# Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

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? – t.b. Jul 26 '11 at 10:54
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. – plusepsilon.de Jul 26 '11 at 12:06

## 1 Answer

"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." – t.b. Jul 26 '11 at 15:16
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. – David Bar Moshe Jul 26 '11 at 15:29
Thanks! These lecture notes (Def. 2.1) make the analogy still a bit clearer than the set of notes you linked to, I think. Very interesting. – t.b. Jul 26 '11 at 15:35
I think Ron Blei's book has some info on the subject too. – Mark Jul 26 '11 at 22:05
I am actually suprised that the answer is yes! Thx. – plusepsilon.de Jul 27 '11 at 5:36