Tsirelson space

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In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l^p space nor a c₀ space can be embedded.

It was introduced by B. S. Tsirelson in 1974. In the same year, Figiel and Johnson published a related article; there they used $T$ for the dual of the Tsirelson space.

1 Construction
2 Properties
3 Derived spaces
4 References

[edit] Construction

Let $P n (x)$ denote the operator which sets all coordinates $x i$ , $i\leq n$ to zero.

We call a sequence $\{x_n\}_{n=1}^N$ block-disjoint, if for each $n$ there are natural numbers $a n$ and $b n$ , so that $(x n) i = 0$ when $i < a n$ or $i > b n$ . Also, $a_1\leq b_1<a_2\leq b_2<\cdots$ .

Define these four properties for a set $A$ :

$A$ is contained in the unit ball of $c 0$ . Every unit vector $e i$ is in $A$ .
$\forall x\in A\ \forall y\in c_0\ (|y|\leq|x|\Rightarrow y\in A)$ (pointwise comparison)
For any $N$ , let $(x_1,\dots,x_N)$ be a block-disjoint sequence in $A$ , then ${1\over2}P_N(x_1+\cdots+x_N)\in A$ .
$\forall x\in A\ \exists n\ 2P_n(x)\in A$ .

We define $T$ as the space with unit ball $V$ , where $V$ is an absolutely convex weakly compact set, for which (1)-(4) hold true. It may be noted that a set with the given properties exists, but is not unique.

[edit] Properties

The Tsirelson space is reflexive, it is uniformly convex and finitely universal. Also, every infinite-dimensional subspace is finitely universal.

[edit] Derived spaces

The symmetric Tsirelson space $S (T)$ is polynomially reflexive and it has the approximation property. As with $T$ , it is reflexive and no $l p$ space can be embedded into it.

Since it is symmetric, it can be defined even on an uncountable supporting set, giving an example of non-separable polynomially reflexive Banach space.

[edit] References

B. S. Tsirelson (1974): Not every Banach space contains an imbedding of $l p$ or $c 0$ . Functional Anal. Appl. 8(1974), 138–141
T. Figiel, W. B. Johnson (1974): A uniformly convex Banach space which contains no $l p$ . Composito Math. 29(1974).
V. Spinka (2002): Smoothness on Banach spaces. Diploma work, Charles University Prague, Department of Mathematical Analysis. (Proof of the polynomial reflexivity of $S (T)$ for both separable and non-separable cases).
Boris Tsirelson's reminiscences on his web page