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Hereditarily finitely decomposable Banach spaces

V. Perenczi — 1997

Studia Mathematica

A Banach space is said to be $H D_{n}$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $H D_{n}$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $H D_{n}$ , then dim $(ℒ (X) / S (X)) \leq n^{2}$ , 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....

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