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

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. 
