The equation $ab+ba=0, a=a^*, b=b^*$ in a $C^*$ algebra

Question

Does the equation $ab+ba=0$, $a=a^*, b=b^*$ in a $C^*$ algebra implies that ab=0? What is the universal unital $C^*$ algebra subject to the above relations with boundedness condition $|a|=|b|=1$

Answer 1

No, it doesn't. For instance take $$ a=\begin{bmatrix} -1&0\\0&1\end{bmatrix},\qquad\qquad b=\begin{bmatrix} 0&1\\1&0\end{bmatrix}. $$ Then $a=a^*$, $b=b^*$, $\|a\|=\|b\|=1$, and $$ ab+ba=\begin{bmatrix} 0&-1\\1&0\end{bmatrix}+\begin{bmatrix} 0&1\\-1&0\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}. $$ I don't know about the universal algebra they generate. But $C^*(a,b)=M_2(\mathbb C)$. And we can replace $1$ and $-1$ with $t$, and you get, for any compact space $X$, $C^*(a(t),b(t))=M_2(C(X))$. But it doesn't look hard to find bigger matrices satisfying this relations, so the universal algebra looks like it should be huge.