Hilbert module

Definition 1.

A (right) pre-Hilbert module over a $C^{*}$ -algebra $A$ is a right $A$ -module $\mathord{\mathcal{E}}$ equipped with an $A$ -valued inner product $\langle-,-\rangle\colon\mathord{\mathcal{E}}\times\mathord{\mathcal{E}}\to A$ , i.e. a sesquilinear pairing satisfying

$\displaystyle\langle u,va\rangle$	$\displaystyle=$	$\displaystyle\langle u,v\rangle a$	(1)
$\displaystyle\langle u,v\rangle$	$\displaystyle=$	$\displaystyle\langle v,u\rangle^{*}$	(2)
$\displaystyle\langle v,v\rangle$	$\displaystyle\geq$	$\displaystyle 0,\mbox{ with\ }\langle v,v\rangle=0\mbox{ iff\ }v=0,$	(3)

for all $u,v\in\mathord{\mathcal{E}}$ and $a\in A$ . Note, positive definiteness is well-defined due to the notion of positivity for $C^{*}$ -algebras. The norm of an element $v\in\mathord{\mathcal{E}}$ is defined by $\|v\|=\sqrt{\|\langle v,v\rangle\|}$ .

Definition 2.

A (right) Hilbert module over a $C^{*}$ -algebra $A$ is a right pre-Hilbert module over $A$ which is complete with respect to the norm.

Example 1 (Hilbert spaces)

A complex Hilbert space is a Hilbert $\mathbb{C}$ -module.

Example 2 ( $C^{*}$ -algebras)

A $C^{*}$ -algebra $A$ is a Hilbert $A$ -module with inner product $\langle a,b\rangle=a^{*}b$ .

Definition 3.

A Hilbert $A$ - $B$ -bimodule is a (right) Hilbert module $\mathord{\mathcal{E}}$ over a $C^{*}$ -algebra $B$ together with a *-homomorphism $\pi$ from a $C^{*}$ -algebra $A$ to $\mathop{\mathrm{End}}\nolimits(\mathord{\mathcal{E}})$ .