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Hilbert module
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(Definition)
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Definition 1 A (right) pre-Hilbert module over a $C^*$ -algebra $A$ is a right $A$ -module $\hilbmod$ equipped with an $A$ -valued inner product $\langle-,-\rangle\colon \hilbmod \times \hilbmod \to A$ , i.e. a sesquilinear pairing satisfying \begin{eqnarray} \langle u,va\rangle & = & \langle u,v\rangle a \\ \langle u,v\rangle & = & \langle v,u\rangle^* \\ \langle v,v\rangle & \geq & 0, \mbox{ with\ } \langle v,v\rangle = 0
\mbox{ iff\ } v = 0, \end{eqnarray}for all $u,v \in \hilbmod$ and $a \in A$ . Note, positive definiteness is well-defined due to the notion of positivity for $C^*$ -algebras. The norm of an element $v \in \hilbmod$ is defined by $\norm{v} = \sqrt{\norm{\langle v,v\rangle}}$ .
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 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 $\hilbmod$ over a $C^*$ -algebra $B$ together with a *-homomorphism $\pi$ from a $C^*$ -algebra $A$ to $\End(\hilbmod)$ .
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"Hilbert module" is owned by mhale.
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Cross-references: *-homomorphism, Hilbert space, complex, complete, norm, well-defined, positive, pairing, inner product, right
There is 1 reference to this entry.
This is version 5 of Hilbert module, born on 2002-08-30, modified 2004-04-16.
Object id is 3401, canonical name is HilbertModule.
Accessed 5605 times total.
Classification:
| AMS MSC: | 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology ) |
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Pending Errata and Addenda
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