Compact operator on a Hilbert C*-module - Imported Knowlpedia Content

Definition

Let $E$ and $F$ be right Hilbert $A$ -modules. For $x\in E$ and $y\in F$ , the rank-one operator $\theta_{y,x}:E\to F$ is

\theta_{y,x}(z)=y\langle x,z\rangle_A.

The space of compact operators from $E$ to $F$ is

\mathcal K_A(E,F)= \overline{\operatorname{span}}\{\theta_{y,x}:x\in E,\ y\in F\},

where the closure is in the operator norm inside the adjointable operators $\mathcal L_A(E,F)$ . The word “compact” names this norm-closed rank-one span; it does not assert that the operator maps bounded sets to relatively compact sets. This construction depends only on the Hilbert-module structures and the given coefficient algebra.

Adjoint and composition

Each rank-one operator is adjointable, with $\theta_{y,x}^*=\theta_{x,y}$ . If $S\in\mathcal L_A(F,G)$ , $T\in\mathcal L_A(D,E)$ , and $k\in\mathcal K_A(E,F)$ , then $Sk\in\mathcal K_A(E,G)$ and $kT\in\mathcal K_A(D,F)$ . These formulas explain why generalized compact operators are stable under composition with adjointable maps.

The algebra

\mathcal K_A(E)

When $E=F$ , one writes $\mathcal K_A(E)$ . It is a $C^*$ -algebra and a closed two-sided ideal of $\mathcal L_A(E)$ . For distinct modules, $\mathcal K_A(E,F)$ is generally only a closed linear space of operators, not an algebra. It appears as an off-diagonal corner in the compact operators on $E\oplus F$ .

Relation to Hilbert-space compactness

For $A=\mathbb C$ , Hilbert $A$ -modules are Hilbert spaces and this definition recovers the usual compact operators. Over a general $C^*$ -algebra, a rank-one operator may have infinite-dimensional range as a complex vector space. Finite-rank approximation therefore refers to module rank-one operators, not to finite-dimensional linear ranges.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Publisher record. Relevant: Chapter 1 on adjointable and compact module operators.
Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace C-Algebras*, American Mathematical Society, 1998. Publisher record. Relevant: Chapter 2 on Hilbert modules and compact operators.