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

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