Tutorial on Rational Rotation $C^*$--Algebras

The rotation algebra Aθ is the universal C ∗–algebra generated by unitary operators U, V satisfying the commutation relation UV = ωV U where ω = e. They are rational if θ = p/q with 1 ≤ p ≤ q − 1, othewise irrational. Operators in these algebras relate to the quantum Hall effect [2, 26, 30], kicked quantum systems [22, 34], and the spectacular solution of the Ten Martini problem [1]. Brabanter [4] and Yin [38] classified rational rotation C∗–algebras up to ∗-isomorphism. Stacey [31] constructed their automorphism groups. They used methods known to experts: cocycles, crossed products, Dixmier-Douady classes, ergodic actions, K–theory, and Morita equivalence. This expository paper defines Ap/q as a C ∗–algebra generated by two operators on a Hilbert space and uses linear algebra, Fourier series and the Gelfand-Naimark-Segal construction [16] to prove its universality. It then represents it as the algebra of sections of a matrix algebra bundle over a torus to compute its isomorphism class. The remarks section relates these concepts to general operator algebra theory. We write for mathematicians who are not C∗–algebra experts. 2020 Mathematics Subject Classification: 15A30; 46L35; 55R15 1 1 Uniqueness of Universal Rational Rotation C∗–algebras N, Z, Q, R, C and T ⊂ C denote the sets of positive integer, integer, rational, real, complex and unit circle numbers. For a Hilbert space H let B(H) be the C–algebra of bounded operators on H. All homomorphisms are assumed to be continuous. We assume famliarity with the material in Section 4. Fix p, q ∈ N with p ≤ q− 1 and gcd(p, q) = 1, define σ := e and ω := σ, and let Cp/q be the set of all C–algebras generated by a set {U, V } ⊂ B(H) satisfying UV = ωV U. Since {U, V } = {V, U}, C(q−p)/q = C(q−p)/q. Mq and the circle subalgebra of L(T) generated by (Uf)(z) := zf(z) and (V f)(z) := f(ωz) belong to C(q−p)/q. The circle algebra is isomorpic to the tensor product C(T) ⊗Mq. Definition 1 A ∈ Cp/q generated by {U, V } ⊂ B(H) satisfying UV = ωV U is called universal if for every A1 ∈ Cp/q generated by {U1, V1} ⊂ B(H1) satisfying U1V1 = ωV1U1, there exists a ∗-homomorphism Ψ : A 7→ A1 satisfying Ψ(U) = U1 and Ψ(V ) = V1. Lemma 1 If A,A1 ∈ Cp/q are both universal, then they are isomorphic. Proof Let U, V, U1, V1 be as in Definition 1. There exists ∗–homomorphisms Ψ : A 7→ A1 and Ψ1 : A1 7→ A with Ψ1 ◦Ψ(U) = U, Ψ1 ◦Ψ(V ) = V, Ψ ◦Ψ1(U1) = U1, Ψ ◦Ψ1(V1) = V1. Since {U, V } generates A, Ψ1 ◦Ψ is the identity map on A. Similarly, Ψ◦Ψ1 is the identity map on A1. Therefore Ψ is a ∗–isomorphism of A onto A1 and A is ∗–isomorphic to A1. This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of Regional Centers for Mathematics Research and Education (Agreement No. 075-02-20201534/1). 2 Construction of Universal Rational Rotation C∗–algebras Define the Hilbert space Hq := L (R,C) consisting of Lebesgue measurable v : R 7→ C satisfying ∫ R vv <∞, equipped with the scalar product < v,w > := ∫ R wv. Define Pq to be the subset of continuous a : R 7→ Mq satisfying a(x1, x2) = a(x1 + q, x2) = a(x1, x2 + q), (x1, x2) ∈ R (1) and regarded as a C–subalgebra of B(Hq) acting by (av)(x) := a(x)v(x), a ∈ Pq, v ∈ Hq. The operator norm of a ∈ Pq satisfies ||a|| = max x∈[0,q] ||a(x)||. (2) Define U, V ∈ Pq by U(x1, x2) := e U0, V (x1, x2) := e V0, (3) where U0, V0 ∈ Mq are defined by (7), and define Ap/q to be the C–subalgebra of Pq generated by {U, V }. Choose r ∈ {1, ..., q− 1} such that pr = 1 mod q. Then r is unique, gcd(r, q) = 1. Define σ := e and ω := ω. Then ω = σ. Theorem 1 If a ∈ Ap/q then a(x1 + 1, x2) = V −r 0 a(x1, x2)V r 0 and a(x1, x2 + 1) = U r 0a(x1, x2)U −1 0 . (4) Conversely, if a ∈ Pq satisfies (4), then a ∈ Ap/q. Proof (3) and (8) give V UV r = σU and U V U = σV. If a = UV , then a(x1+1, x2) = σ a(x1, x2) = V −r 0 a(x1, x2)V r 0 ; a(x1, x2+1) = σ a(x1, x2) = U r 0a(x1, x2)U −r 0 . The first assertion follows since span{UmV n : m,n ∈ Z} is dense in Ap/q. Conversely, if a ∈ Pq, then (1), Lemma 3, and Weierstrass’ approximation theorem implies that there exist unique c(m,n, j, k) ∈ C with