A multiple operator integral (MOI) is an indispensable tool in several branches of noncommutative analysis. However, there are substantial technical issues with the existing literature on the "separation of variables" approach to defining MOIs, especially when the Hilbert spaces on which they act are not separable. In this paper, we provide a detailed development of this approach in a very general setting that resolves existing technical issues. Along the way, we characterize several kinds of "weak" operatorvalued integrals in terms of easily checkable conditions and prove a highly useful Minkowski-type integral inequality for maps with values in a semifinite von Neumann algebra.

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Let M be a von Neumann algebra and a be a self-adjoint operator affiliated with M. We define the notion of an "integral symmetrically normed ideal" of M and introduce a space OC [k] (R) ⊆ C k (R) of functions R → C such that the following result holds: for any integral symmetrically normed ideal I of M and any f ∈ OC [k] (R), the operator function Isa ∋ b → f (a + b) − f (a) ∈ I is k-times continuously Fréchet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if f ∈ Ḃ1,∞. Finally, we prove that all of the following ideals are integral symmetrically normed: M itself, separable symmetrically normed ideals, Schatten p-ideals, the ideal of compact operators, andwhen M is semifinite -ideals induced by fully symmetric spaces of measurable operators.

In a recent paper, the author introduced a rich class N C k (R) of "noncommutative C k " functions R → C whose operator functional calculus is k-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In this paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving a "functional free Itō formula" for noncommutative C 2 functions of self-adjoint free Itō processes with respect to circular Brownian motion. To do this, we extend P. Biane and R. Speicher's theory of free stochastic calculus to allow free Itō processes involving circular Brownian motion and its adjoint -instead of only semicircular Brownian motion -and then reinterpret the quantities in the resultant Itō-type formulas as MOIs. As motivation, we also prove a "functional Itō formula" for C 2 functions of Itō processes with respect to n × n matrix Brownian motion.

In a recent paper, the author introduced a rich class N C k (R) of "noncommutative C k " functions R → C whose operator functional calculus is k-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative C 2 functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher's theory of free stochastic calculus -including their free Itô formula for polynomials -to allow free Itô processes driven by multiple freely independent semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space N C 2 (R). Along the way, we also obtain a useful "traced" Itô formula for arbitrary C 2 scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for C 2 scalar functions of N × N Hermitian matrix Itô processes.