An l(p)-version of von Neumann dimension for Banach space representations of sofic groups

作者：Hayes Ben^*

来源：Journal of Functional Analysis, 2014, 266(2): 989-1040.

DOI：10.1016/j.jfa.2013.09.014

• Sofic groups
• Banach space representations
• l(p)-Dimension

In [5], A. Gournay defined a notion of l(p)-dimension for Gamma-invariant subspaces of l(q) (Gamma)(circle plus n), with F amenable. The number dim(lq), l(p)(Gamma, V) is dim V when p = q, and is preserved by a certain class of Gamma-equivariant bounded linear isomorphisms. We develop a notion of din(lp), Sigma (Y, Gamma) where Y is a Banach space with a uniformly bounded action of a sofic group Gamma and Sigma is a sofic approximation. In particular, our definition makes sense for a large class of non-amenable groups. We also develop a notion of dim(Sp,) (Sigma)(Y, Gamma) with Gamma an R-w-embeddable group and S-P the space of finite-dimensional Schatten p-class operators. These numbers are invariant under bounded Gamma-equivariant linear isomorphisms and under the natural translation action of F, dim(lp) (l(p) (Gamma, V), F) = dim V, and dim(Sp) (l(p) (Gamma, V), Gamma) = dim V for 1 <= p <= 2. In particular, this shows that l(p)(Gamma, V) is not isomorphic to l(P) (Gamma, W) as a representation of Gamma if dim V not equal dim W, and Gamma is R-w-embeddable. We discuss other concrete computations in a follow-up paper, including proving that our dimension agrees with von Neumann dimension for representations contained in a multiple of the left-regular representation.

• 出版日期2014-1-15