Banach *-algebra representation

Definition:

A representation of a Banach *-algebra $𝒜$ is a *-homomorphism $\pi :𝒜⟶ℬ(H)$ of $𝒜$ into the *-algebra of bounded operators on some Hilbert space $H$.

The set of all representations of $𝒜$ on a Hilbert space $H$ is denoted $rep(𝒜,H)$.

Special kinds of representations:

• A subrepresentation of a representation $\pi \in rep(𝒜,H)$ is a representation ${\pi }_0\in rep(𝒜,H_0)$ obtained from $\pi$ by restricting to a closed $\pi (𝒜)$-invariant subspace (http://planetmath.org/InvariantSubspace) $H_0\subseteq H$.

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

  1. $\pi (x)\xi =0\mathit{\hspace{1em}\hspace{1em}}\forall x\in 𝒜⟹\xi =0$, where $\xi \in H$.

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

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

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

• Given two representations ${\pi }_1\in rep(𝒜,H_1)$ and ${\pi }_2\in rep(𝒜,H_2)$, the of ${\pi }_1$ and ${\pi }_2$ is the representation ${\pi }_1\oplus {\pi }_2\in rep(𝒜,H_1\oplus H_2)$ given by ${\pi }_1\oplus {\pi }_2(x):={\pi }_1(x)\oplus {\pi }_2(x),x\in 𝒜$.

  More generally, given a family ${\{{\pi }_i\}}_{i\in I}$ of representations, with ${\pi }_i\in rep(𝒜,H_i)$, their is the representation ${\oplus }_{i\in I}{\pi }_i\in rep(𝒜,{\oplus }_{i\in I}{H}_i)$, in the direct sum of Hilbert spaces ${\oplus }_{i\in I}{H}_i$, such that $\left({\oplus }_{i\in I}{\pi }_i\right)(x):={\oplus }_{i\in I}{\pi }_i(x)$ is the direct sum of the family of bounded operators (http://planetmath.org/DirectSumOfBoundedOperatorsOnHilbertSpaces) ${\{{\pi }_i(x)\}}_{i\in I}$.

• Two representations ${\pi }_1\in rep(𝒜,H_1)$ and ${\pi }_2\in rep(𝒜,H_2)$ of a Banach *-algebra $𝒜$ are said to be unitarily equivalent if there is a unitary $U:H_1⟶H_2$ such that

  $${\pi }_2(a)=U{\pi }_1(a)U^{*}\mathit{\hspace{1em}\hspace{1em}}\forall a\in 𝒜$$

• A representation $\pi \in rep(𝒜,H)$ is said to be if there exists a vector $\xi \in H$ such that the set

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

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

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