interactions

Given a C ∗ -algebra B , a closed * -subalgebra A ⊆ B , and a partial isometry S in B which interacts with A in the sense that S ∗ a S = H ( a ) S ∗ S and S a S ∗ = V ( a ) SS ∗ , where V and H are positive linear operators on A , we derive a few properties which V and H are forced to satisfy. Removing B and S from the picture we define an interaction as being a pair of maps ( V , H ) satisfying the derived properties. Starting with an abstract interaction ( V , H ) over a C ∗ -algebra A we construct a C ∗ -algebra B containing A and a partial isometry S whose interaction with A follows the above rules. We then discuss the possibility of constructing a covariance algebra from an interaction. This turns out to require a generalization of the notion of correspondences (also known as Pimsner bimodules) which we call a generalized correspondence . Such an object should be seen as an usual correspondence, except that the inner-products need not lie in the coefficient algebra. The covariance algebra is then defined using a natural generalization of Pimsner's construction of the celebrated Cuntz–Pimsner algebras.