If such a $d$ exists in $A$, then $d^\ast d$ and $d d^\ast$ are Murray–Von Neumann equivalent projections, such that $d^\ast d + d d^\ast = 1_A$. Hence, a necessary condition for $d$ to exist is that $[1_A] \in K_0(A)$ is divisible by $2$. For instance, this rules out $M_{2n+1}(\mathbb{C})$, since $\operatorname{Tr}([1_{M_{2n+1}(\mathbb{C})}]) = 2n+1$ is odd. With regard to your actual question, this also rules out any UHF algebra $A$ such that $\tfrac{1}{2} \notin K_0(A) \leq (\mathbb{Q},+)$ (i.e., the inductive limit of a sequence $M_{k_1}(\mathbb{C}) \to M_{k_2}(\mathbb{C}) \to \cdots$, where each $k_i$ is odd) or any irrational rotation algebra: in either case, you have have a simple unital $C^\ast$-algebra $A$ together with a normalised trace $\tau$, such that

- $K_0(\tau) : K_0(A) \to \mathbb{R}$ is injective;
- $K_0(\tau)([1_A]) = 1$ (since $\tau$ is normalised);
- $\tfrac{1}{2} \notin K_0(\tau)(K_0(A))$.

3more comments