Definition 7

Let A be a normed algebra. An approximate unit of A is a filter basis $\mathfrak{F}$ on the unit ball of A such that, for every $x$ in A, the filter bases $x\mathfrak{F}$ and $\mathfrak{F}x$ on A converge to $x$, in other words:

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

If A is a *-algebra, an approximate unit $\mathfrak{F}$ is said to be increasing if $\mathfrak{F}$ is a filter basis on $A_+$.

Let A be a *-algebra. We denote by $A^{\leqslant 1}_+$ (resp. $A^{<1}_+$) the set of positive elements of A of norm $\leqslant 1$ (resp. of norm $<1$); these are the Hermitian elements of A whose spectrum is contained in $[0,1]$ (resp. in $[0,1[$).