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On P145 of Murphy's book C$^*$-Algebras and Operator Theory, when he proves a character $\tau$ of an abelian C* algebra has norm one, he uses that $(u_\lambda^2)$ is also an approximate unit when $(u_\lambda) $ is an approximate unit.

But why? An approximate unit of a C* algebra $A$ is an increasing net of positive elements in the closed unit ball such that $u_\lambda a\to a\forall a \in A$. Note that $0\leq u\leq v$ dose not imply that $u^2\leq v^2$. Why is $(u_\lambda^2)$ increasing?

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In a general $C^*$-algebra it is correct that $0\leq u\leq v$ does not imply $0\leq u^2\leq v^2$. But we are considering an abelian $C^*$-algebra. Then the algebra is (up to $*$-isomorphism) $C_0(X)$ for some space $X$, and the implication $0\leq f\leq g$ implies $0\leq f^2\leq g^2$ holds for $f,g\in C_0(X)$.

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