Representation of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and $H$ a Hilbert space. A representation of $A$ on $H$ is a $*$ -homomorphism to $\mathcal B(H)$ , written

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

It is faithful if $\pi$ is injective, and nondegenerate if $\overline{\pi(A)H}=H$ . No unitality is required in the definition. When $A$ is unital, nondegeneracy is equivalent to $\pi(1_A)=I_H$ ; a representation that sends $1_A$ to a proper projection is degenerate rather than unital.

Automatic boundedness

Every $*$ -homomorphism between $C^*$ -algebras is contractive, so no separate continuity hypothesis is necessary. A faithful representation is isometric:

\|\pi(a)\|=\|a\|\qquad(a\in A).

The Gelfand–Naimark theorem says that every $C^*$ -algebra admits a faithful nondegenerate representation on some Hilbert space. Abstract $C^*$ -algebras can therefore always be realized concretely as operator-norm-closed $*$ -algebras of bounded operators.

Degenerate and essential subspaces

For an arbitrary representation, the closed subspace

H_{\mathrm{ess}}=\overline{\pi(A)H}

reduces $\pi$ . The restriction to $H_{\mathrm{ess}}$ is nondegenerate, and $\pi$ is zero on $H_{\mathrm{ess}}^\perp$ . Consequently, degeneracy adds only a zero summand. For a nonunital algebra, nondegeneracy is also equivalent to strong convergence $\pi(e_i)\to I_H$ for any approximate identity $(e_i)$ of $A$ .

Cyclic and irreducible representations

A representation is cyclic if some $\xi\in H$ has $\overline{\pi(A)\xi}=H$ . It is irreducible if its only closed invariant subspaces are $0$ and $H$ , equivalently if its commutant consists only of scalar operators. Every irreducible nonzero representation is nondegenerate. The GNS construction produces cyclic representations from positive linear functionals and states.

Distinction from a group representation

A $C^*$ -representation acts linearly on algebra elements and preserves products, sums, scalar multiplication, and involution. A unitary group representation instead assigns a unitary operator to each group element. For a locally compact group the two settings are connected by the integrated form, but they are not the same definition.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: Chapter 3 on representations and the Gelfand–Naimark theorem.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 3 on representations.