Banach spaces with small spaces of operators. (English) Zbl 0876.46006

Gowers, W. T.; Maurey, B.

Math. Ann. 307, No. 4, 543-568 (1997).

For a certain class of algebras \({\mathcal A}\) we give a method for constructing Banach spaces \(X\) such that every operator on \(X\) is close to an operator in \({\mathcal A}\). This is used to produce spaces with a small amount of structure. We present several applications. Amongst them are constructions of a new prime Banach space, a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square.

Reviewer: B.Maurey (Paris)

MSC:

46B03	Isomorphic theory (including renorming) of Banach spaces
46B20	Geometry and structure of normed linear spaces
47A53	(Semi-) Fredholm operators; index theories
47A99	General theory of linear operators

Keywords:

constructions of a new prime Banach space; space isomorphic to its subspaces of codimension two but not to its hyperplanes; space isomorphic to its cube but not to its square