  Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of
those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\|
\sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise,
given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of
tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum
B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron
$\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild
hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$
normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$
and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant
algebraic representation.
