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On asymptotically symmetric Banach spaces

100%

Junge M., Kutzarova D., Odell E.

Studia Mathematica

2006

tom 173

nr 3

203-231

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences $(x_{j}^{i})_{j=1}^{∞} ⊆ X$, 1 ≤ i ≤ m, for all permutations σ of {1,...,m} and all ultrafilters 𝒰₁,...,𝒰ₘ on ℕ, $lim_{n₁,𝒰₁} ... lim_{nₘ,𝒰ₘ} ||∑_{i=1}^{m} x_{n_i}^{i}|| ≤ C lim_{n_{σ(1)},𝒰_{σ(1)}} ... lim_{n_{σ(m)},𝒰_{σ(m)}} ||∑_{i=1}^{m} x_{n_{i}}^{i}||$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences $(x_{j}^{i})_{j=1}^{∞}$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c₀ then X is w.a.s. We obtain an analogous result if c₀ is replaced by ℓ₁ and also show it is false if c₀ is replaced by $ℓ_{p}$, 1 < p < ∞. We prove that if 1 ≤ p < ∞ and $||∑_{i=1}^{n} x_{i}|| ∼ n^{1/p}$ for all $(x_{i})_{i=1}^{n} ∈ {X}ₙ$, the nth asymptotic structure of X, then X contains an asymptotic $ℓ_{p}$, hence w.a.s. subspace.

Rosenthal operator spaces

100%

Junge M., Nielsen N., Oikhberg T.

Studia Mathematica

2008

tom 188

nr 1

17-55

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_{p}$-space, then it is either an $L_{p}$-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative $L_{p}$-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence σ we prove that most of these spaces are operator $L_{p}$-spaces, not completely isomorphic to previously known such spaces. However, it turns out that some column and row versions of our spaces are not operator $L_{p}$-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.

Ograniczanie wyników

2 Studia Mathematica

2 Junge M.

1 Kutzarova D.

1 Nielsen N.

1 Odell E.

1 Oikhberg T.

1 2008

1 2006

Wyniki wyszukiwania

On asymptotically symmetric Banach spaces

Rosenthal operator spaces