I am currently reading some of Mackey's work on unitary representation.

Given a locally compact group $G$ and a unitary representation $\pi : G\rightarrow U(H)$. As far as I understood it, the representation $\pi$ is primary, if the von Neumann algebra generated by $\pi(g)$ for all $g \in G$ is a factor, see http://mathworld.wolfram.com/PrimaryRepresentation.html. The representation $\pi$ is irreducible, if there does not exist a nontrivial $G$ invariant closed subspace $H' \subset H$, i.e. $\pi(g) h \in H'$ for all $g \in G$ and $h \in H'$.

See the comments: The countable sum of the same irreducible representation is primary.

When is a primary representations quasi equivalent to an irreducible one? Are they the same if the group is of type 1? Does the decomposition of the group von Neumann algbera $L(G)$ into factors correspond to the decomposition (as a direct integral) in isotypic components? Are the some nice examples, which illustrate that this is to much to hope for.

Motivating example is the Peter Weyl theorem, which states that every irreducible is finite dimensional and $$L(G) = \bigoplus_{\pi \; \in irr(G)} M_{dim(\pi)}( \mathbb{C}),$$ where the components $M_{dim(\pi)}( \mathbb{C})$ are the factors. Hence here the factors are quasi equivalent to an irreducible one.

Aside to the original question: Do all unitary representation appear in $L(G)$?