Positive contractive approximate units for C star algebras and ideals

Put $b:={\sum }_{a\in E}(a^{\ast }a+a{a}^{\ast })\in A$. Then $b\ge 0$ by [F1], and the continuous function $f_{\delta }(t):=t(t+\delta {)}^{-1}$ on $[0,\mathrm{\infty })$ satisfies $f_{\delta }(0)=0$ and $0\le f_{\delta }\le 1$. Set $u:=f_{\delta }(b)\in A$ (the vanishing-at-$0$ clause of [F1]); then $0\le u\le 1$, $\mathrm{\parallel }u\mathrm{\parallel }\le 1$, and with $r:=1-u=f_{\delta }^{\mathrm{\prime }}(b)$ for $f_{\delta }^{\mathrm{\prime }}(t):=\delta (t+\delta {)}^{-1}$ one has $\mathrm{\parallel }r\mathrm{\parallel }\le 1$ and, since $rbr$ has calculus transform ${\delta }^{2}t(t+\delta {)}^{-2}$ on $\sigma (b)$, $\mathrm{\parallel }rbr\mathrm{\parallel }={\sup }_{t\in \sigma (b)}{\delta }^{2}t(t+\delta {)}^{-2}\le {\sup }_{t\ge 0}{\delta }^{2}t(t+\delta {)}^{-2}=\delta \mathrm{/}4$ (the supremum being attained at $t=\delta$).

For every $a\in E$: $a{a}^{\ast }\le b$ and $a^{\ast }a\le b$, because $b$ is their sum together with the positive summands attached to the remaining elements of $E$; hence $r(a^{\ast }a)r\le rbr$ and $r(a{a}^{\ast })r\le rbr$ by conjugation positivity [F1]. Using the C*-identity, $\mathrm{\parallel }ar{\mathrm{\parallel }}^2=\mathrm{\parallel }r(a^{\ast }a)r\mathrm{\parallel }\le \mathrm{\parallel }rbr\mathrm{\parallel }\le \delta \mathrm{/}4$, and $\mathrm{\parallel }ra\mathrm{\parallel }=\mathrm{\parallel }{a}^{\ast }r\mathrm{\parallel }$ has square $\mathrm{\parallel }r(a{a}^{\ast })r\mathrm{\parallel }\le \mathrm{\parallel }rbr\mathrm{\parallel }\le \delta \mathrm{/}4$; in particular $\mathrm{\parallel }a-au\mathrm{\parallel }=\mathrm{\parallel }ar\mathrm{\parallel }\le \sqrt{\delta }\mathrm{/}2$ and $\mathrm{\parallel }a-ua\mathrm{\parallel }=\mathrm{\parallel }ra\mathrm{\parallel }\le \sqrt{\delta }\mathrm{/}2$.

Index the pairs $(E,n)$, $E\subseteq A$ finite, $n\ge 1$, by $(E,n)\le (E^{\mathrm{\prime }},n^{\mathrm{\prime }})$ iff $E\subseteq E^{\mathrm{\prime }}$ and $n\le n^{\mathrm{\prime }}$, a directed set, and put $u_{E,n}:=f_{1\mathrm{/}{n}^2}(b_E)$ with $b_E:={\sum }_{a\in E}(a^{\ast }a+a{a}^{\ast })$. By step 1.1 each $u_{E,n}$ is a positive contraction in $A$. For fixed $a\in A$, once $a\in E$ step 2.1 gives $\mathrm{\parallel }a-a{u}_{E,n}\mathrm{\parallel }\le 1\mathrm{/}(2n)$ and $\mathrm{\parallel }a-u_{E,n}a\mathrm{\parallel }\le 1\mathrm{/}(2n)$, so both tend to $0$ along the directed set; hence $(u_{E,n})$ is a two-sided approximate unit of positive contractions. If $A=\{0\}$ the constant zero net serves.

Let $J$ be a closed two-sided ideal of $A$. Then $J^{\ast }:=\{a^{\ast }:a\in J\}$ is a closed two-sided ideal as well, and $D:=J\cap J^{\ast }$ is a closed $\ast$-subalgebra, hence a C*-algebra by [F2]; let $(u_{\lambda })$ be a two-sided approximate unit of $D$ of positive contractions, by step 3.1 applied to $D$. For $a\in J$ one has $a^{\ast }a\in J$ and $a^{\ast }a=(a^{\ast }a{)}^{\ast }\in J^{\ast }$, so $a^{\ast }a\in D$, and $\mathrm{\parallel }a(1-u_{\lambda }){\mathrm{\parallel }}^2=\mathrm{\parallel }(1-u_{\lambda })a^{\ast }a(1-u_{\lambda })\mathrm{\parallel }\le \mathrm{\parallel }{a}^{\ast }a(1-u_{\lambda })\mathrm{\parallel }\to 0$ by the approximate-unit property and $\mathrm{\parallel }1-u_{\lambda }\mathrm{\parallel }\le 1$; taking adjoints gives $\mathrm{\parallel }(1-u_{\lambda })a^{\ast }\mathrm{\parallel }=\mathrm{\parallel }a(1-u_{\lambda })\mathrm{\parallel }\to 0$, so $a^{\ast }={\lim }_{\lambda }{u}_{\lambda }{a}^{\ast }$. Every $u_{\lambda }{a}^{\ast }$ lies in $J$, because $u_{\lambda }\in J$ and $J$ is a right ideal; since $J$ is closed, $a^{\ast }\in J$. Thus $J$ is self-adjoint, and applying step 3.1 to the C*-algebra $J$ produces its two-sided approximate unit of positive contractions inside $J$.

The Axiom of Choice is inherited from the positivity/order calculus and unitization suppliers of [F1]; the construction of the nets uses no further choice (The Axiom of Choice).