C*-algebra of compact operators - Imported Knowlpedia Content

Definition

Let $\mathcal H$ be a complex Hilbert space. The $C^*$ -algebra of compact operators on $\mathcal H$ , denoted $K(\mathcal H)$ , consists of all compact operators on $\mathcal H$ . Equivalently,

K(\mathcal H)=\overline{\{T\in B(\mathcal H): \operatorname{rank}T<\infty\}}^{\lVert\cdot\rVert}.

It is a norm-closed, adjoint-closed [[algebra-rings/two-sided-ideal|two-sided ideal]] in the bounded-operator algebra $B(\mathcal H)$ , and therefore a $C^*$ -algebra. It is unital exactly when $\mathcal H$ is finite-dimensional.

Rank-one generators and approximate units

For $\xi,\eta\in\mathcal H$ , the rank-one operator $\theta_{\xi,\eta}$ is given by $\theta_{\xi,\eta}\zeta=\xi\langle\eta,\zeta\rangle$ . Finite linear combinations of these operators are the finite-rank operators and are norm dense in $K(\mathcal H)$ . If $(P_\lambda)$ is the net of orthogonal projections onto finite-dimensional subspaces, directed by inclusion, then $(P_\lambda)$ is a positive contractive approximate identity for $K(\mathcal H)$ .

Ideal and representation structure

When $\mathcal H\neq 0$ , $K(\mathcal H)$ is simple: it has no nonzero proper closed two-sided ideals. Every nonzero irreducible representation is unitarily equivalent to its defining action on $\mathcal H$ , and every nondegenerate representation is an amplification of that action. These facts make compact-operator algebras the elementary pieces in the representation theory of type I $C^*$ -algebras Davidson, §I.4.

Multiplier algebra and examples

For nonzero $\mathcal H$ , the multiplier algebra of $K(\mathcal H)$ is $B(\mathcal H)$ . If $\mathcal H=\ell^2(\mathbb N)$ , diagonal compact operators correspond to sequences converging to zero, while the identity operator is not compact. In finite dimension, $K(\mathbb C^n)=B(\mathbb C^n)=M_n(\mathbb C)$ , so the distinction between compact and bounded operators disappears.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §4.1 on compact operators and their representations.
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.