An algebra with no von Neumann unitization

Question

What is an example of a non unital $C^*$ algebra which can not be an essential ideal of any von Neumann algebra?

Answer 1

I believe $C_0(\mathbb R)$ is an example.

If $C_0(\mathbb R)$ is an essential ideal in some unital algebra $A$, then $A$ embedds in the multiplier algebra of $C_0(\mathbb R)$, so $A$ must be commutative.

The spectrum of $A$ will then be a compact space containing an open dense copy of $\mathbb R$. Since the closure of a connected set is connected, we see that the spectrum of $A$ is connected.

This implies that $A$ has no nontrivial projections so it cannot be a von Neumann algebra.