TY - UNPB

T1 - Radius of comparison and mean cohomological independence dimension

AU - Hirshberg, Ilan

AU - Phillips, N. Christopher

PY - 2020

Y1 - 2020

N2 - We introduce a notion of mean cohomological independence dimension for
actions of discrete amenable groups on compact metrizable spaces, as a
variant of mean dimension, and use it to obtain lower bounds for the
radius of comparison of the associated crossed product C*-algebras. Our
general theory gives the following for the minimal subshifts constructed
by Dou in 2017. Let G be a countable amenable group, let Z be a
polyhedron, and let T be Dou's subshift of Z^G (which also depends on a
density parameter). Then the radius of comparison of the crossed product
is greater than r (1/2) mdim (T) - 2, in which r depends on the density
parameter and is close to 1 when the density parameter is close to 1. If
Z is even dimensional and has nonvanishing rational cohomology in degree
dim (Z), then the radius of comparison of the crossed product is greater
than (1/2) mdim (T) - 1, regardless of what the density parameter is.

AB - We introduce a notion of mean cohomological independence dimension for
actions of discrete amenable groups on compact metrizable spaces, as a
variant of mean dimension, and use it to obtain lower bounds for the
radius of comparison of the associated crossed product C*-algebras. Our
general theory gives the following for the minimal subshifts constructed
by Dou in 2017. Let G be a countable amenable group, let Z be a
polyhedron, and let T be Dou's subshift of Z^G (which also depends on a
density parameter). Then the radius of comparison of the crossed product
is greater than r (1/2) mdim (T) - 2, in which r depends on the density
parameter and is close to 1 when the density parameter is close to 1. If
Z is even dimensional and has nonvanishing rational cohomology in degree
dim (Z), then the radius of comparison of the crossed product is greater
than (1/2) mdim (T) - 1, regardless of what the density parameter is.

KW - Mathematics - Operator Algebras

KW - Mathematics - Dynamical Systems

KW - 46L05

KW - 46L55

KW - 54H20 (primary)

KW - 37B05

KW - 46L80 (secondary)

M3 - Preprint

BT - Radius of comparison and mean cohomological independence dimension

ER -