Denote by [0,w₁] the compact Hausdorff space consisting of all ordinals not exceeding the first uncountable ordinal w₁, equipped with the order topology. We show that the K-groups of the Banach algebra $B(C[0,w₁]) of bounded operators on the Banach space C[0,w₁] of complex-valued, continuous functions on [0,w₁] are given by K_0(B(C[0,w₁]))=Z and K_1(B(C[0,w₁])) = {0}.

Denote by K the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C be the Banach space of scalar-valued, continuous functions which are defined on K and vanish eventually. We show that a weakly* compact subset of the dual space of C is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval [0,w₁] Using this result, we deduce that a Banach space which is a quotient of C can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C and a subspace of a Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent conditions describing the Loy--Willis ideal, which is the unique maximal ideal of the Banach algebra of bounded, linear operators on C. As a consequence, we find that this ideal has a bounded left approximate identity, thus solving a problem left open by Loy and Willis, and we give new proofs, in some cases of stronger versions, of several known results about the Banach space C and the operators acting on it.

We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E:

1. Does B(E) always contain a maximal left ideal which is not finitely generated?
2. Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} (*) for some non-zero x E?

Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the second question would imply a positive answer to the first. Our main results are:

• Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces;
• Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E);
• the answer to Question (II) is positive for many, but not all, Banach spaces.

We prove that the use of the Continuum Hypothesis in some results of Drewnowski and Roberts concerning the Banach space l_∞/c₀ cannot be avoided. In particular, we prove that in the Cohen model, l_∞(c₀(c)) does not embed isomorphically into l_∞/c₀ where c is the cardinality of the continuum. It follows that consistently l_∞/c₀ is not isomorphically of the form l_∞(X) for any Banach space X.

We construct a continuous image of a Radon-Nikodym compact space which is not Radon-Nikodym compact, solving the problem posed in the 80ties by Isaac Namioka.

Accepted to Duke Mathematical Journal
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We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has l-infinity as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.

Accepted to Algebra Universalis
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In the context of classical associations between classes of Banach spaces and classes of compact Hausdorff spaces we survey known results and open questions concerning the existence and nonexistence of universal Banach spaces and of universal compact spaces in various classes. This gives quite a complex network of interrelations which quite often depend on additional set-theoretic assumptions.

We construct an infinite dimensional Banach space of continuous functions C(K) such that every one-to-one operator on C(K) is onto.

Accepted to Bulletin of the London Mathematical Society
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