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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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    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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    I am actually suprised that the answer is yes! Thx.2011-07-27