Strongly continuous dynamical maps Let's say we have a bipartite system 
$\rho(0)=\rho_A \otimes \rho_B$
The evolution of system $A$ alone will be described by a dynamical map $\Phi_t$, such as: 
$\rho_A(t)=\Phi_t(\rho_A(0))$
If the system is Markovian, then the map $\Phi_t$ forms a quantum dynamical semigroup.
My problem is that I've found different properties about the continuity of this semigroup: some textbooks say only weak continuity is needed, some say strong continuity, others uniform continuity. Does someone know why those differences arise?
Secondly, can someone write down the definition of strong, weak and uniform continuity in terms of dynamical maps acting on $\rho$?
Thank you in advance
 A: For dynamical semigroups on Banach (and Hilbert) spaces, such as the one you are considering, weak and strong continuity are the same, so requiring one or the other is a matter of taste (I think strong continuity is the more common choice). The definitions are the usual: weak continuity means that
$$ w(\Phi_t(x)) \to w(x) \text{ when } t\to 0, \quad \text{for all linear functionals } w, \forall x $$ 
while strong continuity is defined as
$$ \lVert{ \Phi_t(x) - x}\rVert \to 0 \text{ when } t\to 0, \quad \forall x. $$
Both of these are "pointwise": the convergence of $\Phi_t(x)$ to $x$ does not have to be uniform in $x$. If we take a supremum over all possible $x$ first, we get instead the definition of uniform continuity:
$$ \sup_{x: \lVert{x}\rVert=1} \lVert{ \Phi_t(x) - x}\rVert \to 0 \text{ when } t\to 0.$$
So it is easy to see that uniform implies strong convergence. If your system is finite dimensional, then strong convergence implies uniform convergence, so the three notions of continuity you mention are all the same.
In infinite dimensions, things are very different. Uniform and strong continuity are not the same. The most striking difference lays in the generator of the evolution: your semigroup $\Phi_t$ will be of the form $\exp(t L)$ for some (super)-operator $L$ (sometimes called the Liouvillian of the system). If $\Phi_t$ is uniformly continuous, then $L$ is bounded, while if $L$ is unbounded it will generate a strongly continuous but not uniformly continuous semigroup. As you might expect, there are situations where one expect the generator to be unbounded (in the same way we expect other quantum mechanical objects, such as position/momentum/energy operators to be unbounded), so one is forced to work wit strongly but not uniformly continuous semigroups. Uniformly continuous semigroups are easier to study, since they essentially behave in the same way as semigroups in finite dimensions, but they are too restrictive for many applications!
