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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 $φ\mapstoφ(A,B)$ for functions $φ$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$. This functional calculus is linear, the operators $φ(A,B)$ and $ψ(A,B)$ almost commute for $φ,\,ψ\in B_{\infty,1}^1({\Bbb R}^2)$, 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 $B_{\infty,1}^1({\Bbb R}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor products of $L^\infty$ spaces.