Every commutative C star algebra has an approximate unit

In $C_0(X)$ the family $\mathrm{\Lambda }:=\{e\in C_c(X):0\le e\le 1\}$ is nonempty (it contains $0$) and directed by the pointwise order: for $e_1,e_2\in \mathrm{\Lambda }$ the pointwise maximum $e_1\vee e_2$ lies in $C_c(X)$ (the maximum is continuous by $\max (u,v)=(u+v+\mathrm{\mid }u-v\mathrm{\mid })\mathrm{/}2$, and its closed support is contained in the finite union of the compact supports) and satisfies $0\le e_1\vee e_2\le 1$ and $e_j\le e_1\vee e_2$.

Each $e\in \mathrm{\Lambda }$ is a positive contraction of $C_0(X)$: $\mathrm{\parallel }e{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$, and $e=(\sqrt{e}{)}^2=(\sqrt{e}{)}^{\ast }(\sqrt{e})$ with $\sqrt{e}\in C_c(X)$ real, so $e$ is positive; the square root is continuous on $[0,1]$ and has the same nonzero set and support as $e$, so it belongs to $C_c(X)\subset C_0(X)$; after transporting, ${\mathrm{\Gamma }}^{-1}(e)={\mathrm{\Gamma }}^{-1}(\sqrt{e}{)}^{\ast }{\mathrm{\Gamma }}^{-1}(\sqrt{e})$ and is self-adjoint.

For $f\in C_0(X)$ and $ϵ>0$ put $K:=\{\mathrm{\mid }f\mathrm{\mid }\ge ϵ\mathrm{/}2\}$, a compact subset of $X$; by [F3] with $U:=X$ there is $e\in C_c(X)$ with ${\mathbf{1}}_K\le e\le {\mathbf{1}}_X=1$, so $e\in \mathrm{\Lambda }$; for every $e^{\mathrm{\prime }}\in \mathrm{\Lambda }$ with $e^{\mathrm{\prime }}\ge e$ one has $e^{\mathrm{\prime }}=1$ on $K$ and hence $\mathrm{\mid }f-e^{\mathrm{\prime }}f\mathrm{\mid }=0$ on $K$, while off $K$ one has $\mathrm{\mid }f\mathrm{\mid }<ϵ\mathrm{/}2$ and $\mathrm{\mid }1-e^{\mathrm{\prime }}\mathrm{\mid }\le 1$, so $\mathrm{\mid }f-e^{\mathrm{\prime }}f\mathrm{\mid }<ϵ\mathrm{/}2$; thus $\mathrm{\parallel }f-e^{\mathrm{\prime }}f{\mathrm{\parallel }}_{\mathrm{\infty }}\le ϵ\mathrm{/}2<ϵ$ for all $e^{\mathrm{\prime }}\ge e$.

Consequently the net $(e{)}_{e\in \mathrm{\Lambda }}$ indexed by the directed set $\mathrm{\Lambda }$ satisfies $ef\to f$ for every $f\in C_0(X)$, and since $C_0(X)$ is commutative also $fe\to f$; by [step 1.2] the $e$'s are positive contractions, so this is an approximate unit of $C_0(X)$.

Transporting along the isometric $\ast$-isomorphism ${\mathrm{\Gamma }}^{-1}:C_0(X)\to A$ of [F1], the net $({\mathrm{\Gamma }}^{-1}(e){)}_{e\in \mathrm{\Lambda }}$ is an approximate unit of $A$: the explicit factorization in step 1.2 proves positivity and self-adjointness, norms are preserved, and $\mathrm{\parallel }{\mathrm{\Gamma }}^{-1}(e){\mathrm{\Gamma }}^{-1}(f)-{\mathrm{\Gamma }}^{-1}(f)\mathrm{\parallel }=\mathrm{\parallel }ef-f{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$. For the zero algebra the constant net $0$ is an approximate unit by [F2].

Hence every commutative C*-algebra has an approximate unit of positive contractions, and for $A=C_0(X)$ the explicit net of [step 3.1] realises it.