Functions of almost commuting operators and an extension of the Helton–Howe trace formula

Let A and B   be almost commuting (i.e., the commutator AB−BAAB−BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ↦φ(A,B)φ↦φ(A,B) for functions φ   in the Besov class B∞,11(R2). This functional calculus is linear, the operators φ(A,B)φ(A,B) and ψ(A,B)ψ(A,B) almost commute for φ,ψ∈B∞,11(R2), and φ(A,B)=u(A)v(B)φ(A,B)=u(A)v(B) whenever φ(s,t)=u(s)v(t)φ(s,t)=u(s)v(t). We extend the Helton–Howe trace formula for arbitrary functions in B∞,11(R2). The main tool is triple operator integrals with integrands in Haagerup-like tensor products of L∞L∞ spaces.