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Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces?

For the sum we have the notion of a direct integral, here.

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    I doubt there is such a thing. Out of curiosity: why would one want to have such a construction? I can see some (although not much) need for countable tensor products. The utility and applicability of such a construction eludes me. I ask this also in order to clarify your intentions: what kind of properties would you like this gadget to have? Do you want just *something* you could call a tensor product or do you have an application in mind?2011-07-26
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    Not really an application, but some ideas what it'd be. In the theory of stochastic processes, the $\sigma$ algebra often refine with continuous time. For certain processes, I'd think that this can be interpreted as such an infinite tensor product. I want a purely abstract definition here not involving the notion of stochastic processes.2011-07-26

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"Continuous tensor products" have been applied in certain areas such as quantum stochastic processes and quantum field theory, see for example the following paper by Arveson. I think that the definitions in Vershik and Tsilevich are more transparent (in this article the continuous tensor product is mainly referred to as a "factorization").

The basic object which possesses a continuous tensor product structure is the Fock space. However, one can find in the references of Vershik examples of non-Fock factorizations, see for example the following talk. A further application is in the representation theory of current algebras.

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    Okay, I stand corrected to a certain extent. However, I may be misreading what Arveson says, but isn't his point in section 4 that *product systems* may heuristically be thought of "continuous tensor products" but *"While this heuristic picture is often useful, one must be careful not to push it too far."*2011-07-26
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    I think that in the actual work by Averson (the reference I gave is just a lecture note) and also by Vershik (and also by others), the notion of continuous tensor product is rigorously defined.2011-07-26
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    Thanks! [These lecture notes](http://math.berkeley.edu/~arveson/Dvi/prodsys.pdf) (Def. 2.1) make the analogy still a bit clearer than the set of notes you linked to, I think. Very interesting.2011-07-26
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    I think Ron Blei's book has some info on the subject too.2011-07-26
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    I am actually suprised that the answer is yes! Thx.2011-07-27