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 $L(G)$?



What is the difference between a primary representation and a irreducible representation?
I would recommend reading parts of Jacques Dixmier's book: "$C^\ast$-algebras" (North Holland, 1977 - translated from the french version of 1969), especially Chapters 5 (irreducible and factor representations of $C^\ast$-algebras) and Chapter 13 (the analogue for locally compact groups).


The regular representation of a free group is not type I (as the commutant is not type I), while every irreducible representation is type I (since the commutant is $\mathbb{C}$). Hence a factor representation is not necessarily quasi-equivalent to an irreducible. It is true that, for a type I group, every factor representation is quasi-equivalent to an irreducible one (Proposition 5.4.11 in Dixmier).
For the decomposition of the regular representation and the corresponding decomposition of $L(G)$, see Proposition 18.7.7 in Dixmier.


