This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum, while ${\ell }_{\infty }$...

Given a real separable Hilbert space H, we denote with G(H) the geometry of closed linear subspaces of H.The strong convergence of sequences of subspaces is shown to be a L*-convergence and the weak convergence a L-convergence.The smallest L*-convergence containing the weak convergence is found, and the orthogonal image of the strong convergence, which is also a L*-convergence, is defined.

Let E be a Riesz space. By defining the spaces $L{¹}_E$ and $L_E^{\infty }$ of E, we prove that the center $Z\left(L{¹}_E\right)$ of $L{¹}_E$ is $L_E^{\infty }$ and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality $L{¹}_E=Z{\left(E\right)}^{\text{'}}$. Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in $L{¹}_E$ which are different from the representations appearing in the literature.

We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X=T\left[{(\theta ₙ,ₙ)}_{n=1}^{\infty }\right]$: (1) Every block subspace of X contains an $\ell ¹{-}_{\omega }$-spreading model, (2) The Bourgain ℓ¹-index $I_b\left(Y\right)=I\left(Y\right)>{\omega }^{\omega }$ for any block subspace Y of X, (3) $limₘlimsupₙ{\theta }_{m+n}/\theta ₙ>0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions...

Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra ${}_{\mu }$ containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define $L{²}_{\mu }$-Betti numbers and an $L{²}_{\mu }$-Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version of Poincaré...

We construct the category of quotients of $ℒ\Im$-spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.