Abstract
We construct a hereditarily indecomposable Banach space with dual space isomorphic to ℓ 1. Every bounded linear operator on this space is expressible as λI + K, with λ a scalar and K compact.
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Alencar, R., Aron, R.M. & Fricke, G., Tensor products of Tsirelson’s space. Illinois J. Math., 31 (1987), 17–23.
Alspach, D., The dual of the Bourgain–Delbaen space. Israel J. Math., 117 (2000), 239–259.
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.
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.
Argyros, S. A. & Deliyanni, I., Examples of asymptotic l 1 Banach spaces. Trans. Amer. Math. Soc., 349 (1997), 973–995.
Argyros, S. A. & Felouzis, V., Interpolating hereditarily indecomposable Banach spaces. J. Amer. Math. Soc., 13 (2000), 243–294.
Argyros, S. A. & Raikoftsalis, Th., The cofinal property of the reflexive indecomposable Banach spaces. To appear in Ann. Inst. Fourier (Grenoble).
Argyros, S. A. & Todorcevic, S., Ramsey Methods in Analysis. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser, Basel, 2005.
Argyros, S. A. & Tolias, A., Indecomposability and unconditionality in duality. Geom. Funct. Anal., 14 (2004), 247–282.
Aronszajn, N. & Smith, K. T., Invariant subspaces of completely continuous operators. Ann. of Math., 60 (1954), 345–350.
Bourgain, J., New Classes of \( {\mathcal{L}^p} \) -Spaces. Lecture Notes in Mathematics, 889. Springer, Berlin–Heidelberg, 1981.
Bourgain, J. & Delbaen, F., A class of special \( {\mathcal{L}_\infty } \) spaces. Acta Math., 145 (1980), 155–176.
Bourgain, J. & Pisier, G., A construction of \( {\mathcal{L}}_{\infty} \)-spaces and related Banach spaces. Bol. Soc. Brasil. Mat., 14 (1983), 109–123.
Dales, H. G., Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, 24. Oxford University Press, Oxford, 2000.
Daws, M. & Runde, V., Can \( {\mathcal{B}}(l^p) \) ever be amenable? Studia Math., 188 (2008), 151–174.
Emmanuele, G., Answer to a question by M. Feder about K(X, Y). Rev. Mat. Univ. Complut. Madrid, 6 (1993), 263–266.
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.
— On the invariant subspace problem for Banach spaces. Acta Math., 158 (1987), 213–313.
Ferenczi, V., Quotient hereditarily indecomposable Banach spaces. Canad. J. Math., 51 (1999), 566–584.
Gasparis, I., Strictly singular non-compact operators on hereditarily indecomposable Banach spaces. Proc. Amer. Math. Soc., 131 (2003), 1181–1189.
Gowers, W. T., A Banach space not containing c 0, l 1 or a reflexive subspace. Trans. Amer. Math. Soc., 344 (1994), 407–420.
— 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.
Gowers, W. T. & Maurey, B., The unconditional basic sequence problem. J. Amer. Math. Soc., 6 (1993), 851–874.
Grønbæk, N., Johnson, B. E. & Willis, G. A., Amenability of Banach algebras of compact operators. Israel J. Math., 87 (1994), 289–324.
Hagler, J., Some more Banach spaces which contain L 1. Studia Math., 46 (1973), 35–42.
Hagler, J. & Stegall, C., Banach spaces whose duals contain complemented subspaces isomorphic to (C[0, 1])*. J. Funct. Anal., 13 (1973), 233–251.
Haydon, R. G., Subspaces of the Bourgain–Delbaen space. Studia Math., 139 (2000), 275–293.
— Variants of the Bourgain–Delbaen construction. Unpublished conference talk, Caceres, 2006.
James, R. C., Uniformly non-square Banach spaces. Ann. of Math., 80 (1964), 542–550.
Johnson, B. E., Cohomology in Banach Algebras. Memoirs of the American Mathematical Society, 127. Amer. Math. Soc., Providence, RI, 1972.
Lewis, D. R. & Stegall, C., Banach spaces whose duals are isomorphic to l 1(Γ). J. Funct. Anal., 12 (1973), 177–187.
Lindenstrauss, J., Some open problems in Banach space theory. Séminaire Choquet. Initiation `a l’analyse, 15 (1975–1976), Exposé 18, 9 pp.
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.
Maurey, B., Banach spaces with few operators, in Handbook of the Geometry of Banach Spaces, Vol. 2, pp. 1247–1297. North-Holland, Amsterdam, 2003.
Maurey, B. & Rosenthal, H. P., Normalized weakly null sequence with no unconditional subsequence. Studia Math., 61 (1977), 77–98.
Pełczyński, A., On Banach spaces containing L 1(μ). Studia Math., 30 (1968), 231–246.
Read, C. J., A solution to the invariant subspace problem. Bull. London Math. Soc., 16 (1984), 337–401.
— A solution to the invariant subspace problem on the space l 1. Bull. London Math. Soc., 17 (1985), 305–317.
— Strictly singular operators and the invariant subspace problem. Studia Math., 132 (1999), 203–226.
Schlumprecht, T., An arbitrarily distortable Banach space. Israel J. Math., 76 (1991), 81–95.
Tarbard, M., Hereditarily indecomposable, separable \( {\mathcal{L}}_{\infty} \) spaces with ℓ 1 dual having few operators, but not very few operators. Preprint, 2010. arXiv:1011.4776 [math.FA].
Thorp, E. O., Projections onto the subspace of compact operators. Pacific J. Math., 10 (1960), 693–696.
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Argyros, S.A., Haydon, R.G. A hereditarily indecomposable \( {\mathcal{L}_{\infty}} \)-space that solves the scalar-plus-compact problem. Acta Math 206, 1–54 (2011). https://doi.org/10.1007/s11511-011-0058-y
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DOI: https://doi.org/10.1007/s11511-011-0058-y