Admissible representations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 21 November 2012

Representations

A representation of a group $G$, or a $G$-module, is an action of $G$ on a vector space $V$ by automorphisms (invertible linear transformations). A representation of an algebra $A$, or $A$-module, is an action of $A$ on a vector space $V$ by endomorphisms (linear transformations). A morphism $T:V_1\to V_2$ of $A$-modules is a linear transformation such that $T\left(av\right)=aT\left(v\right),$ for all $a\in A$ and $v\in V.$ An $A$-module $M$ is simple, or irreducible, if it has no submodules except $0$ and itself.

A representation of a topological group $G$, or a $G$-module, is an action of $G$ on a topological vector space $V$ by automorphisms (continuous invertible linear transformations) such that the map $$\begin{array}{rcl}G\times V& \to & V\\ \left(gv\right)& \mapsto & gv\end{array}$$ is continuous. When dealing with representations of topological groups all submodules are assumed to be closed subspaces.

A $*$-representation of a $*$-algebra $A$ is an action of $A$ on a Hilbert space $H$ by bounded operators such that $$⟨a{v}_1,v_2⟩=⟨v_1,a*v_2⟩,\text{for all}{v}_1,v_2\in V,a\in A.$$ A $*$-representation of $A$ on $H$ is nondegenerate if $AV=\left\{av\right|a\in A,v\in V\}$ is dense in $V$.

A unitary representation of a topological group $G$, or $G$-module, is an action of $G$ on a Hilbert space $V$ by automorphisms (unitary continuous invertible linear transformations) such that the action $G\times V\to V$ is a continuous map.

An admissable representation of an idempotented algebra $(A,ℰ)$ is an action of $A$ on a vector space $V$ by linear transformations such that

1. $V=\underset{e\in ℰ}{\cup }eV,$
2. each $eV$ is finite dimensional.

A representation of an idempotented algebra is smooth if it satisfies (a).

Notes and References

These are from a subsection entitled Representations of Representation Thery notes Chapter 4, Book2003/chap41.17.03.pdf.

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