Let $A$ be a $C^*$-algebra and consider it's unitization $A_1$ whose underlying vector space is the direct sum $A\oplus \mathbb{C}$. I want to know how does positive elements in $A_1$ look like.
I tried to find out it with the definition (maybe an other criterion is better) but I'm stuck : $a+\lambda 1\in A_1$ positive means, 1. $(a+\lambda 1\in A_1)^*=a+\lambda 1$, and 2. $\sigma(a+\lambda 1)\subseteq [0,\infty)$.
From 1. follows: $a^*=a$ and $\lambda\in\mathbb{R}$. My problem is 2.: Consider $\mu\in \mathbb{C}\setminus \sigma(a+\lambda 1)$. This means $a+\lambda 1-\mu$ is invertible in $A_1$. There exists an element $b+\eta 1\in A_1$ such that $(a+\lambda 1-\mu)(b+\eta 1)=ab+(\lambda-\mu)b+\eta a+(\lambda-\mu)\eta =0+1$. It follows $ab+(\lambda-\mu)b+\eta a=0$ and $(\lambda-\mu)\eta=1$. But it's pointless.
Could you help me to find out, how does positive elements in $A_1$ look like?
Suppose that $a+\lambda 1 \in A_1$ is positive, so that $\sigma(a+\lambda 1) \subseteq [0,\infty)$. Then one can show that $\sigma(a+\lambda 1)=\{ \mu+\lambda: \mu \in \sigma(a) \}$ (when thinking of $\sigma(a)$ as its spectrum in the unitization). Then if $\sigma(a) \subseteq [c,d]$ for some $c,d \in \mathbb{R}$, we have $\sigma(a+\lambda 1) \subseteq [c+\lambda,d+\lambda]$. That is, $a+\lambda 1$ is positive iff $a=a^*$, $\lambda \in \mathbb{R}$ and the function $f(z)=z+\lambda$ is positive (non-negative) on $\sigma(a)$.