  This paper is concerned with the fractional Keller-Segel system in the
temporal and spatial variables. We consider fractional dissipation for the
physical variables including a fractional dissipation mechanism for the
chemotactic diffusion, as well as a time fractional variation assumed in the
Caputo sense. We analyze the fractional heat semigroup obtaining time decay and
integral estimates of the Mittag-Leffler operators in critical Besov spaces,
and prove a bilinear estimate derived from the nonlinearity of the Keller-Segel
system, without using auxiliary norms. We use these results in order to prove
the existence of global solutions in critical homogeneous Besov spaces
employing only the norm of the natural persistence space, including the
existence of self-similar solutions, which constitutes a persistence result in
this framework. In addition, we prove a uniqueness result without assuming any
smallness condition of the initial data.
