The first lemma on p.35 of these notes states that unitary representations are semisimple. Could the same be said of isometries if the space doesn't have an inner product? This topic notes that the assumptions in the notes aren't really used so much as unitarizable although the response by Qiaochu notes you want some completeness conditions. The notes also talk about requisite smoothness, i.e. stabilizers of elements being open subgroups of the compact group, $G$.

So the relevant case and combined question stated precisely is:

1) If I have a representation, $V$, of a compact group, $G$, on a real Banach space (not necessarily Hilbert), which is an isometry, and which contains a dense subset, A, which has the property that $\text{stab}_G(a)$ is an open subgroup of $G$ for every $a\in A$, is V semisimple?

2) On a related note: if we have the same hypotheses as in (1) and additionally assume that there is a dense subspace which is semisimple, can we conclude that $V$ is semisimple?

3) If nothing else, can we deduce a representation, $V$, is smooth if we know that stabilizers of elements of a dense subset are open and admissible if we further know that the representation has that compact open subgroups fix only finite dimensional subspaces of a dense subspace?


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