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finitely generated projective module (Definition)

Let $ R$ be a unital ring. A finitely generated projective right $ R$-module is of the form $ eR^n$, $ n \in \mathbb{N}$, where $ e$ is an idempotent in $ \mathop{\mathrm{End}}\nolimits _R(R^n)$.

Let $ A$ be a unital $ C^*$-algebra and $ p$ be a projection in $ \mathop{\mathrm{End}}\nolimits _A(A^n)$, $ n \in \mathbb{N}$. Then, $ \mathord{\mathcal{E}}= pA^n$ is a finitely generated projective right $ A$-module. Further, $ \mathord{\mathcal{E}}$ is a pre-Hilbert $ A$-module with ($ A$-valued) inner product

$\displaystyle \langle u,v \rangle = \sum_{i=1}^n u_i^* v_i, \quad u,v \in \mathord{\mathcal{E}}. $



"finitely generated projective module" is owned by mhale.
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See Also: Hilbert module

Other names:  finite projective module
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Cross-references: inner product, projection, unital, idempotent, right, finitely generated, unital ring
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This is version 3 of finitely generated projective module, born on 2003-02-26, modified 2003-11-18.
Object id is 4068, canonical name is FinitelyGeneratedProjectiveModule.
Accessed 3383 times total.

Classification:
AMS MSC: 16D40 (Associative rings and algebras :: Modules, bimodules and ideals :: Free, projective, and flat modules and ideals)
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