PlanetMath: Banach *-algebra representation

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Banach *-algebra representation

(Definition)

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

The set of all representations of $\mathcal{A}$ on a Hilbert space 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 ¹ $H_0 \subseteq H$ .

A representation $\pi \in rep(\mathcal{A},H)$ is said to be nondegenerate if one of the following equivalent conditions hold:
1. $\pi(x)\xi = 0 \;\;\;\;\; \forall x\in \mathcal{A}\; \Longrightarrow \; \xi = 0$ , where $\xi \in H$ .
2. 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 irreducible) if the only closed $\pi(\mathcal{A})$ -invariant subspaces of are the trivial ones, $\{0\}$ and .

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

Given two representations $\pi_1 \in rep(\mathcal{A},H_1)$ and $\pi_2 \in rep(\mathcal{A},H_2)$ , the direct sum 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 direct sum 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 $\{\pi_i(x)\}_{i \in I}$ .

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
$\displaystyle \pi_2(a) = U \pi_1(a) U^* \;\;\;\;\; \forall a \in \mathcal{A}$

A representation $\pi \in rep(\mathcal{A},H)$ is said to be cyclic if there exists a vector $\xi \in H$ such that the set
$\displaystyle \pi(A)\,\xi := \{\pi(a)\,\xi : a \in \mathcal{A}\}$
is dense in . Such a vector is called a cyclic vector for the representation $\pi$ .

Footnotes

...¹: by a $\pi(\mathcal{A})$ -invariant subspace we mean a subspace which is invariant under every operator $\pi(a)$ with $a \in \mathcal{A}$

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Also defines:

subrepresentation, cyclic representation, cyclic vector, nondegenerate representation, topologically irreducible, algebrically irreducible, direct sum of representations, unitarily equivalent

Attachments:: representations of Banach *-algebras are continuous (Theorem) by asteroid
criterion for a Banach *-algebra representation to be irreducible (Theorem) by asteroid
topologically irreducible representations are algebrically irreducible for -algebras (Theorem) by asteroid

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Cross-references: unitary, direct sum of Hilbert spaces, vectors, linear span, equivalent, operator, invariant, subspace, closed, Hilbert space, bounded operators, *-algebra, *-homomorphism, Banach *-algebra
There are 9 references to this entry.

This is version 13 of Banach *-algebra representation, born on 2007-08-09, modified 2008-05-07.
Object id is 9843, canonical name is BanachAlgebraRepresentation.
Accessed 1449 times total.

Classification:

AMS MSC:	46H15 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: Representations of topological algebras)
	46K10 (Functional analysis :: Topological algebras with an involution :: Representations of topological algebras with involution)

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