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

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.

### 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)

### Citations:

Zbl 0009.25704; Zbl 0183.405; Zbl 0362.46017; Zbl 0333.46005
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