Derivations Tangential to Compact Groups: The Non‐Abelian Case

Let τ be an action of a compact Lie group G on an unital C * ‐algebra A , and let A F be the G ‐finite elements in A . We show that if τ satisfies a certain spectral condition, then any * ‐derivation δ defined on A F , and mapping A F , is closable and its closure generates a one‐parameter group of * ‐automorphisms. The condition on τ is fulfilled in the cases where τ has full spectrum, or if A separating family ï ∈ Ĝ is associated with Hilbert spaces in A in the sense of Roberts, or if G is abelian and the condition Γ of [7] is fulfilled. The proof is based on an argument of K. Thomsen, who proved the theorem in the case where G is abelian [25]. A more detailed study is made of the cononical action of G = U(n) on the Cuntz's algebra C n , and, in particular, we find that any derivation on A F vanshing on the fixed point algebra for this action is the generator of a one‐parameter subgroup fo the action.