A supplement to my paper on real and complex operator ideals

In this article, we propose a $p$-adic analogue of complex Hilbert space and consider generalizations of some well-known theorems from functional analysis and the basic study of operators on Hilbert spaces. We compute the $K$-theory of the analogue of the algebra of compact operators and the algebra of all bounded operators. This article contains a survey on results from the thesis of the first author.

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X \neq \{0\}$ and $\mathfrak{A}_Y \neq \{0\}$, then $\mathfrak{A}_{X \oplus Y}$ has at least $2$ closed ideals, (ii) there are closed subspaces $X \subset \ell^p$ for $4 < p < \infty$...

The Gleason-Kahane-\.Zelazko theorem states that a linear functional on a Banach algebra that is non-zero on invertible elements is necessarily a scalar multiple of a character. Recently this theorem has been extended to certain Banach function spaces that are not algebras. In this article we present a brief survey of these extensions.

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck $Z_2$ space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of K\"othe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant K\"othe fun...

We consider similarity transformations of a perturbed linear operator $A-B$ in a complex Banach space $\mathcal{X}$, where the unperturbed operator $A$ is a generator of a Banach $L_1(\mathbb{R})$-module and the perturbation operator $B$ is a bounded linear operator. The result of the transformation is a simpler operator $A-B_0$. For example, if $A$ is a differentiation operator and $B$ is an operator of multiplication by an operator-valued function, then $B_0$ is an operator...

In this note, we consider the smallest submaximal space structure {\mu}(X) on a Banach space X. We derive a characterization of {\mu}(X) up to complete isometric isomorphism in terms of a universal property. Also, we show that an injective Banach space has a unique submaximal space structure and we explore some duality relations of {\mu}-spaces. Key Words: operator spaces, maximal operator spaces, submaximal spaces, {\mu}- spaces.

What is an adequate extension of an operator ideal I to the polynomial and multilinear settings? This question motivated the appearance of the concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando el al. We propose a different approach by considering pairs (U_{k},M_{k})_{k=1}^{\infty}, where (U_{k})_{k=1}^{\infty} is a polynomial ideal and (M_{k})_{k=1}^{\infty} is a multi-ideal, instead of...

The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a systematic exposition of this theory; a reference tool for `noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.

It is well known that weakly $p$-summable sequences in a Banach space $E$ are associated to bounded operators from $\ell_{p^*}$ to $E$, and unconditionally $p$-summable sequences in $E$ are associated to compact operators from $\ell_{p^*}$ to $E$. Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, a...

In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admit...