Group algebras

## Group algebras

1. Let $G$ be a group. Then $ℂG$ is the algebra with basis $G$ and multiplication forced by the multiplication in $G$ and the distributive law. A representation of $G$ on a vector space $V$ and this induces an equivalence of categories between the representations of $G$ and the representations of $ℂG.$
2. Let $G$ be a locally compact topological group and fix a Haar measure $\mu$ on $G.$ Let $L 1 Gμ = f:G→ ℂ| f = ∫G f g dμ g < ∞ .$ Then ${L}^{1}\left(G,\mu \right)$ is a $*$-algebra under the operations defined in (???). Any unitary representation of $G$ on a Hilbert space $H$ extends uniquely to a representation of ${L}^{1}\left(G,\mu \right)$ on $H$ by the formula $fv= ∫ G f g gvdμ g , f ∈ L 1 Gμ , g∈ G,$ and this induces an equivalence of categories between the unitary representations of $G$ and the nondegenerate $*$ -representations of ${L}^{1}\left(G,\mu \right).$
3. Let $G$ be a locally compact topological group and fix a Haar measure $\mu$ on $G.$ Let Then ${ℰ}_{c}$ is a ???-algebra under the operations defined in (???). Any representation of the topological group $G$ on a complete locally convex vector space $V$ extends uniquely to a representation of ${ℰ}_{c}$ on $V$ by the formula $μ v= ∫ G gvdμ g , f∈ ℰ c , g ∈ G,$ amd this induces am equivalence of categories between the representations of $G$ on a complete locally convex vector space $V$ and the representations on ${ℰ}_{c}\left(G\right)$ on a complete locally convex vector space $V.$
4. Let $G$ be a totally disconnected locally compact unimodular group and fix a Haar measure $\mu$ on $G.$ Let Then ${C}_{c}\left(G\right)$ is an idempotented algebra with the operations in (???) and with idempotents given by where ${\chi }_{K}$ denotes the characteristic function of the subgroup $K.$ Any smooth representation of $G$ extends uniquely to a smooth representation of ${C}_{c}\left(G\right)$ on $V$ by the formula in (???) and this induces an equivalence of categories between the smooth representations of ${C}_{c}\left(G\right)$ (see Bump Prop 3.4.3 and Prop 3.4.4). This correspondence takes admissible representations for representations for $G$ (see Bump p.425) to admissible representations for ${C}_{c}\left(G\right).$
5. Let $G$ be a Lie group. Let Then ${C}_{c}^{\infty }\left(G\right)$ is a ???-algebra under the operations defined in (???). Any representation of a topological group $G$ on a complete locally connected vector space $V$ extends uniquely to a representation of ${C}_{c}^{\infty }\left(G\right)$ on $V$ by the formula in (???) and this induces an equivalence of categories between the representations of $G$ on a complete locally convex vector space $V$ and the representations of ${C}_{c}^{\infty }\left(G\right)$ on a complete locally convex vector space $V.$
6. Let $G$ be a reductive Lie group and let $K$ be a maximal compact subgroup of $G.$ Let Then $ℰ{\left(G,K\right)}^{\mathrm{fin}}$ is an idempotented algebra with the operations in (???) and with the idempotents given by where ${\chi }_{K}$ denotes the characteristic function of the subgroup $K.$ Any $\left(𝔤,K\right)$ -module extends uniquely to a smooth representation of $ℰ{\left(G,K\right)}^{\mathrm{fin}}$ on $V$ by the formula in (???) and this induces an equivalence of categories between the $\left(𝔤,K\right)$ -modules and the smooth representations of $ℰ{\left(G,K\right)}^{\mathrm{fin}}$ (see Bump Prop 3.4.8). This correspondence takes admissible modules for $G$ (see Bump p.280 and p.193) to admissible modules for $ℰ{\left(G,K\right)}^{\mathrm{fin}}.$ By Knapp and Vogan Cor 1.7.1 $ℰ GK fin = C K fin ⊗ 𝔱 ℂ U 𝔤 ℂ .$
7. Let $G$ be a compact Lie group. Let Then $C{\left(G\right)}^{\mathrm{fin}}$ is an idempotented algebra with idempotents corresponding to the identity on a finite sum of blocks ${⨁}_{\lambda }{G}^{\lambda }\otimes {}^{}$G λ .

The category of representations of $G$ in a Hilbert space $V$ and the category of smooth representations of $C{\left(G\right)}^{\mathrm{fin}}$ are equivalent.

1. Let $𝔤$ be a Lie algebra. The enveloping algebra ${U}_{𝔤}$ of $𝔤$ of the associative algebra with 1 given by The functor $U : Lie algebras → associative algebras 𝔤 ↦ U 𝔤$ is the left adjoint of the functor $L : associative algebras → Lie algebras A• ↦ A [,]$ where $\left(A,\left[,\right]\right)$ is the Lie algebra generated by the vector space $A$ with the bracket $\left[,\right]:A\otimes A\to ℂ$ defined by This means that Let $i:𝔤\to {U}_{𝔤}$ be the map given by $i\left(x\right)=x.$ Then (???) is equivalent to the following universal property satisfied by ${U}_{𝔤}:$

If $\phi :𝔤\to A$ is a map from $𝔤$ to an associative algebra $A$ such that then there exists an algebra homomorphism $\stackrel{~}{\phi }:{U}_{𝔤}\to A$ such that $\stackrel{~}{\phi }\circ i=\phi .$

A representation of $𝔤$ on a vector space $V$ extends uniquely to a representation of ${U}_{𝔤}$ on $V$ and this induces an equivalence of categories between the representations of $𝔤$ and the representations of ${U}_{𝔤}.$

Let G be a Lie group and let $𝔤=ℂ{\otimes }_{ℝ}{𝔤}_{ℝ}$ be the complexification of the Lie algebra ${𝔤}_{ℝ}=\mathrm{Lie}\left(G\right)$ of G. Let $ℰ\left(G,\left\{1\right\}\right)$ be the algebra of distributions $\mu :{C}^{\infty }\left(G\right)\to ℂ$ on G such that $\mathrm{supp}\left(\mu \right)=1.$ Then is an isomorphism of algebras.