Abstract
A dynamical system is a triple ( A , G , α ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α : G → Aut ( A ) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A × is open in A and the inversion map ι : A × → A × , a ↦ a − 1 is continuous at 1 A . Given a dynamical system ( A , G , α ) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A.
Highlights
Let σ : P × G → P be a smooth action of a Lie group G on a manifold P
It follows from ([1], Lemma A.1) that each character χ : C ∞ ( P) → C is an evaluation in some point p ∈ P, that is, of the form δp : C ∞ ( P) → C, f 7→ f ( p)
Given a dynamical system ( A, G, α) with a complete commutative continuous inverse algebra (CIA) A and a compact group G, we show that each character of the corresponding fixed point algebra
Summary
Let σ : P × G → P be a smooth action of a Lie group G on a manifold P. Given a dynamical system ( A, G, α) with a complete commutative continuous inverse algebra (CIA) A and a compact group G, we show that each character of the corresponding fixed point algebra We conclude that Afin · I is a proper left ideal in Afin that contains I.
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