Banach spaces with a unique unconditional basis, up to permutation. (English) Zbl 0575.46011

Bourgain, J.; Casazza, P. G.; Lindenstrauss, J.; Tzafriri, L.

Mem. Am. Math. Soc. 322, 111 p. (1985).

The present Memoir contains a large amount of very interesting material concerning the questions which Banach spaces have a unique unconditional basis (u.b.), up to equivalence and permutation. This question is closely related to the well known result that \(c_ 0\), \(\ell_ 1\) and \(\ell_ 2\) are the only Banach spaces which, up to equivalence, have a unique normalized u.b.[see G. Köthe and O. Toeplitz, J. Reine Angew. Math. 171, 193–226 (1934; Zbl 0009.25704), J. Lindenstrauss and A. Pelczynski, Stud. Math. 29, 275–326 (1968; Zbl 0183.405)]; also, a first result of I. Edelstein and P. Wojtaszczyk [Stud. Math. 56, 263–276 (1976; Zbl 0362.46017)] which proves that the spaces \(c_ 0\oplus \ell_ 1\), \(c_ 0\oplus \ell_ 2\), \(\ell_ 1\oplus \ell_ 2\) and \(c_ 0\oplus \ell_ 1\oplus \ell_ 2\) have a unique normalized u.b., up to equivalence and permutation. Starting with some general properties of u.b. in finite direct sums of Banach spaces (thus, a new and simple proof of the result of Edelstein and Wojtaszczyk is given) the authors consider infinite direct sums of the sequence spaces \(c_ 0,\ell_ 1,\ell_ 2\) in sense of \(c_ 0\) or \(\ell_ p\), \(1\leq p\leq \infty\), which have inevitably surprising behaviours.
The material of this Memoir is divided into the following parts: 0. Introduction. 1. Unconditional bases of finite direct sums of Banach spaces. 2. Infinite direct sums of Hilbert spaces. 3. Infinite direct sums of \(\ell_ 1\)-spaces in sense of \(c_ 0\). Part I. 4. Infinite direct sums of \(\ell_ 1\)-spaces in sense of \(c_ 0\). Part II. 5. Infinite direct sums in this sense of \(\ell_ 2\). 6. Prime spaces. 7. Tsirelson’s space. 8. Complemented subspaces of \((\sum^{\infty}_{n=1}\oplus \ell^ n_ 2)_ 1\) and \((\sum^{\infty}_{n=1}\oplus \ell^ n_{\infty})_ 1\). 9. “Large” subspaces of \((\ell_ q\oplus \ell_ q\oplus... \oplus \ell_ q\oplus...)_ p\). 10. Complemented subspaces of \((\ell_ 2\oplus \ell_ 2\oplus...\oplus \ell_ 2\oplus...)_ 1\) and \((c_ 0\oplus c_ 0\oplus...\oplus c_ 0\oplus...)_ 1\). 11. Open problems.
The authors prove that the spaces \((\ell_ 2\oplus \ell_ 2\oplus...\oplus \ell_ 2\oplus...)\) in the sense of \(c_ 0\) and \(\ell_ 1\), \((\ell_ 1\oplus \ell_ 1\oplus...\oplus \ell_ 1\oplus...)_ 0\) and \((c_ 0\oplus c_ 0\oplus...\oplus c_ 0\oplus...)_ 1\) have a unique normalized u.b., up to equivalence and permutation, while the spaces \((\ell_ 1\oplus \ell_ 1\oplus...\oplus \ell_ 1\oplus...)_ 2\) and \((c_ 0\oplus c_ 0\oplus...\oplus c_ 0\oplus...)_ 2\) fail to have a unique u.b.up to equivalence and permutation.
A central result is obtained in Section 6: A prime space with a unique u.b., up to equivalence and permutation, is isomorphic to either \(\ell_ 1,\ell_ 2\), or \(c_ 0\). Also, the problem considered by the authors is solved for a Tsirelson type space.
In Sections 8–10, the authors discuss the question of uniqueness of u.b.for complemented subspaces of the spaces \((\ell_ 2\oplus \ell_ 2\oplus...\oplus \ell_ 2\oplus...)_ 1\) and \((c_ 0\oplus c_ 0\oplus...\oplus c_ 0\oplus...)_ 1\). In this way, the results of E. Odell type [see Isr. J. Math. 23, 353–367 (1976; Zbl 0333.46005)] concerning the classification of the complemented subspaces having u.b.are obtained.

Reviewer: T. Precupanu (Iaşi)

MSC:

46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B25	Classical Banach spaces in the general theory
46A45	Sequence spaces (including Köthe sequence spaces)

Keywords:

unique unconditional basis; finite direct sums of Banach spaces; infinite direct sums; sequence spaces; Prime spaces; Tsirelson’s space; Complemented subspaces; classification

Citations:

Zbl 0009.25704; Zbl 0183.405; Zbl 0362.46017; Zbl 0333.46005