کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6415238 | 1334972 | 2014 | 52 صفحه PDF | دانلود رایگان |

In [5], A. Gournay defined a notion of lp-dimension for Î-invariant subspaces of lq(Î)ân, with Î amenable. The number dimlqlp(Î,V) is dim V when p=q, and is preserved by a certain class of Î-equivariant bounded linear isomorphisms. We develop a notion of dimlp,Σ(Y,Î) where Y is a Banach space with a uniformly bounded action of a sofic group Î and Σ is a sofic approximation. In particular, our definition makes sense for a large class of non-amenable groups. We also develop a notion of dimSp,Σ(Y,Î) with Î an RÏ-embeddable group and Sp the space of finite-dimensional Schatten p-class operators. These numbers are invariant under bounded Î-equivariant linear isomorphisms and under the natural translation action of Î, dimlp(lp(Î,V),Î)=dimV, and dimSp(lp(Î,V),Î)=dimV for 1⩽p⩽2. In particular, this shows that lp(Î,V) is not isomorphic to lp(Î,W) as a representation of Î if dimVâ dimW, and Î is RÏ-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.
Journal: Journal of Functional Analysis - Volume 266, Issue 2, 15 January 2014, Pages 989-1040