Elementary C*-algebra - Imported Knowlpedia Content

Definition

An elementary $C^*$ -algebra is a $C^*$ -algebra $A$ for which there are a nonzero Hilbert space $H$ and a $*$ -isomorphism

A\longrightarrow K(H).

Thus $A$ is $*$ -isomorphic to the compact-operator algebra on $H$ . The Hilbert space is determined up to unitary isomorphism by $A$ . Some authors include the zero algebra by allowing $H=\{0\}$ ; excluding it makes every elementary algebra nonzero and simple.

Structure

Every elementary $C^*$ -algebra is simple and liminal. Its nonzero irreducible representations are all unitarily equivalent to the defining action of $K(H)$ on $H$ . Minimal projections correspond to rank-one projections, and any one of them generates the whole algebra as a closed two-sided ideal.

The multiplier algebra of $K(H)$ is $B(H)$ . Consequently an infinite-dimensional elementary algebra is nonunital even though its multiplier algebra is unital.

Examples

The matrix algebra $M_n(\mathbb C)$ is elementary because

M_n(\mathbb C)=K(\mathbb C^n).

For infinite-dimensional separable $H$ , the algebra $K(H)$ is an infinite-dimensional elementary algebra. By contrast, $B(H)$ is not elementary when $H$ is infinite-dimensional, and a nontrivial direct sum $K(H_1)\oplus K(H_2)$ is not elementary because it is not simple.

Role in type I theory

Elementary algebras are the irreducible local building blocks of type I and continuous-trace $C^*$ -algebras. In particular, the image of every irreducible representation of a liminal algebra is elementary. Bundles whose fibers are elementary algebras provide a geometric model for continuous-trace algebras and carry the twisting measured by the Dixmier–Douady invariant.

References

Kenneth R. Davidson, C-Algebras by Example*, Fields Institute Monographs 6, American Mathematical Society, 1996. AMS publisher record. Relevant: §I.4 on compact-operator algebras.
Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace C-Algebras*, Mathematical Surveys and Monographs 60, American Mathematical Society, 1998. DOI record. Relevant: Chapter 3 on elementary bundles and continuous trace.