Hilbert module over a quantale $Q$ or a Hilbert $Q$-module is an analogue of a Hilbert module in the setup of quantales and the inner product is valued in $Q$.
Hilbert $Q$-modules have been introduced by Paseka.
In the special case of locale $Q=B$, the Hilbert $B$-modules are studied geometrically in
Pedro Resende, Elias Rodrigues, Sheaves as modules, Appl. Categ. Struct. 18:199–217 (2010) arxiv/0711.4401 doi
Pedro Resende, Groupoid sheaves as Hilbert modules, J. Pure Appl. Algebra 216 (2012) 41-70 arxiv/0807.4848
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