The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL∞（E）. It is proved that there is an injective Lipschitz map φ： （F, ds） --＊ （H,d）, where F is the Thompson＇s group, ds the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that φ（F, ds） → （H, dl） is not a Lipschitz map, where dl is another metric of H.
In this paper,the authors construct a φ-group for n submodules,which generalizes the classical K-theory and gives more information than the classical ones.This theory is related to the classification theory for indecomposable systems of n subspaces.
Abstract This paper generalizes the C＊-dynamical system to the Banach algebra dynam- ical system （A, G, α） and define the crossed product A αG of a given Banach algebra dynamical system （A, G,α）. Then the representation of A α G is described when A ad- mits a bounded left approximate identity. In a natural way, the authors define the reduced crossed product A αG and discuss the question when A α G coincides with A αG.
