Extending traces on fixed point \(C^ *\) algebras under Xerox product type actions of compact Lie groups
On the UHF \(C^*\)-algebra \(A=\otimes M_ n(C)\) of the type \(n^{\infty}\), the (Xerox) product action \(\alpha (g)=\otimes Ad \pi (g)\) of a compact connected Lie group G via a unitary representation \(\pi\) : \(G\to M_ n(C)\) is considered and the fixed point algebras \(A^ G\) and \(A^ T\) are studied, where T is the maximal torus in G. The principal result (Theorem) asserts that every (pure) trace on \(A^ G\) extends to a (pure) trace on \(A^ T\). In the course of the proof, it is shown (Proposition II.2) that \(K_ 0(A^ G)\) is finitely generated as a ring and \(K_ 0(A^ T)\) is finitely generated as a \(K_ 0(A^ G)\)-module.
scientific article; zbMATH DE number 4011211