Trace and Differences of Idempotents in C*-Algebras

Abstract

© 2019, Pleiades Publishing, Ltd. Let φ be atrace on aunital C*-algebra A, let Mφ be the ideal of definition of the trace φ, and let P, Q∈ A be idempotents such that QP = P. If Q∈ Mφ then P∈ Mφ and 0 ≤ φ(P) ≤ φ(Q). If Q− P∈ Mφ then φ(Q − P) ∈ ℝ+. Let A, B∈ A be tripotents. If AB = B and A∈ Mφ, then B∈ Mφ and 0 ≤ φ(B2) ≤ φ(A2) < +∞. Let A be a von Neumann algebra. Then φ(|PQ−QP|)≤min{φ(P),φ(Q),φ(|P−Q|)} for all projections P, Q≤ A. The following conditions are equivalent for a positive normal functional φ on a von Neumann algebra A:(i)φ is a trace;(ii)φ(Q − P) ∈ ℝ+ for all idempotents P, Q∈ A with QP = P;(iii)φ(|PQ − QP|) ≤ min{φ(P), φ(Q)} for all projections P, Q∈ A;(iv)φ(PQ + QP) ≤ φ(PQP + QPQ) for all projections P, Q∈ A;.