Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties:
$\log(L)$ is plurisubharmonic.
$L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda A_r)=|\lambda|\cdot L(A_1,\dots,A_r)$ for $\lambda\in\mathbb{C},A_1,\dots,A_r\in M_n(\mathbb{C})$.
$L(A_1,\dots,A_r)=L(CA_1C^{-1},\dots,CA_rC^{-1})$ whenever $C\in\text{GL}_n(\mathbb{C}),A_1,\dots,A_r\in M_n(\mathbb{C})$.
There are functions $L_1,\dots,L_{n-1}$ where $L_j:M_j(\mathbb{C})^r\rightarrow[0,\infty)$ for $1\leq j\leq n-1$ where if $n=s+t$, then
$$L(\begin{bmatrix} A_1 & B_1 \\ 0 & C_1 \end{bmatrix},\dots,\begin{bmatrix}A_r & B_r \\ 0 & C_r\end{bmatrix})=\max(L_s(A_1,\dots,A_r),L_t(C_1,\dots,C_r))$$ whenever $A_1,\dots,A_r\in M_s(\mathbb{C}),C_1,\dots,C_r\in M_t(\mathbb{C}).$
Then does there exist a Banach algebra $\mathcal{A}$ along with $a_1,\dots,a_r\in\mathcal{A}$ where $L(A_1,\dots,A_r)=\rho(a_1\otimes A_1+\dots+a_r\otimes A_r)$ for all $A_1,\dots,A_r\in M_n(\mathbb{C})$?
If $\mathcal{A}$ is a Banach algebra and $a_1,\dots,a_r\in\mathcal{A}$ and $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ is the function defined by $L(A_1,\dots,A_r)=\rho(a_1\otimes A_1+\dots+a_r\otimes A_r)$, then $L$ satisfies properties 1-4, so I want to know if the converse holds.
