I've been trying to prove that a star-homomorphism T:A to B(H) is continuous, where A is a C*-algebra and B(H) is the bounded linear operators on a Hilbert space. It's proving a real nightmare are there are some proofs which seem solid, but fall through on the details - eg some proofs use spectral radius = norm, but this only holds for normal operators.
Someone please let me know if this is actually worth trying to prove!
More generally, any $^{\ast}$-homomorphism $f : A \to B$ between (edit: unital) C*-algebras is automatically continuous.
Proof. This is a standard argument. If $a - \lambda I$ is invertible then so is $f(a - \lambda I) = f(a) - \lambda I$; it follows that the spectrum satisfies $\sigma(f(a)) \subseteq \sigma(a)$, so the spectral radius satisfies $\rho(f(a)) \le \rho(a)$. Now, for every $a \in A$ we have
$$\| f(a) \| = \sqrt{ \| f(a)^{\ast} f(a) \| } = \sqrt{ \rho(f(a^{\ast} a)) } \le \sqrt{ \rho(a^{\ast} a)} = \| a \|$$
so $f$ has norm $\le 1$ and in particular is continuous. The use of the C*-identity here is what allows us to use the spectral radius even though $a$ and $f(a)$ are not guaranteed to be normal. $\Box$