Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$.
(Both possibly nonunital!)
Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$
Implication: $$\varphi\geq0\implies\|\varphi\|<\infty$$ (Reduction?)
Reference: Boundedness
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If $\|\varphi(A)\|\leq c\,\|A\|$ when $A$ is selfadjoint, then for arbitrary $A$ you have $$ \|\varphi(A)\|=\|\varphi(\text{Re}\,A)+i\varphi(\text{Im}\,A)\| \leq\|\varphi(\text{Re}\,A)\|+\|\varphi(\text{Im}\,A)\|\\ \leq c\,(\|\text{Re}\,A\|+\|\text{Im}\,A)\|\leq 2c\|A\|, $$ since $$ \|\text{Re}A\|=\frac12\,\|A+A^*\|\leq\frac12\,(\|A\|+\|A^*\|)=\|A\|. $$
Martin Argerami
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Ah ok nice!! :) And what about reduction from selfadjoint to positive? – freishahiri Apr 02 '16 at 09:13
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Every selfadjoint is a difference of two positives. – Martin Argerami Apr 02 '16 at 09:31
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Ah ok and we know that: $A_\pm:=\frac12{|A|\pm A}\in\mathcal{A}$ – freishahiri Apr 02 '16 at 10:55
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Yes indeed. $ $ – Martin Argerami Apr 02 '16 at 11:54