Is it possible to construct a C*-algebra \(\displaystyle \mathcal{A}\) such that there exists \(\displaystyle x\in\mathcal{A}\) which cannot be decomposed into \(\displaystyle x=x_1x_2\) with \(\displaystyle x_1\geq0\) and \(\displaystyle x_2\) a partial isometry?
It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some \(\displaystyle B(H)\) and any operator \(\displaystyle T\in B(H)\) can be decomposed using the polar decomposition, or am I missing something?
It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some \(\displaystyle B(H)\) and any operator \(\displaystyle T\in B(H)\) can be decomposed using the polar decomposition, or am I missing something?