### Yolanda Moreno: Complementably univesal space vs Gurarij

Data: segunda-feira, 03 de junho de 2013, às 14h

Sala: 266-A

Resumo: A separable Banach space $\mathcal{K}$ is said to be complementably universal for the class of separable Banach spaces having BAP if every such space is isomorphic to a complemented subspace of $\mathcal K$. The existence of $\mathcal K$ was first proved by Kadec in 1971 using a nice topological approach. A bit later in the same year, Pelczynski and Wojtaszczyk gave another construction of such a space using different techniques. Recently, Garbulinska showed the existence of $\mathcal K$ using categorical language and Fraissé limits.  It is known that $\mathcal K$ is unique up to isomorphisms but we still don't know about uniqueness up to isometries. We will show that the method to construct spaces of universal disposition (Aviles-Cabello-Castillo-Gonzalez-Moreno) allows us to construct a 'version' of $K$, which is isometric to the one constructed by Garbulinska, and where the property defining the space as well as its FDD are explicitly observed during the construction process itself. Through this method we realize that it is possible a local approach to $\mathcal K$ completely analogous to the Gurarij space $\mathcal G$, namely: the space $\mathcal K$ is the corresponding 'Gurarij object' in the category of double arrows (isometric embeddings-norm one projections). Natural questions arise now from comparing the behavior of both spaces: what about the version $\omega_1$ of $\mathcal K$ which would correspond to the space $\mathcal{G}_{\omega_1}$ of universal disposition for finite dimensional spaces? Could we construct through this method a space complementably universal for the class of all separable Banach spaces (we know, by a result of Jonhson-Szankowski, that such a space cannot be separable)? What about extending isometries from separable (other sizes) subspaces to isometries on the whole $\mathcal K$ ($\mathcal K_{\omega_1}$)?