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 $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $\phi \mapsto \phi (A,B)$ for functions $\phi$ in the Besov class $B_{\mathrm{\infty },1}^1({\mathbb{R}}^2)$. This functional calculus is linear, the operators $\phi (A,B)$ and $\psi (A,B)$ almost commute for $\varphi,,\psi\in B_{\infty,1}^1({\Bbb R}^2)$,and$\varphi(A,B)=u(A)v(B)$whenever$\varphi(s,t)=u(s)v(t)$. We extend the Helton–Howe trace formula for arbitrary functions in $B_{\mathrm{\infty },1}^1({\mathbb{R}}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor products of $L^{\mathrm{\infty }}$ spaces.