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A hereditarily indecomposable \(\mathcal L_{\infty}\)-space that solves the scalar-plus-compact problem. (English) Zbl 1223.46007

The scalar-plus-compact problem asks whether there exists an infinite dimensional Banach space \(X\) such that every bounded linear operator \(T: X\to X\) has the form \(T=\lambda\,\text{Id}+K\) with a compact operator \(K\). Informally, one says that such a Banach space has “very few operators”. This long-standing difficult problem is solved in the positive in the paper at hand.
Hitherto, the strongest result in this direction was the theorem by W. T. Gowers and B. Maurey [“The unconditional basic sequence problem”, J. Am.Math.Soc.6, No. 4, 851–874 (1993; Zbl 0827.46008)] that the hereditarily indecomposable spaces constructed in their paper admit “few operators” in the sense that every bounded linear operator \(T: X\to X\) has the form \(T=\lambda\,\text{Id}+S\) with a strictly singular operator \(S\). It is known, though, that the Gowers-Maurey spaces definitely do not have very few operators.
The authors’ construction of a space \({\mathfrak X}_K\) with very few operators uses ideas from the Gowers-Maurey paper, but also another likewise highly nontrivial ingredient, namely the Bourgain-Delbaen construction of “exotic” \({\mathcal L}_\infty\)-spaces; cf.[J. Bourgain and F. Delbaen, “A class of special \({\mathcal L}_\infty\) spaces”, Acta Math.145, 155–176 (1980; Zbl 0466.46024)]. Indeed, the authors’ original intention was to “merely” produce a hereditarily indecomposable space whose dual is isomorphic to \(\ell_1\) and hence very far from this class. (Let us recall that a Banach space \(X\) is hereditarily indecomposable, HI for short, if whenever a closed subspace \(Y\subset X\) is decomposed into a direct sum \(Y=Y_1\oplus Y_2\), then \(Y_1\) or \(Y_2\) is finite dimensional. In particular, \(X\) does not contain an unconditional basic sequence.)
In order to obtain an HI predual of \(\ell_1\), the authors generalise the Bourgain-Delbaen construction in that they base their approach on mixed Tsirelson or Schlumprecht type spaces where originally \(\ell_p\)-spaces were used.
This alone does not guarantee that the resulting space has very few operators; indeed, recent work by M. Tarbard [“Hereditarily indecomposable, separable \({\mathcal{L}}_\infty\) spaces with \(\ell_1\) dual having few operators, but not very few operators,” available from arXiv:1011.4776] gives an example of an HI \({\mathcal L}^\infty\)-space with few, but not very few operators. The authors consider it a piece of serendipity that special features of the Bourgain-Delbaen method allow them to complete their construction of \({\mathfrak X}_K\). The construction itself is highly demanding and technical; suffice it to say that a key notion is, as in the case of the Gowers-Maurey spaces, that of a rapidly increasing sequence of which three important types are identified. The space \({\mathfrak X}_K\) is also shown to be saturated with reflexive subspaces having HI duals.
It is clear from the definition, respectively follows from known results, that a Banach space \(X\) with very few operators also has the following properties: the space of compact operators from \(X\) to \(X\) is complemented in the space of bounded operators \(L(X)\); if one assumes complex scalars, every bounded linear operator on \(X\) has an invariant subspace; the algebra \(L(X)\) is amenable; if \(X^*\) is separable, then \(L(X)\) is separable as well. The space \({\mathfrak X}_K\) – before long to be called Argyros-Haydon space – is also the first (separable) infinite dimensional example of a Banach space with any of the above properties.
As usual with nontrivial constructions of Banach spaces (or indeed any mathematical objects), one has to choose certain parameters and thus even obtains a whole family of examples. A by-product of the authors’ work is the following: There exist continuum many separable Banach spaces \(X_\alpha\) such that, for \(\alpha\neq \beta\), every bounded linear operator from \(X_\alpha\) to \(X_\beta\) is compact. These Banach spaces can be described as being “very incomparable”.
In the last section, the authors pose some open problems and comment on recent, as yet unpublished work. For example, they mention joint work with D. Freeman, E. Odell, Th. Raikoftsalis, Th. Schlumprecht and D. Zisimopoulou showing, in particular, that every separable reflexive space embeds into an \({\mathcal L}_\infty\)-space with very few operators.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B28 Spaces of operators; tensor products; approximation properties
47A15 Invariant subspaces of linear operators
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[1] Alencar, R., Aron, R.M. & Fricke, G., Tensor products of Tsirelson’s space. Illinois J. Math., 31 (1987), 17–23. · Zbl 0587.46016
[2] Alspach, D., The dual of the Bourgain–Delbaen space. Israel J. Math., 117 (2000), 239–259. · Zbl 0956.46017 · doi:10.1007/BF02773572
[3] Androulakis, G., Odell, E., Schlumprecht, T. & Tomczak-Jaegermann, N., On the structure of the spreading models of a Banach space. Canad. J. Math., 57 (2005), 673–707. · Zbl 1090.46004 · doi:10.4153/CJM-2005-027-9
[4] Androulakis, G. & Schlumprecht, T., Strictly singular, non-compact operators exist on the space of Gowers and Maurey. J. London Math. Soc., 64 (2001), 655–674. · Zbl 1015.46007 · doi:10.1112/S0024610701002769
[5] Argyros, S. A. & Deliyanni, I., Examples of asymptotic l 1 Banach spaces. Trans. Amer. Math. Soc., 349 (1997), 973–995. · Zbl 0869.46002 · doi:10.1090/S0002-9947-97-01774-1
[6] Argyros, S. A. & Felouzis, V., Interpolating hereditarily indecomposable Banach spaces. J. Amer. Math. Soc., 13 (2000), 243–294. · Zbl 0956.46014 · doi:10.1090/S0894-0347-00-00325-8
[7] Argyros, S. A. & Raikoftsalis, Th., The cofinal property of the reflexive indecomposable Banach spaces. To appear in Ann. Inst. Fourier (Grenoble). · Zbl 1253.46009
[8] Argyros, S. A. & Todorcevic, S., Ramsey Methods in Analysis. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser, Basel, 2005. · Zbl 1092.46002
[9] Argyros, S. A. & Tolias, A., Indecomposability and unconditionality in duality. Geom. Funct. Anal., 14 (2004), 247–282. · Zbl 1069.46002 · doi:10.1007/s00039-004-0464-9
[10] Aronszajn, N. & Smith, K. T., Invariant subspaces of completely continuous operators. Ann. of Math., 60 (1954), 345–350. · Zbl 0056.11302 · doi:10.2307/1969637
[11] Bourgain, J., New Classes of $ {\(\backslash\)mathcal{L}\^p} $ -Spaces. Lecture Notes in Mathematics, 889. Springer, Berlin–Heidelberg, 1981. · Zbl 0476.46020
[12] Bourgain, J. & Delbaen, F., A class of special $ {\(\backslash\)mathcal{L}_\(\backslash\)infty } $ spaces. Acta Math., 145 (1980), 155–176. · Zbl 0466.46024 · doi:10.1007/BF02414188
[13] Bourgain, J. & Pisier, G., A construction of $ {\(\backslash\)mathcal{L}}_{\(\backslash\)infty} $ -spaces and related Banach spaces. Bol. Soc. Brasil. Mat., 14 (1983), 109–123. · Zbl 0586.46011 · doi:10.1007/BF02584862
[14] Dales, H. G., Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, 24. Oxford University Press, Oxford, 2000. · Zbl 0981.46043
[15] Daws, M. & Runde, V., Can $ {\(\backslash\)mathcal{B}}(l\^p) $ ever be amenable? Studia Math., 188 (2008), 151–174. · Zbl 1145.47056 · doi:10.4064/sm188-2-4
[16] Emmanuele, G., Answer to a question by M. Feder about K(X, Y). Rev. Mat. Univ. Complut. Madrid, 6 (1993), 263–266.
[17] Enflo, P., On the invariant subspace problem in Banach spaces, in Séminaire Maurey–Schwartz (1975–1976), Espaces L p , applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14–15, 7 pp. Centre Math., École Polytech., Palaiseau, 1976.
[18] – On the invariant subspace problem for Banach spaces. Acta Math., 158 (1987), 213–313. · Zbl 0663.47003 · doi:10.1007/BF02392260
[19] Ferenczi, V., Quotient hereditarily indecomposable Banach spaces. Canad. J. Math., 51 (1999), 566–584. · Zbl 0934.46015 · doi:10.4153/CJM-1999-026-4
[20] Gasparis, I., Strictly singular non-compact operators on hereditarily indecomposable Banach spaces. Proc. Amer. Math. Soc., 131 (2003), 1181–1189. · Zbl 1019.46006 · doi:10.1090/S0002-9939-02-06657-1
[21] Gowers, W. T., A Banach space not containing c 0, l 1 or a reflexive subspace. Trans. Amer. Math. Soc., 344 (1994), 407–420. · Zbl 0811.46014
[22] – A remark about the scalar-plus-compact problem, in Convex Geometric Analysis (Berkeley, CA, 1996), Math. Sci. Res. Inst. Publ., 34, pp. 111–115. Cambridge Univ. Press, Cambridge, 1999.
[23] Gowers, W. T. & Maurey, B., The unconditional basic sequence problem. J. Amer. Math. Soc., 6 (1993), 851–874. · Zbl 0827.46008 · doi:10.1090/S0894-0347-1993-1201238-0
[24] Grønbæk, N., Johnson, B. E. & Willis, G. A., Amenability of Banach algebras of compact operators. Israel J. Math., 87 (1994), 289–324. · Zbl 0806.46058 · doi:10.1007/BF02773000
[25] Hagler, J., Some more Banach spaces which contain L 1. Studia Math., 46 (1973), 35–42. · Zbl 0251.46023
[26] Hagler, J. & Stegall, C., Banach spaces whose duals contain complemented subspaces isomorphic to (C[0, 1])*. J. Funct. Anal., 13 (1973), 233–251. · Zbl 0265.46019 · doi:10.1016/0022-1236(73)90033-5
[27] Haydon, R. G., Subspaces of the Bourgain–Delbaen space. Studia Math., 139 (2000), 275–293. · Zbl 0967.46013
[28] – Variants of the Bourgain–Delbaen construction. Unpublished conference talk, Caceres, 2006.
[29] James, R. C., Uniformly non-square Banach spaces. Ann. of Math., 80 (1964), 542–550. · Zbl 0132.08902 · doi:10.2307/1970663
[30] Johnson, B. E., Cohomology in Banach Algebras. Memoirs of the American Mathematical Society, 127. Amer. Math. Soc., Providence, RI, 1972. · Zbl 0256.18014
[31] Lewis, D. R. & Stegall, C., Banach spaces whose duals are isomorphic to l 1({\(\Gamma\)}). J. Funct. Anal., 12 (1973), 177–187. · Zbl 0252.46021 · doi:10.1016/0022-1236(73)90022-0
[32] Lindenstrauss, J., Some open problems in Banach space theory. Séminaire Choquet. Initiation ‘a l’analyse, 15 (1975–1976), Exposé 18, 9 pp.
[33] Lomonosov, V. I., Invariant subspaces of the family of operators that commute with a completely continuous operator. Funktsional. Anal. i Prilozhen., 7 (1973), 55–56 (Russian); English translation in Funct. Anal. Appl. 7 (1973), 213–214.
[34] Maurey, B., Banach spaces with few operators, in Handbook of the Geometry of Banach Spaces, Vol. 2, pp. 1247–1297. North-Holland, Amsterdam, 2003. · Zbl 1044.46011
[35] Maurey, B. & Rosenthal, H. P., Normalized weakly null sequence with no unconditional subsequence. Studia Math., 61 (1977), 77–98. · Zbl 0357.46025
[36] Pełczyński, A., On Banach spaces containing L 1({\(\mu\)}). Studia Math., 30 (1968), 231–246. · Zbl 0159.18102
[37] Read, C. J., A solution to the invariant subspace problem. Bull. London Math. Soc., 16 (1984), 337–401. · Zbl 0566.47003 · doi:10.1112/blms/16.4.337
[38] – A solution to the invariant subspace problem on the space l 1. Bull. London Math. Soc., 17 (1985), 305–317. · Zbl 0574.47006 · doi:10.1112/blms/17.4.305
[39] – Strictly singular operators and the invariant subspace problem. Studia Math., 132 (1999), 203–226. · Zbl 0929.47004
[40] Schlumprecht, T., An arbitrarily distortable Banach space. Israel J. Math., 76 (1991), 81–95. · Zbl 0796.46007 · doi:10.1007/BF02782845
[41] Tarbard, M., Hereditarily indecomposable, separable $ {\(\backslash\)mathcal{L}}_{\(\backslash\)infty} $ spaces with 1 dual having few operators, but not very few operators. Preprint, 2010. arXiv:1011.4776 [math.FA]. · Zbl 1257.46013
[42] Thorp, E. O., Projections onto the subspace of compact operators. Pacific J. Math., 10 (1960), 693–696. · Zbl 0119.31904 · doi:10.2140/pjm.1960.10.693
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