Does this C*-algebra embed into a simple nuclear C*-algebra?

Question

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.

Answer 1

There is an exact sequence $$ 0 \to \oplus_n M_n(\mathbb C) \to A \to \mathcal K \to 0.$$ Thus, $A$ is nuclear as an extension of nuclear $C^*$-algebras, see vor example $IV.3.1.3$ in [Bruce Blackadar, Operator Algebras: Theory of $C^*$-Algebras and von Neumann Algebras].

Now, any nuclear embeds into $O_2$ by Kirchberg’s Embedding Theorem (which even applies to exact algebras), see [Eberhard Kirchberg, N. Christopher Phillips, Embedding of exact $C^∗$-algebras in the Cuntz algebra $O_2$, Journal für die reine und angewandte Mathematik 525 (2000), 17– 53.].

The algebra $O_2$ is well-known to unital, simple, and nuclear, see [Joachim Cuntz, Simple $C^*$-algebras generated by isometries. Comm. Math. Phys. 57 (1977), no. 2, 173–185.].

Answer 2

Your algebra is Type I and residually finite dimensional (RFD) Therefore it's unitization is also Type I and RFD. Let's call $B$ it's unitization. Since $B$ is Type I it satisfies the UCT and therefore embeds into a unital simple AF algebra by Huaxin Lin's 2000 paper in Proc. AMS (RFD algebras and AF embeddings).

You may also want to check out Marius Dadarlat's 2000 paper "Nonnuclear subalgebras of AF algebras." His restricted Bratelli diagrams give a very nice straightforward way to embed any unital RFD nuclear C*-algebra into a unital simple nuclear monotracial C*-algebra.