Traces of C⁎-algebras of connected solvable groups

We give an explicit description of the tracial state simplex C ⁎ -algebra ( G ) arbitrary connected, second countable, locally compact, solvable group . show that every lifts from a abelianized group, and intersection kernels all states is proper ideal unless abelian. As consequence, connected nonabelian Lie cannot embed into simple unital AF-algebra.