K-theory for algebras of operators on Banach spaces

It is proved that, for each pair (m,n) of non-negative integers, there is a Banach space X for which the group K_0(B(X)) is isomorphic to m copies of the integers and the group K_1(B(X)) is isomorphic to n copies of the integers. Along the way we compute the K-groups of all closed ideals of operators contained in the ideal of strictly singular operators, and we derive some results about the existence of splittings of certain short exact sequences

Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space

We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space $E$ that we use was originally introduced by Read (J. London Math. Soc. 1989).Comment: to appear in C.R. Math. Acad. Sci. Pari

A weak*-topological dichotomy with applications in operator theory

Denote by $[0,\omega_1)$ the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let $C_0[0,\omega_1)$ be the Banach space of scalar-valued, continuous functions which are defined on $[0,\omega_1)$ and vanish eventually. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,\omega_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,\omega_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,\omega_1)$ can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of $C_0[0,\omega_1)$ 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 $C_0[0,\omega_1)$. As a consequence, we find that this ideal has a bounded left approximate identity, thus resolving 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 $C_0[0,\omega_1)$ and the operators acting on it.Comment: accepted to Transactions of the London Mathematical Societ

Extensions and the weak Calkin algebra of Read's Banach space admitting discontinuous derivations

Read produced the first example of a Banach space E such that the associated Banach algebra B(E) of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalize Read's main theorem about B(E) from which he deduced this conclusion, as well as the key technical lemmas that his proof relied on, by constructing a strongly split-exact sequence {0}→W(E)→B(E)→l2~→{0}, W(E) where W(E) denotes the ideal of weakly compact operators on E, while l2~ is the unitization of the Hilbert space l2, endowed with the zero product

Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

We show that for each of the following Banach spaces~$X$, the quotient algebra $\mathscr{B}(X)/\mathscr{I}$ has a unique algebra norm for every closed ideal $\mathscr{I}$ of $\mathscr{B}(X)\colon$ - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}$\quad and its dual,\quad $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}$, - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}\oplus c_0(\Gamma)$\quad and its dual, \quad $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}\oplus\ell_1(\Gamma)$,\quad for an uncountable cardinal number~$\Gamma$, - $X = C_0(K_{\mathcal{A}})$, the Banach space of continuous functions vanishing at infinity on the locally compact Mr\'{o}wka space~$K_{\mathcal{A}}$ induced by an uncountable, almost disjoint family~$\mathcal{A}$ of infinite subsets of~$\mathbb{N}$, constructed such that $C_0(K_{\mathcal{A}})$ admits "few operators". Equivalently, this result states that every homomorphism from~$\mathscr{B}(X)$ into a Banach algebra is continuous and has closed range. The key step in our proof is to show that the identity operator on a suitably chosen Banach space factors through every operator in $\mathscr{B}(X)\setminus\mathscr{I}$ with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest

Ideal structure of the algebra of bounded operators acting on a Banach space

We construct a Banach space Z such that the Banach algebra B(Z) of bounded operators on Z contains exactly four non-zero, proper closed ideals, including two maximal ideals. We then determine which kinds of approximate identities (bounded/left/right), if any, each of these four ideals contains, and we show that one of the two maximal ideals is generated as a left ideal by two operators, but not by a single operator, thus answering a question left open in our collaboration with Dales, Kochanek and Koszmider (Studia Math. 2013). In contrast, the other maximal ideal is not finitely generated as a left ideal. The Banach space Z is the direct sum of Argyros and Haydon's Banach space XAH which has very few operators and a certain subspace Y of XAH. The key property of Y is that every bounded operator from Y into XAH is the sum of a scalar multiple of the inclusion map and a compact operator.non-ze