# 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 $\mathop{\mathrm{End}}\nolimits(\mathord{\mathcal{E}})$.

Title Hilbert module HilbertModule 2013-03-22 13:01:01 2013-03-22 13:01:01 mhale (572) mhale (572) 8 mhale (572) Definition msc 46C05 $C^{*}$-module HilbertSpace FinitelyGeneratedProjectiveModule pre-Hilbert module