Say we have an n-dimensional row vector with entries in a C*-algebra . We construct the matrix

How do we show that this matrix is positive?

Here is my idea:

If we have a representation for some Hilbert space , we can inflate this homomorphism to

with norm which makes into a C*-algebra.

Furthermore we know that is isomorphic to where .

We know that , so it is self adjoint. Now is we can show that where then the matrix is positive.

So

so when we have positivity, but what about the terms where ?

Or am I completely on the wrong track?