# Hereditarily finitely decomposable Banach spaces

Studia Mathematica (1997)

- Volume: 123, Issue: 2, page 135-149
- ISSN: 0039-3223

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topPerenczi, V.. "Hereditarily finitely decomposable Banach spaces." Studia Mathematica 123.2 (1997): 135-149. <http://eudml.org/doc/216383>.

@article{Perenczi1997,

abstract = {A Banach space is said to be $HD_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $HD_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $HD_n$, then dim $(ℒ(X)/S(X)) ≤ n^2$, where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring.},

author = {Perenczi, V.},

journal = {Studia Mathematica},

keywords = {hereditarily indecomposable spaces; strictly singular operators; quaternionic division ring},

language = {eng},

number = {2},

pages = {135-149},

title = {Hereditarily finitely decomposable Banach spaces},

url = {http://eudml.org/doc/216383},

volume = {123},

year = {1997},

}

TY - JOUR

AU - Perenczi, V.

TI - Hereditarily finitely decomposable Banach spaces

JO - Studia Mathematica

PY - 1997

VL - 123

IS - 2

SP - 135

EP - 149

AB - A Banach space is said to be $HD_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $HD_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $HD_n$, then dim $(ℒ(X)/S(X)) ≤ n^2$, where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring.

LA - eng

KW - hereditarily indecomposable spaces; strictly singular operators; quaternionic division ring

UR - http://eudml.org/doc/216383

ER -

## References

top- [E] P. Enflo, A counterexample to the approximation property in Banach spaces, Acta Math. 130 (1973), 309-317. Zbl0267.46012
- [F1] V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc., to appear.
- [F2] V. Ferenczi, Quotient hereditarily indecomposable Banach spaces, preprint.
- [G1] W. T. Gowers, A new dichotomy for Banach spaces, preprint.
- [G2] W. T. Gowers, Analytic sets and games in Banach spaces, preprint.
- [GM] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. Zbl0827.46008
- [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York, 1977. Zbl0362.46013
- [R] C. E. Richart, General Theory of Banach Algebras, D. Van Nostrand, Princeton, N.J., 1960.

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