arXiv subjects
Publications and source records attributed to Niels Jakob Laustsen.
We present two new examples of reflexive Banach spaces $X$ for which $\mathscr{B}(X)$ is not a Grothendieck space, namely $X = T$ (the Tsirelson space) and $X = B_p$ (the $p^{\text{th}}$ Baernstein space) for $p\in(1,\infty)$.
Read produced the first example of a Banach space $E_{\text{R}}$ such that the associated Banach algebra $\mathscr{B}(E_{\text{R}})$ of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalise Read's main theorem about $\mathscr{B}(E_{\text{R}})$ 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} --> $\mathscr{W}(E_{\text{R}})$ --> $\mathscr{B}(E_{\text{R}})$--> $\tilde{\ell_2}$-->{0}, where $\mathscr{W}(E_{\text{R}})$ denotes the ideal of weakly compact operators on $E_{\text{R}}$, while $\tilde{\ell_2}$ is the unitization of the Hilbert space $\ell_2$, endowed with the zero product.
Let $E$ be the Banach space constructed by Read (J. London Math. Soc. 1989) such that the Banach algebra $\mathscr{B}(E)$ of bounded operators on $E$ admits a discontinuous derivation. We show that $\mathscr{B}(E)$ has a singular, admissible extension which splits algebraically, but does not split strongly. This answers a natural question going back to the work of Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), and complements recent results of Laustsen and Skillicorn (C. R. Math. Acad. Sci. Paris, to appear).
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).
We construct a Banach space $Z$ such that the lattice of closed two-sided ideals of the Banach algebra $\mathscr{B}(Z)$ of bounded operators on $Z$ is as follows: $$ \{0\}\subset \mathscr{K}(Z)\subset\mathscr{E}(Z) \raisebox{-.5ex}% {\ensuremath{\overset{\begin{turn}{30}$\subset$\end{turn}}% {\begin{turn}{-30}$\subset$\end{turn}}}}\!\!% \begin{array}{c}\mathscr{M}_1\\[1mm]\mathscr{M}_2\end{array}\!\!\!% \raisebox{-1.25ex}% {\ensuremath{\overset{\raisebox{1.25ex}{\ensuremath{\begin{turn}{-30}$\subset$\end{turn}}}}% {\raisebox{-.25ex}{\ensuremath{\begin{turn}{30}$\subset$\end{turn}}}}}}\,\mathscr{B}(Z) $$ We then determine which kinds of approximate identities (bounded/left/right), if any, each of the four non-trivial closed ideals of $\mathscr{B}(Z)$ contain, and we show that the maximal ideal $\mathscr{M}_1$ 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 (\emph{Studia Math.} 2013). In contrast, the other maximal ideal $\mathscr{M}_2$ is not finitely generated as a left ideal. The Banach space $Z$ is the direct sum of Argyros and Haydon's Banach space $X_{\text{AH}}$ which has very few operators and a certain subspace $Y$ of $X_{\text{AH}}$. The key property of~$Y$ is that every bounded operator from $Y$ into $X_{\text{AH}}$ is the sum of a scalar multiple of the inclusion mapping and a compact operator.
A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_p}$ and $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_1^n\bigr)_{\ell_p}$ whenever $p\in(1,\infty)$.
Denote by $[0,ω_1)$ the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let $C_0[0,ω_1)$ be the Banach space of scalar-valued, continuous functions which are defined on $[0,ω_1)$ and vanish eventually. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,ω_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,ω_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,ω_1)$ can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of $C_0[0,ω_1)$ 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 $C_0[0,ω_1)$. 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 $C_0[0,ω_1)$ and the operators acting on it.
Let $X$ and $Y$ be Banach spaces such that the ideal of operators which factor through $Y$ has codimension one in the Banach algebra $\mathscr{B}(X)$ of all bounded operators on $X$, and suppose that $Y$ contains a complemented subspace which is isomorphic to $Y\oplus Y$ and that $X$ is isomorphic to $X\oplus Z$ for every complemented subspace $Z$ of $Y$. Then the $K_0$-group of $\mathscr{B}(X)$ is isomorphic to the additive group $\mathbb{Z}$ of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that $K_0(\mathscr{B}(C([0,ω_1])))\cong\mathbb{Z}$, where $C([0,ω_1])$ denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal $ω_1$, endowed with the order topology.
Denote by $[0,ω_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,α]$ for $α$ countable, and consider the Banach spaces $C_0[0,ω_1)$ and $C_0(L_0)$ consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces $[0,ω_1)$ and $L_0$, respectively, and which vanish eventually. Our main result states that a bounded operator $T$ between any pair of these two Banach spaces fixes a copy of $C_0(L_0)$ if and only if the identity operator on $C_0(L_0)$ factors through $T$, if and only if the Szlenk index of $T$ is uncountable. This implies that the set $\mathscr{S}_{C_0(L_0)}(C_0(L_0))$ of $C_0(L_0)$-strictly singular operators on $C_0(L_0)$ is the unique maximal ideal of the Banach algebra $\mathscr{B}(C_0(L_0))$ of all bounded operators on $C_0(L_0)$, and that $\mathscr{S}_{C_0(L_0)}(C_0[0,ω_1))$ is the second-largest proper ideal of $\mathscr{B}(C_0[0,ω_1))$. Moreover, it follows that the Banach space $C_0(L_0)$ is primary and complementably homogeneous.
We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ 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: (i) Question (Q1) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (Q2) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (Q2) is positive for many, but not all, Banach spaces.
Let $ω_1$ be the first uncountable ordinal. By a result of Rudin, bounded operators on the Banach space $C([0,ω_1])$ have a natural representation as $[0,ω_1]\times 0,ω_1]$-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on $[0,ω_1]$ defines a maximal ideal of codimension one in the Banach algebra $\mathscr{B}(C([0,ω_1]))$ of bounded operators on $C([0,ω_1])$. We give a coordinate-free characterization of this ideal and deduce from it that $\mathscr{B}(C([0,ω_1]))$ 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 $\mathscr{B}(C([0,ω_1]))$.
A theorem of Giesy and James states that $c_0$ is finitely representable in James' quasi-reflexive Banach space $J_2$. We extend this theorem to the $p$th quasi-reflexive James space $J_p$ for each $p \in (1,\infty)$. As an application, we obtain a new closed ideal of operators on $J_p$, namely the closure of the set of operators that factor through the complemented subspace $(\ell_\infty^1 \oplus \ell_\infty^2 \oplus...\oplus \ell_\infty^n \oplus...)_{\ell_p}$ of $J_p$.
We prove two dichotomy theorems about sequences of operators into $L_1$ given by random matrices. In the second theorem we assume that the entries of each random matrix form a sequence of independent, symmetric random variables. Then the corresponding sequence of operators either uniformly factor the identity operators on $\ell_1^k$ $(k\in\mathbb N$) or uniformly approximately factor through $\mathrm{c}_0$. The first theorem has a slightly weaker conclusion still related to factorization properties but makes no assumption on the random matrices. Indeed, it applies to operators defined on an arbitrary sequence of Banach spaces. These results provide information on the closed ideal structure of the Banach algebra of all operators on the space $\big(\bigoplus_{n=1}^\infty\ell_1^n \big)_{\mathrm{c}_0}$.
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.