Let $A$ be a complex associative unital involutive algebra (*-algebra). Consider all *-representations of $A$ on pre-Hilbert spaces by possibly unbounded operators. Is there a neat criterion on whether there is a faithful such representation? I guess that this is the same as asking whether the universal representation (direct sum of all cylic representations) is faithful. Is there a literature on unbounded * -representations? I have the feeling that my C*-intuition may be misleading.
Thank you.