Hilbert module

Definition 1.

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

$⟨u,va⟩$$=$$⟨u,v⟩a$(1)

$⟨u,v⟩$$=$${⟨v,u⟩}^{*}$(2)

$⟨v,v⟩$$\ge$$0,\text{ with }⟨v,v⟩=0\text{ iff }v=0,$(3)

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

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 $\mathrm{C}$-module.

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

A $C^{\mathrm{*}}$-algebra $A$ is a Hilbert $A$-module with inner product$\mathrm{⟨}a\mathrm{,}b\mathrm{⟩}\mathrm{=}{a}^{\mathrm{*}}\mathit{}b$.

Definition 3.

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