Theorem 3 (Larsen’s Alternative)

Assume that G is infinite. Let $\pi$ be a faithful unitary representation of G in a Hilbert space E of finite dimension $\geqslant 2$.

a) Assume that the derived group of G is infinite. Then $M_4(\pi )\geqslant 2$ with equality if and only if G contains $\mathbf{S}\mathbf{U}(E)$;

b) Assume that dim(E) $\geqslant 3$, that $\pi$ is of orthogonal or symplectic type, and that dim(E) $\not = 4$ if $\pi$ is of orthogonal type. Then $M_4(\pi )\geqslant 3$ with equality if and only if the complexified Lie algebra of G is equal to the Lie algebra of a non-degenerate G-invariant bilinear form on E. This is the case if and only if the identity component $G_0$ of G is a maximal compact subgroup of the automorphism group of such a bilinear form.

We shall use the following lemmas in the proof.