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∗aS = H(a)S∗S and SaS∗ = 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. © 2006 Elsevier Inc. All rights reserved.