Prove in a C* algebra that a*a is positive

Question

I am trying to find a reference to the following "obvious facts" (not sure if they are true or not, but should have some comparable similar results) regarding a non-commutative $C^\ast$ algebra $A$.

1. For $a\in A,$ let $\Phi_A$ be the set of all multiplicative linear functionals $A \to \mathbb C.$ Then the spectrum $\sigma_A(a) = \{\varphi(a): \varphi\in \Phi_A\}.$
2. $a^\ast a$ is positive in the sense that its spectrum is a subset of nonnegative real numbers.
3. If $a$ is positive and invertible, and $b$ is positive, then $a+b$ is positive and invertible.

Is there a place where I can find the proof of these results?

Answer 1

1. Is false in general. Take $A=M_2(\mathbb C)$, then the only multiplicative linear functional is zero. The assertion is true for unital commutative C$^*$-algebras, where you take $\Phi_A$ to be the nonzero multiplicative linear functionals. As you wrote it, the assertion is true for non-unital commutative C$^*$-algebras.

2. The proof of "$a^*a$ is positive" that I know is a bit technical. The problem one has is that the statement needs to be proven early in the theory, before functional calculus and representations are available.

3. If $a$ is positive and invertible, then $\sigma(a)\subset[c,\infty)$ for some $c>0$. It follows that $a\geq cI$, since $a-cI$ is positive. Then $a+b\geq a\geq cI$, so $a+b-cI\geq0$, which implies that $\sigma(a+b)\subset[c,\infty)$ and so $a+b$ is invertible.

As for books, this stuff will be found in all C$^*$-algebra books (Murphy, Davidson, Kadison-Ringrose, even Conway's Functional Analysis, to mention a few; as well as older ones like Sakai and Dixmier), though the facts might be distributed differently. Depending among other things on how positivity is defined.