unitary representation UnitaryRepresentation 2007-03-24 20:41:24 2007-03-24 20:41:24 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $G$ be a topological group. A \emph{unitary representation} of $G$ is a pair $(\pi, H)$ where $H$ is a Hilbert space and $\pi: G \to U(H)$ is a homomorphism such that the mapping of $G \times H \to H$ that sends $(g,v)$ to $\pi(g)v$ is continuous. Here $U(H)$ denotes the set of unitary operators of $H$. The group $G$ is said to act unitarily on $H$ or sometimes, $G$ is said to act by unitary representation on $H$.