The first Hochschild cohomology groups of ultrapowers of Banach algebra with coefficients in some special Banach bimodules

We consider a Banach algebra A, a nonzero element phi in Delta(A) boolean OR {0}, and a Banach A-bimodule X. We investigate ultrapowers denoted as (A)U and (phi), along with treating (X)U as a Banach (A)U-bimodule. We analyze H-1((A)U, (X)U), with the constraint that (X)U is an element of SM(phi)(A)U . Moreover, we establish a connection between H-1((A)U, C) vanishing and H1((A)U, (X)U) vanishing. Subsequently, we relax the symmetry conditions of SM(phi)(A)U and explore character contractibility and character amenability of (A)U, which is referred to as ultra-character contractibility and ultra-character amenability of A. In particular, we verify the ultra-character amenability for Lau products and group algebras