Edited question:


Are there any other non-trivial \*-homomorphisms between matrix algebras apart from the unitary homomorphisms? 


Original question:


Does there exist a surjective (but not bijective) \*-homomorphism between matrix algebras over the complex numbers? If so, are there any nice examples?


(I had not realized the matrix algebras were simple but since they are, the answer to the original question is indeed obvious)



*-homomorphisms between matrix algebras
The algebra $M\_n(\mathbb{C})$ of $n \times n$ complex matrices is Morita equivalent to $\mathbb{C}$. Which implies: Every unital representation of $M\_n(\mathbb{C})$ is isomorphic to a direct sum of copies of the defining representation. Thus your homomorphism $\rho:M\_n \to M\_k$ exists precisely when $k$ is a multiple of $n$, and after a change of basis in the target, $\rho(A)$ is just copies of $A$ on the diagonal. This is so for all homomorphisms, whether or not they are \*-homomorphisms, but the answer for \*-homomorphisms is the same. The only difference is that instead of a general change of basis, everything is equivalent up to a unitary change of basis.


The nonunital homomorphisms are not much more general. Up to a change of basis, you can pad a unital homomorphism with extra rows and columns that are all 0.


There is a similar result for a direct sum of matrix algebras. It is summarized in the concept of a "Bratteli diagram" to describe a homomorphism between two direct sums of matrix algebras. The homomorphism can be thought of as a bin packing -- packing items in bins --- with allowed repetition of the items. The Bratteli diagram shows how many copies of each item (matrix summand of the domain) goes into each bin (matrix summand of the target). In the unital case, the bins have to be filled exactly.


