On a class of smooth Frechet subalgebras of C * -algebras

An averaging trick for smooth actions of compact quantum groups on manifolds

“…Also denote (d⊗id)α (F (g 1 , ..., g n )) by S ′ and (d⊗id)(α(χ)) by T ′ . So we have T 2 S = T S and by (2) we have T ′ T = T T ′ and S ′ S = SS ′ . F (g 1 , ..., g n ))) = α(χ 2 ) n i=1 α(∂ i F (g 1 , ..., g n ))(d ⊗ id)(α(g i )) (by (4)…”

“…We end this subsection with an example of Hopf-algebra (of non compact type) having coaction on a coordinate algebra of an algberaic variety which violates the condition (2). However as mentioned in the introduction, we don't have any example of a smooth CQG action which violates the condition (2).…”

“…We actually need a slightly smaller subclass of such algebras, to be called 'nice algebra' for convenience. This is one of special class of examples of Fréchet (D * ∞ )-subalgebra of C * algebra treated in [2]. Definition 2.1 A unital Fréchet * algebra A will be called a 'nice' algebra if there is a C *norm · on A and the the underlying locally convex topology of A comes from the family of seminorms · α , with α = (i 1 , ..., i k ) being any finite multi index (including the empty index, i.e.…”

“…of India) and also acknowledges the Fields Institute, Toronto for providing hospitality for a brief stay when a small part of this work was done. 2 Acknowledges support from CSIR. noncommutative geometry ( [4]) such lifts are going to be important when we are going to treat CQG as 'symmetry' objects.…”