Bounded star-representation - Imported Knowlpedia Content

Definition

Let $A$ be an involutive algebra and $H$ a Hilbert space. A bounded $*$ -representation of $A$ on $H$ is an algebra homomorphism into the bounded-operator algebra $\mathcal B(H)$ :

\pi:A\longrightarrow\mathcal B(H)

such that $\pi(a^*)=\pi(a)^*$ for every $a\in A$ . Here “bounded” says that every represented element is a bounded, everywhere-defined operator; no norm or topology on $A$ is assumed. The representation is nondegenerate when $\overline{\pi(A)H}=H$ , and faithful when $\pi$ is injective. Unitality is an additional condition when $A$ has an identity.

Relation to C*-representations

When $A$ is a $C^*$ -algebra, this is the usual $C^*$ -representation. Its map into $\mathcal B(H)$ is then automatically contractive, so elementwise boundedness and continuity require no separate hypotheses. For a general normed $*$ -algebra, a representation by bounded operators need not be continuous as a map from $A$ ; continuity must be imposed if the phrase “bounded representation” is intended in the normed-linear-map sense.

Essential subspace and unitality

The essential subspace $H_{\mathrm{ess}}=\overline{\pi(A)H}$ is reducing for $\pi$ . Restriction to it is nondegenerate, while $\pi$ vanishes on $H_{\mathrm{ess}}^\perp$ . If $A$ is unital, a unital representation is nondegenerate. Conversely, a nondegenerate representation satisfies $\pi(1_A)=I_H$ , so the two conditions agree in the unital case. A degenerate representation may send $1_A$ to a proper orthogonal projection.

Distinction from unbounded representations

An unbounded $*$ -representation assigns operators on a common dense domain and requires explicit domain invariance and adjoint conditions. Those requirements are absent here because $\mathcal B(H)$ is already an involutive algebra of everywhere-defined operators. Thus the adjective “bounded” distinguishes the codomain, not a bound uniform over all $a\in A$ .

References

Konrad Schmüdgen, Unbounded Operator Algebras and Representation Theory, De Gruyter, 1990. DOI record. Relevant: Part II, §8 on $*$ -representations and the bounded-operator case.
Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Chapter 3 on representations of $C^*$ -algebras and automatic contractivity.