Assume we have a homomorphism $\phi: C(S^{1},M\_{n}(\mathbb{C}))\rightarrow C(S^{1},M\_{m}(\mathbb{C}))$ where $n$ divides $m$. Under what conditions does $\phi$ send constant functions to constant functions?



Homomorphisms preserving constant functions
This is perhaps just a partial answer; but I feel that Will's answer is incomplete, and this is a more interesting question that he is suggesting.


Given a \*-homomorphism $\phi$, composing with point-evaluation at any point $x \in S^1$ gives us a representation
$$\phi\_x: C(S^1, M\_n) \to M\_m, $$
which is therefore of the form
$$ \phi\_x(f) = \alpha\_x(\text{diag}(f(y(x,1)), \dots, f(y(x,k)))), $$
where $\alpha\_x$ is an automorphism of $M\_m$ and $y(x,1),\dots,y(x,k) \in S^1$ (and $k=m/n$).


Now, $\phi$ sends constant functions to constant functions if and only if $\alpha\_x \circ \alpha\_y^{-1}$ acts as the identity on the $M\_m \otimes 1\_k \subset M\_n$.


I'm not sure if there is a more explicit characterization.
Some remarks:


* Certainly, $\phi$ preserving constant functions doesn't imply that $\alpha\_x = \alpha\_y$.
* $\alpha\_x$ isn't uniquely determined (when there are repeats among $y(x,1),\dots,y(x,k)$, and related to this, needn't be continuous.


