For which simple unital $C^\*$ algebras does the following functional equation have a solution:


$$ d^2=0,\;{(d+d^\*)}^2=1$$


The Calkin algebra and $M\_{2n}(\mathbb{C})$ are some examples. It is not solvable in $M\_{2n+1}(\mathbb{C})$.
What infinite dimensional simple $C^\*$ algebra does not admit an element $d$ with the above property?



Solvability of a certain functional equation in simple $C^*$ algebras
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


1. $K\_0(\tau) : K\_0(A) \to \mathbb{R}$ is injective;
2. $K\_0(\tau)([1\_A]) = 1$ (since $\tau$ is normalised);
3. $\tfrac{1}{2} \notin K\_0(\tau)(K\_0(A))$.


