In this talk, we will show that the quotient of a
Banach algebra that acts on an L^p space cannot in
general be represented on an L^p space whenever p
is not equal to 2. This answers a 20 year old question
of Le Merdy, and should be contrasted with Junge's
partial (affirmative) result for subspaces of L^p spaces.
The construction of the (counter)example is not hard,
but the proof that it is indeed a counterexample relies
on our earlier study of Banach algebras generated by
invertible isometries of L^p spaces.
This is joint work with Hannes Thiel from the University
of Muenster, and a preprint containing the result can
be found at: http://arxiv.org/abs/1412.3985