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**Mauritzvdworm** 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?