The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property?
The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101].
Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists.
The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question.
We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications.
The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators.