operator on $\mathcal A$. Then $f'(\alpha)$ is completely positive, $\displaystyle\lim_{\stackrel{z
    \longrightarrow0}{{\sphericalangle}}}f(\alpha+zv):=f(\alpha)$ exists, does not depend on $v$ and is selfadjoint, and $$f'(\alpha)(v)={\rm so\text{-}}
    \lim_{\stackrel{ z\longrightarrow0}{{
    \sphericalangle}}}\frac{\Im f(\alpha+iyv)}{y}\quad\text{for any }
    v>0.$$

Unfortunately, unlike in the classical case of Theorem \[JC\], and similar to the case of functions of several complex variables [@Rudin; @Abate], item (1’) above cannot be improved upon. Indeed, it was observed in [@AMcY] that for analytic functions of two complex variables on the bidisk with values in the unit disk, there exist examples that satisfy the commutative equivalent of for the bidisk, and yet the gradient map does not have a nontangential limit. The equivalent of condition implies the existence of all [*directional*]{} derivatives in permissible directions, but these directional derivatives do not necessarily “add up” to a linear map. This commutative example has a natural noncommutative extension, as shown in [@PTD]. It is enough for our purposes to treat a simplified version of this extension. It is shown in [@ATY] that any Loewner map from the $n$-dimensional upper half-plane $(\mathbb C^+)^n$ to $\mathbb C^+$ has a certain operatorial realization: for any such $h\colon(\mathbb C^+)^n\to\mathbb C^+$ there exist Hilbert spaces $\mathcal N,\mathcal M$, a selfadjoint densely defined operator 