I have been reading the online lecture notes by Fiona Murnaghan
http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf
The first lemma in p.35 says that every unitary representation of locally compact group $G$ is semisimple. In her notes, she defines a semisimple representation to be a representation which is a direct sum of irreducible representations.
On the other hand, people often say that the right regular representation of $G$ on $L^2(G)$, which is unitary, does not decompose into a direct sum of irreducible representations but it is a direct integral.
But if the above lemma by Murnaghan is correct, it seems that $L^2(G)$ must be a direct sum of irreducible representations.
I read the proof of the lemma in her note carefully, and noticed that even though she stated the lemma under the assumption that the group $G$ is $p$-adic and the representation is smooth, the proof goes through for any locally compact group and any uniterizable representation. Indeed, the proof only uses the property that any subrepresentation has the orthogonal complement and Zorn's lemma.
What am I missing?
Best Answer
Probably the version of "semi-simplicity" relevant in the context in those notes refers to a repn of $G$ that is admissible with respect to a compact subgroup $K$, in the sense that it is assumed to decompose with finite multiplicities over $K$. This assumption holds for repns induced from cuspidal repns on the compact itself, for example, as observed by Jacquet already in 1971. In such a context, the otherwise-too-glib remarks about complete-reducibility still reach a correct conclusion, even without completeness in a Hilbert-space sense.
The more down-to-earth situations, like $L^2(\mathbb R)$, or subspaces generated by rough functions on the circle, do not satisfy suitable "admissibility" conditions, perhaps-oddly.