Can a finite dimension operator space faithfully represented in a finite dimensional operator algebra?

Question

Suppose that a operator space (system) $E$, of finite dimension, is included in a $C^\ast$-algebra. Can we find another finite dimensional Hilbert space $H$ such that $E\subset B(H)$?

This hold in the case $E$ itself is a $C^\ast$-algebra, but I am not sure it still hold in the general case.

Thanks for all helps!

Answer 1

No it doesn't. Consider $S\in B(\ell^2(\mathbb N))$ the unilateral shift. Let $$E=\{\alpha I+\beta S:\ \alpha,\beta\in\mathbb C\}.$$ Then $E$ is $2$-dimensional, but its C$^*$-envelope is $C(\mathbb T)$, infinite-dimensional. So $E$ cannot be embedded in any finite-dimensional C$^*$-algebra.