) ≤ φ(Q). If \(Q-P\in\mathfrak{M}_\varphi\) then φ(Q − P) ∈ ℝ⁺. Let \(A,B\in\mathcal{A}\) be tripotents. If AB = B and \(A\in\mathfrak{M}_\varphi\), then \(B\in\mathfrak{M}_\varphi\) and 0 ≤ φ(B²) ≤ φ(A²) < +∞. Let \(\mathcal{A}\) be a von Neumann algebra. Then

\(\varphi(|PQ-QP|)\le {\rm{min}}\{\varphi(P),\varphi(Q),\varphi(|P-Q|)\}\)

for all projections \(P,Q\le\mathcal{A}\). The following conditions are equivalent for a positive normal functional φ on a von Neumann algebra \(\mathcal{A}\):

1. φ is a trace;

2. φ(Q − P) ∈ ℝ⁺ for all idempotents \(P,Q\in\mathcal{A}\) with QP = P;

3. φ(|PQ − QP|) ≤ min{φ(P), φ(Q)} for all projections \(P,Q\in\mathcal{A}\);

4. φ(PQ + QP) ≤ φ(PQP + QPQ) for all projections \(P,Q\in\mathcal{A}\);.