Im doing an exercise on C*-algebras where I have two find a C*-algebra $\mathscr{A}$ and two positive (i.e. self-adoint(=hermitian) and spectrum contained in $[0,\infty)$) elements $a,b\in\mathscr{A}$ such that $ab$ is not positive.
Since one can prove that $ab$ is positive when $a$ and $b$ commute, my guess was to find two non-commuting positive matrices $A$, $B$ in the non-abelian C*-algebra $\mathscr{A}:=M_{2}(\mathbb{C})$ such that $AB$ is not positive.
Note that a matrix $A$ is positive iff $A=\overline{A}^{T}$ and the eigenvalues of $A$ lie in $[0,\infty)$.
Since $A=\overline{A}^{T}$ is always satisfied for symmetric matrices with real coefficients, I was hoping two find two such matrices. However, I did not succeed. In particular I tried $$A:=\begin{pmatrix}1&-1\\-1&1\end{pmatrix},$$ which has eigenvalues $0$ and $2$. But I can't find a $B$ that provides a counterexample together with this $A$.