Noncommutative Torus: Weyl Quantization & Aθ
The algebra generated by U,V with UV = e^{2πiθ} VU — the rotation algebra at angle θ
UV = e^{2πiθ} VU | θ = 0 (commutative torus T²)
Noncommutative torus Aθ: C*-algebra generated by unitaries U,V satisfying UV = e^{2πiθ}VU. At θ=0: commutative = C(T²). At θ∈ℚ: matrix algebra × torus. At θ irrational: simple C*-algebra.
Weyl quantization: Map f(x,y) → Op(f) = Σ f̂_{m,n} U^m V^n. The operator product law picks up a phase: f★g has extra e^{iπθmn'} twist.
Moyal product: (f★g)(x) = f(x)g(x) + iπθ{f,g} + O(θ²). This is the quantum mechanics star-product in disguise.
Connes' framework: Noncommutative geometry replaces spaces with algebras. The NC torus has a Dirac operator, cyclic cohomology, and K-theory K₀(Aθ) = Z+Zθ ⊂ ℝ (a dense subgroup!).