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We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g.,$B({\mathcal H})$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the$k{\mathrm {th}}$directional derivative of any NC function at a scalar point is ak-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function$f$in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type$f=\varphi g$, where$g$is cyclic,$\varphi$is a contractive multiplier, and$\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.