Relation between positive elements of a $C^{\star}$ algebra and positive elements of its unitization

Question

Is there any relation between positive elements of a $C^{\star}-$ algebra $A$ and positive elements of unitization of $A$?

I have recently learned about positive elements so can’t see it. Any ideas?

Answer 1

The elements of the unitization are $(a,\lambda)$ with $a\in A$, $\lambda\in\mathbb C$.

One characterization of positive elements is that they are always of the form $c^*c$. So, if $c^*c\in A$ is positive, then $$ (c^*c,0)=(c,0)^*(c,0) $$ is also positive in the unitization.

There are lots of "new" positive elements in the unitization, though. For starters, all $(0,\lambda)$ with $\lambda\geq0$. Positive elements are selfadjoint, so if $(a,\lambda)\geq0$ then $a=a^*$ and $\lambda\in\mathbb R$. You have $$ (a,\lambda)=(a,0)+(0,\lambda)=(a,0)+\lambda I. $$ So $$ \sigma((a,\lambda))=\sigma((a,0)+\lambda I)=\sigma((a,0))+\lambda. $$ Thus $(a,\lambda)$ will be positive precisely when $\sigma(a)\subset[-\lambda,\infty)$.