Nijenhuis operators on homogeneous spaces related to C∗-algebras

For a unital non-simple [Formula: see text]-algebra [Formula: see text], we consider its Banach–Lie group [Formula: see text] of invertible elements. For a given closed ideal [Formula: see text] in [Formula: see text], we consider the embedded Banach–Lie subgroup [Formula: see text] of [Formula: see text] of elements differing from the unit element by an element in [Formula: see text]. We study Banach vector bundle maps of the tangent space of the homogeneous space [Formula: see text], induced by an admissible bounded operator on [Formula: see text]. In particular, we discuss when this Banach vector bundle map is a Nijenhuis operator in [Formula: see text]. The special case of almost complex structures in [Formula: see text] is also addressed. Examples for particular classes of [Formula: see text]-algebras are presented, including the Toeplitz algebra and crossed products by [Formula: see text].