Invariant States on Noncommutative Tori

For any number h such that ℏ:=h/2π is irrational and any skew-symmetric, non-degenerate bilinear form σ:ℤ2g×ℤ2g→ℤ, let be hg,σ be the twisted group ∗-algebra ℂ[ℤ2g] and consider the ergodic group of ∗-automorphisms of hg,σ induced by the action of the symplectic group Sp(ℤ2g,σ). We show that the only Sp(ℤ2g,σ)-invariant state on hg,σ is the trace state τ.