Hilbert C*-module - Knowlpedia

Let $A$ be a $C^*$ -algebra. A right Hilbert $A$ -module is a right $A$ -module $E$ with a map $\langle\cdot,\cdot\rangle_A:E\times E\to A$ , complex-linear in the second variable, such that, for $x,y,z\in E$ and $a\in A$ ,

\langle x,y+z\rangle_A=\langle x,y\rangle_A+\langle x,z\rangle_A,\qquad \langle x,ya\rangle_A=\langle x,y\rangle_Aa,\qquad \langle x,y\rangle_A=\langle y,x\rangle_A^*,

$\langle x,x\rangle_A$ is positive, and $\langle x,x\rangle_A=0$ only when $x=0$ . Finally, $E$ is complete for $\|x\|=\|\langle x,x\rangle_A\|^{1/2}$ . Thus the inner product is linear in the second variable under this convention.

Standard examples

The algebra $A$ is a Hilbert $A$ -module over itself with $\langle a,b\rangle_A=a^*b$ . The column module $A^n$ has

\langle(a_i),(b_i)\rangle_A=\sum_{i=1}^{n}a_i^*b_i.

A Hilbert space is exactly a Hilbert module over $\mathbb C$ . Unlike a Hilbert space, a Hilbert $A$ -module need not have an orthonormal basis, and a closed submodule need not possess an orthogonal complement.

Adjointable and compact operators

An $A$ -linear map $T:E\to F$ is adjointable if there is an $A$ -linear map $T^*:F\to E$ satisfying $\langle Tx,y\rangle_A=\langle x,T^*y\rangle_A$ . Adjointable maps are automatically bounded, but bounded $A$ -linear maps need not be adjointable. The compact operators on $E$ are the norm closure of the span of $\theta_{x,y}(z)=x\langle y,z\rangle_A$ ; “compact” here is a module-theoretic notion and need not mean compact as a Banach-space operator.

Conventions and scope

Some authors use left modules and inner products linear in the first variable; all displayed formulas must then be reversed consistently. Fullness, meaning that the closed span of $\langle E,E\rangle_A$ equals $A$ , is an additional condition rather than part of the definition. Self-dual modules over a von Neumann algebra form a more restrictive $W^*$ -module theory and should not be assumed for a general Hilbert $C^*$ -module.

References

E. C. Lance, Hilbert $C^*$ -Modules: A Toolkit for Operator Algebraists, Cambridge University Press, 1995. Publisher record. Relevant: Chapter 1 on Hilbert modules, adjointable maps, and compact module operators.