Equivalent definitions of 'strictly positive' in C*-algebra

Question

The answer in this post shows that the definition of Takesaki implies the definition of Murphy. However, I don't see why the converse is true. Any suggestions are greatly appreciated!

Answer 1

If $aAa$ is dense and $\tau(a)=0$, then for any $b\in A_+$ you have$$0\leq \tau(aba)\leq\|b\|\,\tau(a^2)\leq\|b\|\,\|a\|\,\tau(a)=0.$$ As positive elements span $A$, you get $aAa\subset \ker\tau$. By density and continuity, $\tau=0$, a contradiction. So $\tau(a)>0$ for each state $\tau$.