BanachAlgebraRepresentation 2007-08-09 11:23:33 2008-11-08 14:19:31 Definition subrepresentation cyclic representation cyclic vector nondegenerate representation topologically irreducible algebrically irreducible direct sum of representations unitarily equivalent % 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 \PMlinkescapeword{representation} \PMlinkescapeword{representations} \PMlinkescapeword{nondegenerate} \subsection*{Definition:} A {\bf representation} of a Banach *-algebra $\mathcal{A}$ is a *-homomorphism $\pi : \mathcal{A} \longrightarrow \mathcal{B}(H)$ of $\mathcal{A}$ into the *-algebra of bounded operators on some Hilbert space $H$. The set of all representations of $\mathcal{A}$ on a Hilbert space $H$ is denoted $rep(\mathcal{A},H)$. \subsection*{Special kinds of representations:} \begin{itemize} \item A {\bf subrepresentation} of a representation $\pi \in rep(\mathcal{A},H)$ is a representation $\pi_0 \in rep(\mathcal{A},H_0)$ obtained from $\pi$ by restricting to a closed $\pi(\mathcal{A})$-\PMlinkname{invariant subspace}{InvariantSubspace} \footnote{by a $\pi(\mathcal{A})$-\PMlinkescapetext{invariant subspace} we \PMlinkescapetext{mean} a subspace which is invariant under every operator $\pi(a)$ with $a \in \mathcal{A}$} $H_0 \subseteq H$. \end{itemize} \begin{itemize} \item A representation $\pi \in rep(\mathcal{A},H)$ is said to be {\bf nondegenerate} if one of the following equivalent conditions hold: \begin{enumerate} \item $\pi(x)\xi = 0 \;\;\;\;\; \forall x\in \mathcal{A}\; \Longrightarrow \; \xi = 0$, where $\xi \in H$. \item $H$ is the closed linear span of the set of vectors $\pi(\mathcal{A})H := \{\pi(x)\xi : x \in \mathcal{A}, \xi \in H\}$ \end{enumerate} \end{itemize} \begin{itemize} \item A representation $\pi \in rep(\mathcal{A},H)$ is said to be {\bf topologically irreducible} (or just \PMlinkescapetext{{\bf irreducible}}) if the only closed $\pi(\mathcal{A})$-invariant \PMlinkescapetext{subspaces} of $H$ are the trivial ones, $\{0\}$ and $H$. \end{itemize} \begin{itemize} \item A representation $\pi \in rep(\mathcal{A},H)$ is said to be {\bf algebrically irreducible} if the only $\pi(\mathcal{A})$-invariant \PMlinkescapetext{subspaces} of $H$ (not necessarily closed) are the trivial ones, $\{0\}$ and $H$. \end{itemize} \begin{itemize} \item Given two representations $\pi_1 \in rep(\mathcal{A},H_1)$ and $\pi_2 \in rep(\mathcal{A},H_2)$, the \PMlinkescapetext{{\bf direct sum}} of $\pi_1$ and $\pi_2$ is the representation $\pi_1 \oplus \pi_2 \in rep(\mathcal{A}, H_1 \oplus H_2)$ given by $\pi_1 \oplus \pi_2 (x) := \pi_1 (x) \oplus \pi_2 (x), \;\;\; x \in \mathcal{A}$. More generally, given a family $\{\pi_i\}_{i \in I}$ of representations, with $\pi_i \in rep(\mathcal{A}, H_i)$, their \PMlinkescapetext{{\bf direct sum}} is the representation $\bigoplus_{i \in I} \pi_i \in rep(\mathcal{A}, \bigoplus_{i \in I}H_i)$, in the direct sum of Hilbert spaces $\bigoplus_{i \in I}H_i$, such that $\left(\bigoplus_{i \in I} \pi_i\right) (x):= \bigoplus_{i \in I} \pi_i(x)$ is the \PMlinkname{direct sum of the family of bounded operators}{DirectSumOfBoundedOperatorsOnHilbertSpaces} $\{\pi_i(x)\}_{i \in I}$. \end{itemize} \begin{itemize} \item Two representations $\pi_1 \in rep(\mathcal{A},H_1)$ and $\pi_2 \in rep(\mathcal{A},H_2)$ of a Banach *-algebra $\mathcal{A}$ are said to be {\bf unitarily equivalent} if there is a unitary $U : H_1 \longrightarrow H_2$ such that \begin{displaymath} \pi_2(a) = U \pi_1(a) U^* \;\;\;\;\; \forall a \in \mathcal{A} \end{displaymath} \end{itemize} \begin{itemize} \item A representation $\pi \in rep(\mathcal{A},H)$ is said to be {\bf \PMlinkescapetext{cyclic}} if there exists a vector $\xi \in H$ such that the set \begin{displaymath} \pi(A)\,\xi := \{\pi(a)\,\xi : a \in \mathcal{A}\} \end{displaymath} is \PMlinkname{dense}{Dense} in $H$. Such a vector is called a {\bf cyclic vector} for the representation $\pi$. \end{itemize}