criterion for a Banach *-algebra representation to be irreducible

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

Proof : $(⟹)$

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

Suppose $\pi {(𝒜)}^{\prime }\ne ℂI$. Then the dimension of $\pi {(𝒜)}^{\prime }$ 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 $P\in \pi {(𝒜)}^{\prime }$ such that $P\ne 0$ and $P\ne I$.

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

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

$(⟸)$

Conversely, suppose that $\pi$ is not an irreducible representation. There exists a closed $\pi (𝒜)$-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 $a\in 𝒜$. 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 $a\in 𝒜$.

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

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