Problem
Given a C*-algebra with unit $1\in\mathcal{A}$.
Define positive elements as: $$A\geq0\iff\sigma(A)\geq0\quad(A=A^*)$$
Positive elements can be characterized by: $$A\geq0\iff A=B^*B$$
Attempts
One direction easily follows from the continuous calculus: $$A\geq0\implies A=\sqrt{A}\sqrt{A}$$
For operator algebras the numerical range becomes accessible: $$A=B^*B\implies\mathcal{W}(A)\geq0\implies\sigma(A)\geq0$$
For general C*-algebras one has a faithful representation : $$\pi:\mathcal{A}\to\mathcal{B}(\mathcal{H}):\quad\ker\pi=(0)$$
But how to show the above without exploiting advanced tools like Gelfand Naimark?