I am reading Conway's book, A Course in Operator Theory. I don't understand the definition of representation $\rho$ here. In the start of this section, he only defined the representation from a $C^*$-algebra to $B(H)$. But in 6.10 Proposition, the ideal isn't required to be closed, thus it may be not a $C^*$-algebra, then we don't know the representation means what?
So, i want to ask if the ideal in Proposition should be added a condition "closed"
It doesn't really matter. Any bounded operator on a subspace can be uniquely extended (as a bounded operator) to the closure of the subspace. If said bounded operator is a $*$-homomorphism, so is the extension. That $*$-homomorphisms are bounded was shown earlier in the book (Proposition 1.7).