## Annals of Mathematics |

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abelian group Af(X alternative algebra arbitrary assume automorphism Banach bounded C*-algebra canonical coboundary cochain cocycle coefficients commutative complete condition connected consider contains continuous converges coprime Corollary corresponding coset cup product defined definition denote derivatives determined dimension dimensional direct sum equation equivalent exists extension fixed follows formula function given group G harmonic field Hence holds homology homomorphism homotopy groups Hopf ideal identity implies induced integral intersection interval isomorphism Jordan algebra kernel L)-group Lemma Let G Lie algebra Lie group linear locally compact group manifold mapping Math matrix maximal mod q n-sphere neighborhood normal subgroup obtain operator homomorphism orthogonal positive projection proof properties prove r(X)-measurable representation respectively result satisfies sequence solution space subgroup of G subset subspace Suppose tensor Theorem theory topological topological group transformation tube unitary space vector zero