Functions of Almost Commuting Operators and an Extension of the Helton–howe Trace Formula

Abstract. Let A and B be almost commuting (i.e., the commutator AB−BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ 7→ φ(A,B) for functions φ in the Besov class B ∞,1(R ). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ, ψ ∈ B ∞,1(R ), and φ(A,B) = u(A)v(B) whenever φ(s, t) = u(s)v(t). We extend the Helton–Howe trace formula for arbitrary functions in B ∞,1(R ). The main tool is triple operator integrals with integrands in Haagerup-like tensor products of L∞ spaces.