# 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:

• 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 $a\in\mathcal{A}$ $H_{0}\subseteq H$.

• 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 $\pi(\mathcal{A})H:=\{\pi(x)\xi:x\in\mathcal{A},\xi\in H\}$

• 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 $H$.

• 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 $H$.

• 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) $\{\pi_{i}(x)\}_{i\in I}$.

• 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

 $\pi_{2}(a)=U\pi_{1}(a)U^{*}\;\;\;\;\;\forall a\in\mathcal{A}$
• 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 $\pi$.