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?
Edit: I had meant to ask: does it embed into a simple nuclear unital stably finite C*-algebra?
Andreas Thom's example, like $\mathcal Q \otimes \mathcal K$ is nuclear and simple, but has no (bounded) trace.