  We establish operator structure identities for quantum channels and their
error-correcting and private codes, emphasizing the complementarity
relationship between the two perspectives. Relevant structures include
correctable and private operator algebras, and operator spaces such as
multiplicative domains and nullspaces of quantum channels and their
complementary maps. For the case of privatizing to quantum states, we also
derive related dimension inequalities that may be viewed as operator algebra
uncertainty relations.
