Tsirelson space

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.

Construction

Let $P_n(x)$ denote the linear 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 $A$:

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

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.

Properties

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

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.

References

• B. S. Tsirelson (1974): Not every Banach space contains an imbedding of $l_p$ or $c_0$. Functional Anal. Appl. 8, 138–141
• T. Figiel, W. B. Johnson (1974): A uniformly convex Banach space which contains no $l_p$. Composito Math. 29.
• Casazza, Peter G.; Shura, Thaddeus J. (1989). Tsirelson's Space. Springer. ISBN 978-3-540-50678-2. 

External linksd

• Boris Tsirelson's reminiscences on his web page