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At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the nonunital case by using unitarization, for example the Gelfand-Naimark-theorems and the continuous functional-calculus-theorems, see for example what I asked here continuous functional calculus for nonunital $c^*$-algebras .
As I wrote here Equivalence of categories ($c^*$ algebras <-> topological spaces), I sometimes try to formulate some results in the language of category theory to have a better overview of the results in the theory of $c^*$-algebras, but I know almost nothing about category theory.
I know, that unitarization of $c^*$-algebras means compactification of topological Hausdorff spaces. As I wrote in question above (Equivalence of categories ($c^*$ algebras <-> topological spaces)), I know that the categories $(c^*com_1)^{op}$ and $KTOP$ are equivalent and I learned there that $(c^*com)^{op}$ and $LKTOP$ are not equivalent.

Now my question is: Nevertheless, is it possible to prove such theorems in the nonunital case, which you know for unital $c^*$-algebras and which you prove with unitarization, in a categorial way? For example, does there exists a proof of the continuos functional calculus theorem for nonunital $c^*$-algebras which is more categorial as the mentioned proof here continuous functional calculus for nonunital $c^*$-algebras ? ( I think if yes, the arguments will be similar but the language will be different).
I found something helpful: "Categories of $C^*$-algebras" by Ivo Dell'Ambrogio Regards

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    $\begingroup$ The category of commutative C$^*$-algebras with *non-degenerate* $*$-homomorphisms is anti-equivalent to the category of locally compact Hausdorff spaces with proper continuous maps. $\endgroup$ – Martin Brandenburg May 15 '15 at 8:46

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