Recent results in the theory of infinite-dimensional Banach spaces
W. T. Gowers (University College London, England)
Many of the best known questions about infinite-dimensional Banach spaces are special cases of the following general question. If $E$ is a Banach space, $A$ is a given algebra of operators known to be continuous on $E$ and ${\scr S}$ is a given class of operators, must $B(E)$ contain an operator from ${\scr S}$? (All spaces and subspaces mentioned in this abstract are presumed infinite-dimensional and separable unless it is stated otherwise.) When $A$ is just the set of multiples of the identity, one is asking whether there are certain kinds of operators (for example, projections or isomorphisms to subspaces) admitted by a general Banach space.
In 1991 Maurey and the author independently constructed a Banach space $X$ which contains no subspace with an unconditional basis. (A Schauder basis $(x_n)^\infty_{n=1}$ is unconditional if, for every set $A\subset {\bf N}$ the projection $\sum^\infty_{n=1}a_nx_n\mapsto \sum_{n\in A}a_nx_n$ is continuous.) The space was a development of spaces constructed by Tsirelson [8] and Schlumprecht [7]. It was then observed by W. B. Johnson that $X$ is hereditarily indecomposable. That is, if $Y$ is a subspace of $X$ and $P\colon Y\to Y$ is a continuous projection then $PY$ is either finite-dimensional or finite-codimensional. In other words, no subspace of $X$ admits a non-trivial projection.
These results appear in [4]. It is shown that the space $X$ gives a fairly complete answer to the question about operators on a general space. Indeed, every operator on $X$ is of the form $\lambda I+S$, where $\lambda$ is a scalar, $I$ is the identity and $S$ is strictly singular. (This means that for no subspace $Y$ of $X$ is the restriction of $S$ to $Y$ an isomorphism.) This implies, using standard results about Fredholm operators, that $X$ is not isomorphic to any proper subspace of itself, answering the so-called hyperplane problem of Banach [1]. In general, it shows that the space of operators on a Banach space may be surprisingly small. It is not known whether there is a space for which the result above holds with ``strictly singular'' replaced by ``compact''.
Maurey and the author generalized the methods of [4] to solve problems associated with larger algebras $A$. For example, there is a space with an unconditional basis but no isomorphism to a proper subspace [2], a space with a basis such that the right shift is an isometry but there are no non-trivial projections [5], and a space $Z$ isomorphic to $Z\oplus Z\oplus Z$ but not to $Z\oplus Z$ [5].
A second question of Banach [1] turned out to be related to these results. he asked whether (separable) Hilbert space is characterized, up to isomorphism, by the property that it is isomorphic to every closed subspace of itself. The answer is yes, and it follows easily from three recent results. The first, due to Komorowski and Tomczak-Jaegermann [6], states that every Banach space contains either $l_2$ or a subspace without an unconditional basis. The second [3] is that every Banach space contains either a subspace with an unconditional basis or a hereditarily indecomposable subspace. The third [4] (easier than the other two) is that a hereditarily indecomposable space is isomorphic to no proper subspace of itself.
References
[1] S. Banach, ``Th\'eorie des operations lin\'eaires'', Warszawa, 1932.
[2] W. T. Gowers, {\it A solution to Banach's hyperplane problem}, Bull.\ L.M.S., (to appear).
[3] W. T. Gowers, {\it A new dichotomy for Banach spaces}, (preprint).
[4] W. T. Gowers and B. Maurey, {\it The unconditional basic sequence problem}, Jour.\ A.M.S. {\bf 6} (1993), 851--874.
[5] W. T. Gowers and B. Maurey, {\it Banach spaces with few operators}, preprint.
[6] R. Komorowski and N. Tomcz ak-Jaegermann, {\it Banach spaces without local unconditional structure}, preprint.
[7] T. Schlumprecht, {\it An arbitrarily distortable Banach space}, Israel J. Math., {\bf 76} (1991), 81--95.
[8] B. S. Tsirelson, {\it Not every Banach space contains $l_p$ or $c_0$}, Funct.\ Anal.\ Appl.\ {\bf 8} (1974), 138--141.