∗-representation of the complex group ring

Invariant subspaces of ∗-representations and unitary representations coïncide

Consider a mutually inducing pair of a Unitary representation $Γ : 𝐺 \to G L ()$ and a ∗-representation $Γ ℂ [𝐺] : ℂ [𝐺] \to G L (𝑉)$ . Then every invariant subspace under $Γ$ is an invariant subspace of $Γ ℂ [𝐺]$ and vice-versa. #m/thm/rep Thus $Γ$ is an irrep iff $Γ ℂ [𝐺]$ is irreducible, i.e. has no non-trivial invariant subspace.

Proof

Let $𝑈 \subseteq 𝑉$ be an invariant subspace of $Γ$ . Then $𝑉$ is also an invariant subspace of $Γ ℂ [𝐺]$ , because for any $𝑢 \in 𝑈$ and $𝑎 \in ℂ [𝐺]$
$Γ ℂ [𝐺] (𝑎) 𝑢 = \sum 𝑔 \in 𝐺 𝑎 (𝑔) Γ (𝑔) 𝑢 ⏟ \in 𝑈 \in 𝑈$
Likewise if $𝑈 \subseteq 𝑉$ is an invariant subspace of $Γ ℂ [𝐺]$ then it is also an invariant subspace of $Γ$ , because for any $𝑢 \in 𝑈$ and $𝑔 \in 𝐺$
$Γ (𝑔) 𝑢 = Γ ℂ [𝐺] (𝛿 𝑔) 𝑢 \in 𝑈$
as required.

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Invariant subspaces of ∗-representations and unitary representations coïncide

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