n}
(S)$ the set of all matrices with $m$ rows and $n$ columns having entries from $S$. For simplicity, we let $M_n(S):=M_{n\times n}(S)$. Given C${}^*$-algebra $\mathcal A$, a [*noncommutative set*]{} is a family $\Omega:=(\Omega_n)_{n\in\mathbb N}$ such that

1.  for each $n\in\mathbb N$, $\Omega_n\subseteq M_n
    (\mathcal A);$

2.  for each $m,n\in\mathbb N$, we have $\Omega_m\oplus
    \Omega_n\subseteq\Omega_{m+n}$.

The noncommutative set $\Omega$ is called [*right admissible*]{} if in addition the condition (c) below is satisfied:

1.  for each $m,n\in\mathbb N$ and $a\in\Omega_m,c\in
    \Omega_n,w\in M_{m\times n}(\mathcal A)$, there is an $\epsilon>0$ such that $\left(\begin{array}{cc}
    a & zw\\
    0 & c\end{array}\right)\in\Omega_{m+n}$ for all $z\in\mathbb C,
    |z|<\epsilon$.

Given C${}^*$-algebras $\mathcal{A,C}$ and a noncommutative set $\Omega\subseteq\coprod_{n=1}^\infty M_n(\mathcal A)$, a [ *noncommutative function*]{} is a family $f:=(f_n)_{n\in\mathbb N}$ such that $f_n\colon\Omega_n\to M_n(\mathcal C)$ and

1.  $f_m(a)\oplus f_n(c)=f_{m+n}(a\oplus c)$ for all $m,n\in\mathbb N$, $a\in\Omega_m,c\in\Omega_n$;

2.  for all $n\in\mathbb N$, $f_n(T^{-1}aT)=T^{-1}f_n(a)T$ whenever $a\in\Omega_n$ and $T\in GL_n(\mathbb C)$ are such that $T^{-1}aT$ belongs to the domain of definition of $f_n$.

These two conditions are equivalent to the single condition

1.  For any $m,n\in\mathbb N$, $a\in\Omega_m,c\in\Omega_n$, $S\in M_{m\times n}(\mathbb C)$, one has $$aS=Sc\implies f_m(a)S=Sf_n(c).$$

We shall refer to the indices $n$ of $\Omega_n$ or of $f_n$ as the “levels” of the noncommutative 