Some C∗C∗-algebras which are coronas of non-C∗C∗-Banach algebras

We present results and motivating problems in the study of commutants of hermitian nn-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗C∗-algebras which arise in this context as coronas of non-C∗C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.