a careful reading of the proofs of [@ATY Propositions 3.4 and 3.5] allows one to observe that they adapt without modification to show that $\Im M(a_1,\dots,
a_n)>0$ for $(a_1,\dots,a_n)\in\{a\in M_k(\mathbb C)\colon \Im a>0\}^n$.) The extension of $h$ becomes $$h_k(a)=s\otimes1_k+(\varphi_v\otimes{\rm Id}_k)(M(a)),$$ for all $ a=(a_1,\dots,a_n)\in
\{a\in M_k(\mathbb C)\colon \Im a>0\}^n.$ For $n=2$ [*any*]{} analytic function $h\colon\mathbb C^+\times\mathbb C^+\to\mathbb C^+$ admits such an operatorial realization, and hence it has a noncommutative extension as described above (see [@AMcY; @ATY; @ATY2]). Considering the counterexample $h$ provided in [@AMcY], the map $H\colon H^+(\mathbb C^2)\to H^+(\mathbb C^2)$ defined by $H(a)=(h(a),h(a))$ shows that we cannot dispense of item (1’) in Theorem \[Main\]. However, observe that the noncommutative structure of the function $f$ in Theorem \[Main\] (1) allows for a slightly stronger conclusion than in classical case of [@AMcY; @Abate]: the “directional derivative” becomes a bounded linear operator defined on all of $\mathcal A$.

As noted above, a classical analytic function is itself the first level of a noncommutative function, via the classical analytic functional calculus applied to matrices over $\mathbb C$. Relations , , are the obvious consequences of relations and in this context. Thus the statements of Theorem \[Main\] are anything but surprising. Indeed, if $f$ 