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I would recommend reading parts of Jacques Dixmier's book: "$C^$-algebras" $C^\ast$-algebras" (North Holland, 1977 - translated from the french version of 1969), especially Chapters 5 (irreducible and factor representations of $C^$-algebras) 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.
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I would recommend reading parts of Jacques Dixmier's book: "$C^$-algebras" (North Holland, 1977 - translated from the french version of 1969), especially Chapters 5 (irreducible and factor representations of $C^$-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.