Nijenhuis operators on homogeneous spaces related to $C^*$-algebras
Tomasz Goliński, Gabriel Larotonda, Alice Barbora Tumpach
TL;DR
The paper develops a framework to study Nijenhuis operators on tangent bundles of homogeneous spaces $G/K$ arising from unital non-simple $C^*$-algebras, formulating admissible operators on the Banach–Lie algebra and deriving explicit torsion criteria. It then specializes to four canonical $C^*$-algebras ($\mathcal B(\mathcal H)$, $C(X)$, the Toeplitz algebra, and crossed products) to classify when rank-one, left/right multiplication, and adjoint actions yield Nijenhuis operators and, in some cases, integrable almost complex structures. Key contributions include an explicit torsion condition in terms of commutators, a concrete identifications between $G/K$ and quotient groups, and a compendium of examples showing when Nijenhuis operators exist or are trivial across standard operator-algebra contexts. The results illuminate connections between Banach–Lie homogeneous geometry and operator-algebra dynamics, with potential implications for integrable structures and Poisson–Nijenhuis theory in noncommutative settings.
Abstract
For a unital non-simple $C^*$-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 $G/K$, induced by an admissible bounded operator on $\mathcal A$. In particular, we discuss when this vector bundle map is a Nijenhuis operator in $G/K$. The special case of almost complex structures in $G/K$ is also addressed. Examples for particular classes of $C^*$-algebras are presented, including the Toeplitz algebra and crossed products by $\mathbb Z$.