$A$ on $\mathcal M$, a real number $s$ an orthogonal decomposition $P=
\{P_1,\dots,P_n\}$ of $\mathcal N\oplus\mathcal M$ (i.e. $P_iP_j=P_jP_i=\delta_{ij}P_j=\delta_{ij}P_j^*$ and $P_1+\cdots+P_n=1_{\mathcal M\oplus\mathcal N}$) and a vector state $\varphi_v(\cdot)=\langle\cdot v,v\rangle$ on the von Neumann algebra of bounded linear operators on $\mathcal N\oplus\mathcal M$ such that $$h(z)=s+\varphi_v(M(z)), \quad z=(z_1,\dots,z_n)\in(\mathbb C^+)^n,1\leq j\leq n,$$ where $$\begin{aligned}
M(z) & = & \left(\begin{array}{cc}
-i & 0 \\
0 & 1-iA
\end{array}\right)\left(\left(\begin{array}{cc}
1 & 0 \\
0 & A
\end{array}\right)-(z_1P_1+\cdots+z_nP_n)
\left(\begin{array}{cc}
0 & 0 \\
0 & 1
\end{array}\right)\right)^{-1}\\
& & \mbox{}\times\left((z_1P_1+\cdots+z_nP_n)
\left(\begin{array}{cc}
1 & 0 \\
0 & A
\end{array}\right)+
\left(\begin{array}{cc}
0 & 0 \\
0 & 1
\end{array}\right)\right)
\left(\begin{array}{cc}
-i & 0 \\
0 & 1-iA
\end{array}\right)^{-1}.\end{aligned}$$ The $2\times2$ matrix decomposition is realized with respect to the canonical orthogonal decomposition of $\mathcal N\oplus\mathcal M$. We observe that such maps $M\colon(\mathbb C^+)^n\to B(\mathcal N\oplus\mathcal M)$ have a natural noncommutative extension to $H^+(\mathbb C^n):=
\coprod_{k\ge1}\{a\in M_k(\mathbb C)\colon \Im a>0\}^n$ given by replacing $(z_1P_1+\cdots+z_nP_n)$ in the above formula of $M(z)$ by $$\sum_{j=1}^n
(P_j\otimes1_k) a_j 
(P_j\otimes 1_k).$$ (While it is not obvious from its formula that $\Im M$ is positive when evaluated on $(\mathbb C^+)^n$, and even less when its amplification is evaluated on $\{a\in M_k(\mathbb C)\colon \Im a>0\}^n$, 