I am confused about one part in a proof on page 78 of Murphy's book. Let $A$ be a C$^{*}$-algebra and let $\Lambda=\{a\in A:a\geq 0\text{ and }\|a\|<1\}$. While proving that $A$ is upwards directed, Murphy states the following:
Observe that if $a\in A^{+}$, then $a(1_{\widetilde{A}}+a)^{-1}$ belongs to $\Lambda$ (use the Gelfand representation applied to the C$^{*}$-algebra generated by $1_{\widetilde{A}}$ and $a$).
I understand the hint: using the spectral mapping theorem, I can prove that $\sigma_{\widetilde{A}}(a(1_{\widetilde{A}}+a)^{-1})\subset[0,1]$, but this shows that $a(1_{\widetilde{A}}+a)^{-1}$ is positive in $\widetilde{A}$. I do not know how to prove that $a(1_{\widetilde{A}}+a)^{-1}\in A^{+}$. In fact, this seems false since $a(1_{\widetilde{A}}+a)^{-1}$ may not even be in $A$. I know in general that $A^{+}=\widetilde{A}^{+}\cap A$, but I am unsure how or if this is being used here. I would really appreciate some clarification regarding why Murphy is able to pass to the unitisation here.
Thank you!
When you consider the unitization, $A$ sits inside it as an ideal. Also, since $a$ and $(1_{\tilde A}+a)^{-1}$ commute, you have $$ a(1_{\tilde A}+a)^{-1}=a^{1/2}(1_{\tilde A}+a)^{-1}a^{1/2}\geq0. $$ So $a(1_{\tilde A}+a)^{-1}\in A_+$.