Last update: 21 November 2012
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 $$\u27e8a{v}_{1},{v}_{2}\u27e9=\u27e8{v}_{1},a*{v}_{2}\u27e9,\phantom{\rule{2em}{0ex}}\text{for all}\phantom{\rule{2em}{0ex}}{v}_{1},{v}_{2}\in V,a\in A.$$ A $*$-representation of $A$ on $H$ is nondegenerate if $AV=\left\{av\phantom{\rule{.5em}{0ex}}\right|\phantom{\rule{.5em}{0ex}}a\in A,v\in V\}$ 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 $(A,\mathcal{E})$ is an action of $A$ on a vector space $V$ by linear transformations such that
These are from a subsection entitled Representations of Representation Thery notes Chapter 4, Book2003/chap41.17.03.pdf.