Unitary group with operator-norm topology

Core idea

For a complex Hilbert space $H$ , the group

U(H)=\{U\in B(H):U^*U=UU^*=I\}

of unitary operators, equipped with the operator norm, is a real Banach–Lie group. Its Lie algebra is

\mathfrak u(H)=\{B\in B(H):B^*=-B\}, \qquad [B,C]=BC-CB,

the real Banach space of bounded skew-adjoint operators. Here $B^*$ is the bounded-operator adjoint.

Local charts

The operator series $\exp B=\sum_{n\geq0}B^n/n!$ maps skew-adjoint operators to unitaries. Near $I$ , its inverse is the norm-convergent logarithm

\log U=\sum_{n\geq1}\frac{(-1)^{n+1}}n(U-I)^n \qquad(\|U-I\|<1).

This logarithm is skew-adjoint. Translating the local chart around the group gives its manifold structure; multiplication and inversion are smooth in these charts.

Why this is stronger than an orbit-map condition

For a smooth homomorphism $\pi:G\to U(H)$ into this Banach–Lie group, differentiation produces bounded operators on all of $H$ . A strongly continuous unitary representation only makes $g\mapsto\pi(g)v$ continuous for each $v$ . Its derivatives can be unbounded and require the common smooth-vector domain. The modulation group exhibits the distinction explicitly.

References

Jesús Espinoza and Bernardo Uribe, Topological properties of the unitary group, §1, p. 2, discussion before Theorem 1.2. The exponential and logarithm give the local construction used above.