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Derivations Tangential to Compact Groups: The Non‐Abelian Case
Author(s) -
Bratteli Ola,
Evans David E.
Publication year - 1986
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-52.2.369
Subject(s) - mathematics , abelian group , action (physics) , pure mathematics , generator (circuit theory) , automorphism , closure (psychology) , spectrum (functional analysis) , fixed point , center (category theory) , group (periodic table) , discrete mathematics , algebra over a field , combinatorics , mathematical analysis , law , power (physics) , chemistry , physics , organic chemistry , quantum mechanics , political science , crystallography
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.