Positive Elements: Decomposition?

Question

Problem

Given a C*-algebra with unit $1\in\mathcal{A}$.

Then every selfadjoint element decomposes into positives ones: $$A=A^*:\quad A=\frac12(|A|-A)-\frac12(|A|-A)\quad\left(\frac12(|A|\pm A)\geq0\right)$$

However, one also has the special dcomposition: $$A=A^*:\quad A=\frac12(\|A\|1+A)-\frac12(\|A\|1-A)\quad\left(\frac12(\|A\|1\pm A)\geq0\right)$$

In general, they differ as $|A|\neq\|A\|1$. But the positive decomposition is unique, isn't it?

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Answer 1

The missing ingredient is orthogonality: $A_+A_-=A_-A_+=0$

The former ones are: $$\frac12(|A|\pm A)\frac12(|A|\mp A)=\frac{1}{4}(|A|^2-A^2)=0$$ while the latter ones aren't: $$\frac12(\|A\|1\pm A)\frac12(\|A\|1\pm A)=\frac12(\|A\|^21-A^2)\neq0$$ So only the first ones correspond to the unique decomposition!