# criterion for a Banach *-algebra representation to be irreducible

Theorem - Let $\mathcal{A}$ be a Banach *-algebra, $H$ an Hilbert space^{} and $I$ the identity operator in $H$. A representation (http://planetmath.org/BanachAlgebraRepresentation) $\pi :\mathcal{A}\u27f6H$ is topologically irreducible if and only if $\pi {(\mathcal{A})}^{\prime}=\u2102I$, i.e. if and only if the commutant of $\pi (\mathcal{A})$ consists of scalar multiples of the identity operator.

Proof : $(\u27f9)$

As $\pi (\mathcal{A})$ is selfadjoint, $\pi {(\mathcal{A})}^{\prime}$ is a von Neumann algebra^{}.

Suppose $\pi {(\mathcal{A})}^{\prime}\ne \u2102I$. Then the dimension^{} of $\pi {(\mathcal{A})}^{\prime}$ is greater than one.

It is known that von Neumann algebras of dimension greater than one contain non-trivial projections, so there is a projection $P\in \pi {(\mathcal{A})}^{\prime}$ such that $P\ne 0$ and $P\ne I$.

As $P\in \pi {(\mathcal{A})}^{\prime}$, $P$ commutes with every operator $T\in \pi (\mathcal{A})$, that is $PT=TP$.

Thus $RanP$ is an invariant subspace of every $T\in \pi (\mathcal{A})$. Therefore $\pi $ is not an irreducible representation.

$(\u27f8)$

Conversely, suppose that $\pi $ is not an irreducible representation. There exists a closed $\pi (\mathcal{A})$-invariant subspace different from $\{0\}$ and $H$.

Let $P$ be the projection onto that closed invariant subspace.

Invariance can be expressed as: $\pi (a)P=P\pi (a)P$ for every $a\in \mathcal{A}$. It follows that

$$P\pi (a)={(\pi {(a)}^{*}P)}^{*}={(\pi ({a}^{*})P)}^{*}={(P\pi ({a}^{*})P)}^{*}=P\pi {({a}^{*})}^{*}P=P\pi (a)P=\pi (a)P$$ |

for every $a\in \mathcal{A}$.

We conclude that $P$ commutes with every element of $\pi (\mathcal{A})$, i.e. $P\in \pi {(\mathcal{A})}^{\prime}$.

Thus $\pi {(\mathcal{A})}^{\prime}\ne \u2102I\mathrm{\square}$

Title | criterion for a Banach *-algebra representation to be irreducible^{} |
---|---|

Canonical name | CriterionForABanachalgebraRepresentationToBeIrreducible |

Date of creation | 2013-03-22 17:27:43 |

Last modified on | 2013-03-22 17:27:43 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 9 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46K10 |