The Banach space \(D(0,1)\) is primary. (English) Zbl 1101.46006
A Banach space \(X\) is said to be primary if whenever \(X = Y \oplus Z\), then either \(Y\) or \(Z\) is isomorphic to \(X\). It is well-known that \(C[0,1]\), \(c_0\), \(\ell_p\) \((1 \leq p \leq \infty)\) and \(L^p [0,1]\) \((1 < p < \infty)\) are primary. Other primary Banach spaces have been obtained by various authors [see P. G.Casazza, Isr.J.Math.26, 294–305 (1977; Zbl 0344.46045); P. G.Casazza, C. A.Kottman and B.–L.Lin, Can.J.Math.29, 856–873 (1977; Zbl 0338.46017); D. Alspach, P. Enflo and E. Odell, Stud.Math.60, 79–90 (1977; Zbl 0343.46017); P. Billard, Stud.Math.62, 143–162 (1978; Zbl 0416.46015); M. Capon, Isr.J.Math.36, 346–364 (1980; Zbl 0445.46013); Isr.J.Math.42, 87–98 (1982; Zbl 0491.46012); Proc.Lond.Math.Soc., III. Ser.45, 113–130 (1982; Zbl 0486.46015); Trans.Am.Math.Soc.276, 431–487 (1983; Zbl 0511.46012); J. Bourgain, Isr.J.Math.45, 329–336 (1983; Zbl 0551.46031); A. D.Andrew, Pac.J.Math.108, 3–17 (1983; Zbl 0535.46011); L. Drewnowski and J. W.Roberts, Proc.Am.Math.Soc.112, No. 4, 949–957 (1991; Zbl 0758.46022); L. Drewnowski, Rev.Mat.Univ.Complutense Madr.2, No. Suppl., 119–127 (1989; Zbl 0734.46012); G. Blower, Bull.Lond.Math.Soc.22, No. 2, 176–182 (1990; Zbl 0664.46052); P. F. X.Müller, Ill.J.Math.38, No. 4, 554–573 (1994; Zbl 0824.46027)]).
In the paper under review, it is shown that \(D(0,1)\) is primary, where \(D(0,1)\) is the space of all scalar functions from \([0,1)\) which are right continuous at each point of \([0,1)\) and have left limits at each point of \((0,1]\), equipped with the uniform convergence topology. This space is isometrically isomorphic to the space of all continuous scalar functions on the two arrows space \(L_0\) and the proof uses the topological properties of \(L_0\), as well as Pełczyński’s decomposition method. In the first step of the proof, the author shows that for every continuous linear operator \(J \colon D(0,1) \to D(0,1)\) with nonseparable range, there exists a subspace \(Y\) of \(D(0,1)\) which is isomorphic to \(D(0,1)\) and for which \(J_{\mid Y}\) is an isomorphism.
In the paper under review, it is shown that \(D(0,1)\) is primary, where \(D(0,1)\) is the space of all scalar functions from \([0,1)\) which are right continuous at each point of \([0,1)\) and have left limits at each point of \((0,1]\), equipped with the uniform convergence topology. This space is isometrically isomorphic to the space of all continuous scalar functions on the two arrows space \(L_0\) and the proof uses the topological properties of \(L_0\), as well as Pełczyński’s decomposition method. In the first step of the proof, the author shows that for every continuous linear operator \(J \colon D(0,1) \to D(0,1)\) with nonseparable range, there exists a subspace \(Y\) of \(D(0,1)\) which is isomorphic to \(D(0,1)\) and for which \(J_{\mid Y}\) is an isomorphism.
Reviewer: Daniel Li (Lens)
MSC:
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46B25 | Classical Banach spaces in the general theory |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
