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Version 13 |
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| \subsection*{Definition:} |
\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 $H$. |
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 $H$. |
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| The set of all representations of $\mathcal{A}$ on a Hilbert space $H$ is denoted $rep(\mathcal{A},H)$. |
The set of all representations of $\mathcal{A}$ on a Hilbert space $H$ is denoted $rep(\mathcal{A},H)$. |
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| {\bf Remarks:} |
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| Recall that a Hilbert space is a Banach space in the norm induced by the inner product, and also that |
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| a Banach *-algebra is a Banach algebra endowed with an involution (*); as the definition of a Banach |
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| algebra also involves a Banach space with additional algebraic properties, the representation of a |
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| Banach *-algebra on a Hilbert space is natural in the sense that the space underlying such a representation |
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| is a Banach space both for the domain and range (codomain) of the *-homomorphism that defines $\pi$. |
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| \subsection*{Special kinds of representations:} |
\subsection*{Special kinds of representations:} |
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| \begin{itemize} |
\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 $a \in \mathcal{A}$} $H_0 \subseteq H$. |
\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} |
\end{itemize} |
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| \begin{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: |
\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} |
\begin{enumerate} |
| \item $\pi(x)\xi = 0 \;\;\;\;\; \forall x\in \mathcal{A}\; \Longrightarrow \; \xi = 0$, where $\xi \in H$. |
\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 $\pi(\mathcal{A})H := \{\pi(x)\xi : x \in \mathcal{A}, \xi \in H\}$ |
\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{enumerate} |
| \end{itemize} |
\end{itemize} |
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| \begin{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 $H$. |
\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} |
\end{itemize} |
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| \begin{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 $H$. |
\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} |
\end{itemize} |
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| \begin{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 $\pi_1 \oplus \pi_2 (x) := \pi_1 (x) \oplus \pi_2 (x), \;\;\; x \in \mathcal{A}$. |
\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 $\pi_1 \oplus \pi_2 (x) := \pi_1 (x) \oplus \pi_2 (x), \;\;\; x \in \mathcal{A}$. |
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| 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} $\{\pi_i(x)\}_{i \in I}$. |
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} |
\end{itemize} |
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| \begin{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 |
\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} |
\begin{displaymath} |
| \pi_2(a) = U \pi_1(a) U^* \;\;\;\;\; \forall a \in \mathcal{A} |
\pi_2(a) = U \pi_1(a) U^* \;\;\;\;\; \forall a \in \mathcal{A} |
| \end{displaymath} |
\end{displaymath} |
| \end{itemize} |
\end{itemize} |
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| \begin{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 |
\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} |
\begin{displaymath} |
| \pi(A)\,\xi := \{\pi(a)\,\xi : a \in \mathcal{A}\} |
\pi(A)\,\xi := \{\pi(a)\,\xi : a \in \mathcal{A}\} |
| \end{displaymath} |
\end{displaymath} |
| is \PMlinkname{dense}{Dense} in $H$. Such a vector is called a {\bf cyclic vector} for the representation $\pi$. |
is \PMlinkname{dense}{Dense} in $H$. Such a vector is called a {\bf cyclic vector} for the representation $\pi$. |
| \end{itemize} |
\end{itemize} |
