In C*-algebras, there is an invariant of interest called the "radius
of comparison", and in the case of $C*$-algebras associated to dynamical
systems, the radius of comparison is conjectured to be equal to half
of the dynamical system's mean dimension. However, this problem has
been difficult to approach, with the best current upper bound of the
radius of comparison (as of two years ago) being roughly 36 times the
mean dimension. In 2020, Hirshberg and Phillips defined the notion of
mean cohomological independence dimension, which has proved useful in
refining this upper bound. It is a cohomological variant of mean
dimension, and it is likely that it agrees with mean dimension in most
cases. In addition to helping the radius of comparison question for
$C*$-algebras, mean cohomological independence dimension is easier to
work with in a variety of ways. For example, the proof for the product
formula $mcid ( X^n , G ) = n \cdot mcid ( X , G )$ is much simpler than
what is known in the mean dimension case.
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