Positive Elements: Norm (Decomposition)

Question

Given a C*-algebra $\mathcal{A}$.

Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$

Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$

But what about the converse: $\|Z_\alpha\|\leq\|Z\|$

By continuous calculus it holds: $$A=A^*:\quad \|A\|=\max\{\|A_+\|,\|A_-\|\}$$

So it remains to observe why: $\|X\|,\|Y\|\leq\|Z\|$

Here, I cannot make use of: $a,b\leq\sqrt{a^2+b^2}\quad(a,b\geq0)$

So how can I proceed then?

Answer 1

Given $Z=X+iY$ with $X$ and $Y$ self-adjoint, note that $X=\frac12(Z+Z^*)$, and $Y=\frac1{2i}(Z-Z^*)$. Take norms, apply the triangle inequality, and recall that $\|Z^*\|=\|Z\|$ to conclude that $\|X\|\leq \|Z\|$ and $\|Y\|\leq \|Z\|$.