items (1), (2) above (see [@ncfound Sections 2.3–2.5] for details). For convenience, from now on we shall suppress the indices denoting the levels for noncommutative functions, as it will almost always be obvious from the context.

We provide three examples of noncommutative sets:

1.  The noncommutative upper half-plane $H^+(\mathcal A)=(H^+(M_n(\mathcal A)))_{n\in\mathbb N}$, where $H^+(M_n(\mathcal A))=\{b\in M_n(\mathcal A)\colon\Im b>0\}$ (here $\Im b=\frac{b-b^*}{2i},\Re b=\frac{b+b^*}{2}$),

2.  The set of nilpotent matrices with entries from $\mathcal A
    $, and

3.  The unit ball $(B(M_n(\mathcal A)))_{n\in\mathbb N}$, where $B(M_n(\mathcal A))=\{b\in M_n(\mathcal A)\colon\|b\|<1\}$.

Our paper will focus on the first example.

As the domains we consider in this paper are mostly noncommutative subsets of von Neumann algebras given by an order relation, we give below a few of the well-known results we use systematically in the rest of the paper. For them, we refer to [@Bruce; @Paulsen; @SZ]. First, recall that for any C${}^*$-algebra (hence, in particular, von Neumann algebra) $\mathcal A$, if $x\in\mathcal A$, then $\|x\|^2=\|x^*\|^2=\|x^*x\|=\|xx^*\|$. For a selfadjoint element $x=x^*\in\mathcal A$, $\|x\|$ is equal to the spectral radius of $x$. We say that $x\ge0$ in $\mathcal A$ if $x=x^*$ and the spectrum of $x$ is included in $[0,+\infty)$. Equivalently, if $\mathcal H$ is the 