# Banach *-algebra representation

## Definition:

A 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 $rep(\mathcal{A},H)$.

## Special kinds of representations:

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A 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 $\pi(\mathcal{A})$-invariant subspace (http://planetmath.org/InvariantSubspace) 11by a $\pi(\mathcal{A})$- we a subspace which is invariant under every operator $\pi(a)$ with $a\in\mathcal{A}$ $H_{0}\subseteq H$.

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A representation $\pi\in rep(\mathcal{A},H)$ is said to be nondegenerate if one of the following equivalent conditions hold:

1. (a)

$\pi(x)\xi=0\;\;\;\;\;\forall x\in\mathcal{A}\;\Longrightarrow\;\xi=0$, where $\xi\in H$.

2. (b)

$H$ is the closed linear span of the set of vectors $\pi(\mathcal{A})H:=\{\pi(x)\xi:x\in\mathcal{A},\xi\in H\}$

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A representation $\pi\in rep(\mathcal{A},H)$ is said to be topologically irreducible (or just ) if the only closed $\pi(\mathcal{A})$-invariant of $H$ are the trivial ones, $\{0\}$ and $H$.

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A representation $\pi\in rep(\mathcal{A},H)$ is said to be algebrically irreducible if the only $\pi(\mathcal{A})$-invariant of $H$ (not necessarily closed) are the trivial ones, $\{0\}$ and $H$.

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Given two representations $\pi_{1}\in rep(\mathcal{A},H_{1})$ and $\pi_{2}\in rep(\mathcal{A},H_{2})$, the 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 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 direct sum of the family of bounded operators (http://planetmath.org/DirectSumOfBoundedOperatorsOnHilbertSpaces) $\{\pi_{i}(x)\}_{i\in I}$.

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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 unitarily equivalent if there is a unitary $U:H_{1}\longrightarrow H_{2}$ such that

 $\pi_{2}(a)=U\pi_{1}(a)U^{*}\;\;\;\;\;\forall a\in\mathcal{A}$
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A representation $\pi\in rep(\mathcal{A},H)$ is said to be if there exists a vector $\xi\in H$ such that the set

 $\pi(A)\,\xi:=\{\pi(a)\,\xi:a\in\mathcal{A}\}$

is dense (http://planetmath.org/Dense) in $H$. Such a vector is called a for the representation $\pi$.

Linked file: http://aux.planetmath.org/files/objects/9843/BanachAlgebraRepresentation.pdf

 Title Banach *-algebra representation Canonical name BanachalgebraRepresentation Date of creation 2013-03-22 17:27:37 Last modified on 2013-03-22 17:27:37 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 23 Author asteroid (17536) Entry type Definition Classification msc 46H15 Classification msc 46K10 Defines subrepresentation Defines cyclic representation Defines cyclic vector Defines nondegenerate representation Defines topologically irreducible Defines algebrically irreducible Defines direct sum of representations Defines unitarily equivalent