# One-parameter semigroups of bimodules

Suppose M is a von Neumann algebra. Consider a monoidal category of bimodules over M. Here a bimodule is a Hilbert space with two normal representations of M. The monoidal structure is given by Connes' fusion. Alternatively we can take right W*-correspondences (right Hilbert W*-modules over M with a normal left action of M) together with the completed algebraic tensor product.

What can we say about one-parameter “semigroups” of such bimodules? More precisely, consider a family E of bimodules parametrized by a real number t > 0 such that E_s ⊗ E_t = E_{s+t}. Equality here denotes isomorphism. Does it have an “infinitesimal generator”? What is the “exponent” then? Do we need any “continuity” conditions to guarantee good properties of such “semigroups”? What kind of additional restrictions we can obtain on M provided that such a family exists?

It appears to me that such objects are known as “continuous tensor product systems”. Any references on this matter will be appreciated.

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If I am not mistaken, there is a conspicuous absence of results in any but the type I setting? I'd be very interested if any of what Orr Shalit said below had analogues, for example in the type $II_{1}$ case. –  Jon Bannon Jan 15 '11 at 21:11
@Jon: As far as I can tell, for any von Neumann algebra A and for any A-A-bimodule F there is a canonical example of a one-parameter semigroup of A-A-bimodules t↦G_t, where G_t is a categorification of exp(tF). All other examples that I am aware of have A=C (the algebra of complex numbers) and there are very few of them. –  Dmitri Pavlov Jan 15 '11 at 23:32

Such structures have been investigated at depth (welcome to the club!). Let me try to answer some of your questions.

1) First, let me suggest that you look at Bill Arveson's book on this subject, Noncommutative Dynamics and E-semigroups. Arveson is considered the pioneer of modern work on these structures, and you will enjoy his book.

2) An "infinitesemal generator" can mean various things, but in some sense, remarkably, it doesn't always exist. Arveson's book contains some treatment of this issue, known as the existence of type III examples, which are due to R. T. Powers and B. Tsirelson (there is also recet work of M. Izumi).

3) Do we need continuity conditions? Measurability conditions suffice. The condition is that the bundle {E_t}_{t>0} be isomorphic, as a measurable bundle of Hilbert spaces, to the trivial bundle (0,infinity) X H_0, with H_0 some fixed Hilbert space, plus compatibility of the measurable structure with addition, multiplication, etc.

4) Extensive research has been carried out also in the case where the E_t are Hilbert bimodules (C^*-correspondences). Search the ArXiv for works of Michael Skeide, or Paul Muhly and Baruch Solel.

5) I should also mention: these product systems arise naturally in the study of, give rise to, and are in a one-to-one correspondence with semigroups of *-endomorphisms on von Neumann algebras.

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