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In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.

We study the nonnegative part \overline {G_ {> 0}} of the De Concini-Procesi compactification of a semisimple algebraic group \overline {G_ {> 0}} , as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of \overline {G_ {> 0}} . This answers the question of Lusztig in Total positivity and canonical bases, Algebraic groups and Lie groups (ed. GI Lehrer), Cambridge Univ. Press, 1997, pp. 281-295. We will also prove that \overline {G_ {> 0}} has a cell decomposition which was conjectured by Lusztig.

A family of coordinates h for the Teichmller space of a compact surface with boundary was introduced by the second author. Mondello showed that the coordinate 0 can be used to produce a natural cell decomposition of the Teichmller space, invariant under the action of the mapping class group. In this paper, we show that, for any h 0, the coordinate h produces a natural cell decomposition of the Teichmller space.