Applications of some results of infinite-dimensional topology to the topological classification of operator images
Volume 387 / 2000
Abstract
This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T.~Banakh, “The topological and Borel classification of operator images” by T.~Banakh, T.~Dobrowolski and A.~Plichko, and “Operator images homeomorphic to ${\mit\Sigma}^\omega$” by T. Banakh. The articles summarize investigations that has been done by these authors for the past 10 years. All that started in the late 80s with the following question by T. Dobrowolski: {\it Is the topological type of an operator image fully determined by its Borel type?}\/ Let us recall that an {\it operator image} is a space of the form $TX$, where $T:X\to Y$ is a continuous linear operator between Fréchet spaces (operator images often appear in analysis and topology, for example, the space $C_{\rm p}^*(X)$ of bounded continuous functions on a countable space $X$ with the topology of pointwise convergence can be considered as an operator image of the Banach space $C(\beta X)$). In the early 90s T. Dobrowolski obtained a positive answer to the above question for operator images of low Borel complexity. Namely, he showed that every infinite-dimensional separable $\sigma$-complete operator image is homeomorphic to one of the spaces: $l^2$, ${\mit\Sigma}$, or ${\mit\Sigma}\times l^2$, where ${\mit\Sigma}$ is the linear hull of the standard Hilbert cube in the Hilbert space $l^2$ (a space is $\sigma$-{\it complete} if it is a countable union of closed completely-metrizable subspaces). The same result was proved independently by T.~Banakh. The topological structure of operator images of higher Borel complexity remained unclear. However, it was known that the topological type of $C_{\rm p}^*(X)$ is determined by its Borel type for the case of the second multiplicative Borel class. More precisely, each absolute $F_{\sigma\delta}$-space $C_{\rm p}^*(X)$ over a nondiscrete space $X$ is homeomorphic to ${\mit\Sigma}^\omega$. Thus the conjecture appeared: {\it The spaces $l^2$, ${\mit\Sigma}$, ${\mit\Sigma}\times l^2$ and ${\mit\Sigma}^\omega$ exhaust all possible topological types of infinite-dimensional operator images that are absolute $F_{\sigma\delta}$-spaces.} Numerous attempts to confirm this conjecture were unsuccessful (though many of those attempts lead to very fruitful developments in infinite-dimensional topology). Finally, in 1998, T.~Banakh found a counterexample to the above conjecture. The counterexample came from the study of the weak topology of the closed unit balls of Banach spaces. It turned out that the topological type of an operator image $TX$ depends much on the geometric properties of the Fréchet space $X$ as well as on the properties of the weak topology of $X$. This topic, which is of independent interest, is considered in detail in the first article of this volume, that is, “The topological classification of weak unit balls of Banach spaces”. An example of a pathological Banach space constructed in the last section of this article is applied in the remaining two articles,\break strictly devoted to studying operator images. The first of them, “The topological and Borel classification of operator images”, deals with some general questions in the area, and also with operator images of high Borel complexity, while the second one is restricted to the study of operator images homeomorphic to ${\mit\Sigma}^\omega$. We refer the reader to Introductions that the three articles start with for more detailed information on their contents.
