Interactions

Given a C*-algebra B, a closed *-subalgebra A contained in 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.