Niels Jakob Laustsen

Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])

Let ω1 be the smallest uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C([0,ω1) have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).

A weak*-topological dichotomy with applications in operator theory

Denote by [0 ,ω 1) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C0[0 ,ω 1) be the Banach space of scalar-valued, continuous functions which are defined on [0 ,ω 1) and vanish eventually. We show that a weak ∗ compact subset of the dual space of C0[0 ,ω 1) is either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval [0 ,ω 1]. This dichotomy yields a unifying approach to most of the existing studies of the Banach space C0[0 ,ω 1) and the Banach algebra B(C0[0 ,ω 1)) of bounded, linear operators acting on it, and it leads to several new results, as well as to stronger versions of known ones. Specifically, we deduce that a Banach space which is a quotient of C0[0 ,ω 1) can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C0[0 ,ω 1) and a subspace of a Hilbert-generated Banach space; and we obtain several equivalent conditions describing the Loy–Willis ideal M , which is the unique maximal ideal of B(C0[0 ,ω 1)), including the following: an operator belongs to M if and only if it factors through the Banach space ( � α<ω1 C[0 ,α ])c0 . Among the consequences of these characterizations of M is that M has a bounded left approximate identity; this resolves a problem left open by Loy and Willis.

Maximal left ideals of the Banach algebra of bounded operators on a Banach space

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: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) 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 in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.

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 such that: - the Banach algebra B(E) of bounded, linear operators on E has a singular extension which splits algebraically, but it does not split strongly; - the homological bidimension of 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).

ON RING-THEORETIC (IN)FINITENESS OF BANACH ALGEBRAS OF OPERATORS ON BANACH SPACES

Let B ( X ) denote the Banach algebra of all bounded linear operators on a Banach space X . We show that B ( X ) is finite if and only if no proper, complemented subspace of X is isomorphic to X , and we show that B ( X ) is properly infinite if and only if X contains a complemented subspace isomorphic to X [oplus ] X . We apply these characterizations to find Banach spaces X 1 , X 2 , and X 3 such that B ( X 1 ) is finite, B ( X 2 ) is infinite, but not properly infinite, and B ( X 3 ) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces D 1 and D 2 such that B ( D 1 ) and B ( D 2 ) are infinite without being properly infinite, B ( D 1 ) has a continued bisection of the identity, and B ( D 2 ) has no continued bisection of the identity. Finally, we exhibit a unital

Uniqueness of the maximal ideal of operators on the ℓ p -sum of ℓ∞ n (n ∈ N ) for 1 < p< ∞

C^\ast

Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals

-algebra which is finite and has a continued bisection of the identity.

Banach spaces whose algebra of bounded operators has the integers as their K0-group

A recent result of Leung (Proceedings of the American Mathematical Society 2015) states that the Banach algebra B(X) of bounded, linear operators on the Banach space X which is the l1-direct sum of l∞n for n=1,2,... contains a unique maximal ideal. We show that the same conclusion holds true for the Banach space X which is the lp-direct sum of l∞n for n=1,2,... and its dual space X* whenever 1

A singular, admissible extension which splits algebraically, but not strongly, of the algebra of bounded operators on a Banach space

Denote by

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

[0,\omega_1)