By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's Operator theory :
My question: In the first theorem, the author claims that for every representation the theorem holds while in the second just shows that for nondegenerate representation holds. By their proofs, I think the first is proved for nondegenerate representations, too. Please help me to understand that this theorem always holds or just for nondegenerate representations. Thanks in advance.
A cyclic representation is nondegenerate. A direct sum of nondegenerate representations is nondegenerate. Hence, the theorem could only hold for nondegenerate representations.
You are right that in Conway's proof it can be seen that the algebra must be nondegenerate, as we have $\mathscr{H}_0$ defined to be a subset of $\mathrm{cl}[\pi(A)\mathscr{H}]$, and see later that $\mathscr{H_0}=\mathscr{H}$. If I had my copy of Conway handy I would check whether it states somewhere earlier that representations would be assumed nondegenerate unless stated otherwise.