This is from the textbook "An introduction to K-theory for C*alebgra" :
So I don't have a question about the problem itself but am more interested in the fact that we can define the norm for the subalgebra of diagonal matrices using the max norm because this subalgebra is isometrically isomorphic to $A \oplus A$ which carries the max norm. So just to be clear, the norm for matrices which gives us a C* algebra is actually very complicated and definitely not something you want to work with, but for the subalgebra of diagonal matrices, we have a much simpler norm. So my question is if we were working with a subalgebra of a C* algebra and "discovered" another norm for the subalgebra which satisfies all the C* properties, would that norm in fact be equal to the original norm? This is because the norm for a C* algebra has to be unique right?