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THE BANACH SPACE S IS COMPLEMENTABLY MINIMAL AND SUBSEQUENTIALLY PRIME
 

Summary: THE BANACH SPACE S IS COMPLEMENTABLY MINIMAL AND
SUBSEQUENTIALLY PRIME
G. ANDROULAKIS AND TH. SCHLUMPRECHT
Abstract We first include a result of the second author showing that the Banach space S is
complementably minimal. We then show that every block sequence of the unit vector basis of
S has a subsequence which spans a space isomorphic to its square. By the Pelczy´nski decom-
position method it follows that every basic sequence in S which spans a space complemented
in S has a subsequence which spans a space isomorphic to S (i.e. S is a subsequentially
prime space).
1. Introduction
The Banach space S was introduced by the second author as an example of an arbitrarily
distortable Banach space [14]. In [8] the space S was used to construct a Banach space which
does not contain any unconditional basic sequence. In this paper we are concerned with the
question whether or not S is a prime space. We will present two partial results: In Section 2
we include a result proved by the second author some time ago but not published until now
[15], that S is complementably minimal, and thereby answer a question of P. G. Casazza,
who asked whether or not p, 1 p , and c0 are the only complementably minimal
spaces. In Section 3 we prove that S is subsequentially prime.
Let us recall the above notions. A Banach space X is called prime [12] if every comple-
mented infinite dimensional subspace of X is isomorphic to X. A. Pelczy´nski [13] showed

  

Source: Androulakis, George - Department of Mathematics, University of South Carolina

 

Collections: Mathematics