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Maximal left ideals in Banach algebras

Let A be a Banach algebra, and let M be a maximal left ideal in A. It is easy to see that often M must be closed and that sometimes it can be dense, with codimension 1. It has been conjectured that these are the only two possibilities. This is true for many classes of Banach algebras, including all commutative ones. But we construct a counter-example: given n ∈ N, there is a Banach algebra with a dense maximal left ideal of codimension n. I do not know if we can find an example where the maximal left ideal has infinite codimension. This is joint work with M. Cabrera Garcoia and A. Rodriguez Palacios of Granada.