This [paper][1] proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.


PS : it is not clear from the statement in the paper if one need a functors that extend the gelfand duality on object or on object and morphism to obtain the obstruction. but it seems to be using only very specific commutative algebra.


  [1]: https://www.cs.ox.ac.uk/people/chris.heunen/publications/2012/nogo/nogo.pdf