Banach Poisson-Lie group structure on \(\mathrm{U}(\mathcal{H})\) (Q6606764)

Let \(\mathcal{H}\) be a complex separable Hilbert space, and let \(U(\mathcal{H})\) denote the Banach Lie group of bounded unitary operators. The authors define a Banach Poisson-Lie group structure on \(U(\mathcal{H})\) defined on the pre-cotangent bundle \(T^*U(\mathcal{H})\).\N\NA core issue being addressed is whether certain subspaces of the dual space \(\mathfrak{u}^*(\mathcal{H})\) of \(\mathfrak{u}(\mathcal{H})\), the Banach Lie algebra of \( U(\mathcal{H}) \) consisting of skew-Hermitian bounded operators, can be used to define a Poisson-Lie group structure. The first candidate subspace \(\mathfrak{b}_1^+(\mathcal{H})\), a certain Banach Lie algebra of upper triangular trace-class operators, fails due to a lack of continuity in the coadjoint action. The authors then find a workaround by embedding \(\mathfrak{b}_1^+(\mathcal{H})\) densely into a more appropriate subspace of \(\mathfrak{u}^*(\mathcal{H})\) where continuity is preserved.\N\NAnother key ingredient in the construction is the decomposition of the space of Hilbert-Schmidt operators into two Lie algebras: the Lie algebra of skew-Hermitian operators, which is invariant under conjugation by a unitary operator, and the Lie algebra of upper triangular operators with real diagonal entries, which is not invariant under conjugation. The conjugation properties of the corresponding continuous projections of the decomposition provide the foundation for establishing the Poisson-Lie group structure.\N\NFor the entire collection see [Zbl 1531.53004].