Proposition 17

Let A be a *-algebra, and let $\mathfrak{F}$ be a filter basis on $A^{\leqslant 1}_+$. For $\mathfrak{F}$ to be an increasing approximate unit of A, it is necessary and sufficient that one have

lim$_{f,\mathfrak{F}}f x=x$

for every positive element $x$ of A.

The condition is obviously necessary; let us prove that it is sufficient. Let $\widetilde{A}$ be the *-algebra deduced from A by adjunction of a unit element. Let $x$ be an element of A and let $f\in A^{\leqslant 1}_+$. We have

because $\|f-1\|\leqslant 1$ (Lemma 12, c) of I, p. 116). We therefore have

lim sup$_{f,\mathfrak{F}}\|f x-x\|^2\leqslant$ lim sup$_{f,\mathfrak{F}}\|f xx^*-xx^*\|$. Since $xx^*$ is positive (Theorem 2 of I, p. 118), the assumption implies that

lim sup$_{f,\mathfrak{F}}\|f x-x\|^2\leqslant \|xx^*-xx^*\|= 0$,

so that

lim$_{f,\mathfrak{F}}f x=x$.

Since the involution of A is continuous and since $\mathfrak{F}$ is a filter basis on $A_h$, we have

lim$_{f,\mathfrak{F}}xf=$ lim$_{f,\mathfrak{F}}(f^*x^*)^*=$ lim$_{f,\mathfrak{F}}(f x^*)^*= (x^*)^*=x$.

The proposition follows.