$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras.

Question

Is it true that $A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$.

I read in Murphy 2.1.11 that this is true if $B$ is unital and $A$ contains the unit of $B$.

Does it make sense to look at unitizations of $A$ and $B$ if they are non-unital ?

Answer 1

You don't need to look at the spectra. You can characterize postive elements as those of the form $z^*z$. So $a\in A^+$. then $a=z^*z$ for some $z\in A\subset B$. So $a$ is positive in $B$ too.

Answer 2

Yes, that makes sense. Murphy does this at the beginning of section 2.2, on pages 44 and 45. In the non-unital case we get $\sigma_B(b) \cup \{0\} = \sigma_A(b) \cup \{0\}$, but this does not mess up positivity.

In order to understand the answer Martin Argerami gives, see Murphy theorem 2.2.5(1).