# Banach *-algebra representation

## Definition:

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

## Special kinds of representations:

• 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 $H$.

• 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 $\pi$.

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