Let $\mathcal K$ denote the C-algebra of compact operators, and fix an embedding $\phi_n:M_n(\mathbb C) \to \mathcal K$ for each $n\in \mathbb N$. Define the C-algebra

$A := \{(a_n)_{n=1}^\infty \in \prod_n M_n(\mathbb C): \lim_n \phi_n(a_n)$ exists in $\mathcal K\}$.

Does A embed into a simple unital nuclear C*-algebra?

The answer is almost yes:

• it embeds into a non-unital simple AF algebra, namely $\mathcal Q \otimes \mathcal K$ where $\mathcal Q$ is the universal UHF algebra;
• it embeds into a simple unital non-nuclear C*-algebra, namely the hyperfinite II$_1$ factor;
• it embeds into an ultrapower of a UHF algebra, i.e., it almost embeds into a UHF algebra).

But my feeling is that the answer should be no. Can we even show that A doesn't embed into a simple unital AF algebra?