  We examine the question of whether the Calkin algebra of a Banach space must
have a unique complete algebra norm. We present a survey of known results, and
make the observation that a recent Banach space construction of Argyros and
Motakis (preprint, 2015) provides the first negative answer. The parallel
question for the weak Calkin algebra also has a negative answer; we demonstrate
this using a Banach space of Read (J. London Math. Soc. 1989).
