Element $a$ of a C*-algebra such that $f(a^*a) = 0$ but $f(aa^*) \neq 0$ for some positive linear functional $f$

Question

Could somebody please give me an example, assuming one exists, of an element $a$ of a C*-algebra $A$ such that $f(a^*a) = 0$ but $f(aa^*) \neq 0$ for some positive linear functional $f:A \to \mathbb{C}$? I tried to prove there was no example, but that didn't seem to work out so now I'm guessing it's false. Thanks in advance.

Answer 1

Let $e_0=(1,0,0,0,0,\ldots)\in\ell^2$, let $f:B(\ell^2)\to\mathbb C$ be defined by $f(T)=\langle Te_0,e_0\rangle$, and let $S\in B(\ell^2)$ be the "backward shift" defined by $S(a_0,a_1,a_2,a_3,\ldots)=(a_1,a_2,a_3,a_4,\ldots)$. Then $f(SS^*)=1$ but $f(S^*S)=0$.

Similar but simpler: $A=M_2(\mathbb C)$, $f(T)=\left\langle T\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}1\\0\end{bmatrix} \right\rangle$, $a=\begin{bmatrix}0&1\\0&0\end{bmatrix}$.