3 edited tags
2 added 140 characters in body

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.

1

# 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?