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It seems to be trivial but I am not sure about monotonicity of the norm in the non-commutative case:

Is every C*-algebra a Banach lattice with respect to its natural positive cone?

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    If you are asking whether $0\leq a\leq b$ implies $\|a\|\leq \|b\|$, the answer is that that is always true. I could elaborate in an answer, but I'm not sure if that's what you're looking for. Could you please say precisely and explicitly what it is you want to prove about a C*-algebra?2011-11-19

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