Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group of automorphisms of $\mathscr B(\mathcal H)$, meaning simply that $\sigma $ is a group homomorphism $$ \sigma :{\mathbb R}\to \text{Aut}\big (\mathscr B(\mathcal H)\big ), $$ where $ \text{Aut}( \mathscr B(\mathcal H) ) $ is the group formed by all *-automorphisms of $\mathscr B(\mathcal H)$.
In quantum mechanics such one-parameter groups are crucial ingredients since they are seen as the time evolution of observables and they almost always have the form $$ \sigma _t(A) = e^{itH}Ae^{-itH},\quad (t\in {\mathbb R}),\quad (A\in \mathscr B(\mathcal H)), $$ where $H$ is a (possibly unbounded, densely defined) self-adjoint operator on $\mathcal H$, sometimed called the Hamiltonian of the system.
It is almost foklore that all "nice" one-parameter groups are of the above form, and in fact Blackadar states precisely this in Theorem (I.7.4.10) [B. Blackadar, Operator Algebras Theory of C*-Algebras and von Neumann Algebras, Operator Algebras and Non-Commutative Geometry III, Encyclopaedia of Mathematical Sciences Volume 122, 2006], where "nice" is taken to be "strongly continuous". However the meaning of "strongly continuous" here is a bit unclear since it could either be taken to be:
(1) for every $A\in \mathscr B(\mathcal H)$, the map $$ t\in {\mathbb R} \mapsto \sigma _t(A)\in \mathscr B(\mathcal H) $$ is continuous in the norm topology of $\mathscr B(\mathcal H)$, or
(2) for every $A\in \mathscr B(\mathcal H)$, and every $\xi \in \mathcal H$, the map $$ t\in {\mathbb R} \mapsto \sigma _t(A)\xi \in \mathcal H $$ is continuous in the norm topology of $\mathcal H$ (this should be called pointwise-strongly continuous).
Unfortunately Blackadar makes no further comments on the meaning of "strongly continuous" and neither does he give a reference for this fact.
Even though (1) seems to be the natural interpretation of "strongly continuous" in Theorem (I.7.4.10), given the context and the results discussed right before this result, it should be said that this is not a very interesting condition since it fails for the one-parameter groups defined in terms of an unbounded Hamiltonian, as above.
As this is something I need for a project I'm involved in, I spent some time searching for a reference, but I could not find it! This was indeed was a big surprise, given the fundamental importance of this result not only in quantum mechanics, but also in many other applications.
Frustrated with my bad luck I ended up proving this result by myself, using an even weaker condition, namely
(3) for every $A\in \mathscr B(\mathcal H)$, and every $\xi , \eta \in \mathcal H$, the map $$ t\in {\mathbb R} \mapsto \langle \sigma _t(A)\xi , \eta \rangle \in {\mathbb R} $$ is continuous (this should be called pointwise-weakly continuous).
My proof turned out to be rather convoluted, and in it I used Theorem 4.13 of [R. Kadison, Transformations of states in operator theory and dynamics, Topology Vol. 3, Suppl. 2, pp. 177-198], which requires a different continuity condition, namely that $\sigma $ be continuous in the bounded-weak topology of $\text{Aut}(\mathscr B(\mathcal H))$, meaning that $$ \lim_{t\to t_0}\ \sup_{\buildrel{\scriptstyle A\in \mathscr B(\mathcal H)}\over{\Vert A\Vert \leq 1}} \Big |\big \langle \big (\rho _t(A)-\rho _{t_0}(A)\big )\xi ,\xi \big \rangle \Big | = 0, $$ for every $t_0$ in ${\mathbb R}$, and every $\xi $ in $\mathcal H$.
Despite my long years pretending to be a mathematician, I am embarrassed to admit that I have never heard of this topolopgy before!
In a nutshel, I think I proved that pointwise-weakly continuous one-parameter groups are automatically bounded-weakly continuous. My questions therefore are:
Question 1. Is there a reference for the fact that every pointwise-weakly continuous one-parameter group of automorphisms is given by a Hamiltonian, as above?
Question 2. Where else is the bounded-weak topolopgy on $\text{Aut}(\mathscr B(\mathcal H))$ used nowadays?
EDIT 1. Based on some comments below, it is perhaps worth mentioning that the main topologies mentioned above, namely the pointwise-strong, pointwise-weak, and bounded-weak topologies, are all topologies on the set of bounded linear maps from $\mathscr B(\mathcal H)$ to $\mathscr B(\mathcal H)$, rather than from $\mathcal H$ to $\mathcal H$.
EDIT 2. I should have said that (2) and (3) are in fact equivalent, the reason being that the weak-operator topology is well known to coincide with the strong-operator topology on unitary operators, and also that it is enough to test these conditions when $A$ is a unitary operator (as these linearly span $\mathscr B(\mathcal H)$).
