Nijenhuis operators on homogeneous spaces related to $C^{\ast }$-algebras

For a unital non-simple $C^{\ast }$-algebra $\mathcal{A}$ we consider its Banach–Lie group $G$ of invertible elements. For a given closed ideal $\mathfrak{k}$ in $\mathcal{A}$, we consider the embedded Banach–Lie subgroup $K$ of $G$ of elements differing from the unit element by an element in $\mathfrak{k}$. We study vector bundle maps of the tangent space of the homogeneous space $G\mathrm{/}K$, induced by an admissible bounded operator on $\mathcal{A}$. In particular, we discuss when this vector bundle map is a Nijenhuis operator in $G\mathrm{/}K$. The special case of almost complex structures in $G\mathrm{/}K$ is also addressed. Examples for particular classes of $C^{\ast }$-algebras are presented, including the Toeplitz algebra and crossed products by $\mathbb{Z}$.