# Morita equivalence of noncommutative tori *Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).* Translations and modulations on the real line can represent two different rotation algebras at once. The two actions commute, but the module linking them has a dimension which need not be an integer. We construct this equivalence, calculate its trace normalization, and then use the ordered K-group to prove the classification of irrational rotation algebras up to Morita equivalence. ## 1. Generators, traces and the convention For an irrational real number \(\theta\), let \(A_\theta\) be the universal unital C*-algebra generated by unitaries \(u,v\) with \[ vu=e^{2\pi i\theta}uv. \tag{1.1} \] Put \(b_{m,n}=u^mv^n\). Its smooth algebra consists of series with rapidly decreasing coefficients in these monomials. We use *Connections and curvature from symmetries of an algebra*, Theorem 6.1, for this smooth-algebra description. That lesson writes \(UV=e^{2\pi i\theta}VU\); the identification here is \(U=v,V=u\). A right action reverses operator composition: \(R_{ab}=R_bR_a\). This reversal will determine the translation sign. **Lemma 1.1.** The constant coefficient defines the unique normalized trace \(\tau_\theta\) on \(A_\theta\). It is faithful, and \(A_\theta\) is simple. *Proof.* The gauge action sends \(u\) to \(zu\) and \(v\) to \(wv\), for \((z,w)\in\mathbb T^2\). Its average \(F\) sends a polynomial to its constant coefficient times identity. Density gives \(F(a)=\tau_\theta(a)1\). This is positive and unital. It is faithful: if \(a\geq0\) and \(F(a)=0\), then for every state \(\omega\) the continuous nonnegative function \(\omega(\alpha_{z,w}(a))\) has integral zero, hence vanishes everywhere. At \((1,1)\) this gives \(\omega(a)=0\) for every state, so \(a=0\). The monomial product has phase \(e^{2\pi i\theta nr}\) in \(b_{m,n}b_{r,s}\). A constant coefficient occurs only for \((r,s)=(-m,-n)\), and the reverse product has the same phase. Thus \(\tau_\theta\) is tracial on polynomials, hence on the completion. In any trace, conjugation by \(u\) forces the coefficient of \(b_{m,n}\) to vanish if \(n\ne0\); conjugation by \(v\) does so if \(m\ne0\), since \(\theta\) is irrational. This proves uniqueness. Finally, successive Cesàro averages of conjugation by powers of \(u\) and \(v\) converge on every monomial to its constant coefficient. They are contractions, so polynomial approximation gives norm convergence to \(F(a)\) for every \(a\). A nonzero closed ideal contains a nonzero positive \(a\). Faithfulness makes \(\tau_\theta(a)>0\); all those conjugation averages belong to the ideal, hence so does \(\tau_\theta(a)1\). The ideal is all of \(A_\theta\). ∎ The ordered K-theory foundation is the written programme prerequisite *Irrational rotation algebras*, Theorems 2.1, 3.1 and 5.1, in *K-theory of C*-algebras*. Its required statement, for irrational real \(\theta\) with the generator convention fixed above, is: \[ \begin{gathered} \tau_{\theta*}:K_0(A_\theta)\cong G_\theta,\\ G_\theta=\mathbb Z+\theta\mathbb Z,\\ K_0(A_\theta)^+\longleftrightarrow G_\theta\cap[0,\infty),\\ [1]\longleftrightarrow1. \end{gathered} \tag{1.2} \] Those proofs are written. Theorem 5.1 realizes every positive trace value explicitly by a Powers–Rieffel projection for the fractional part and diagonal units for the integer part; it does not use the separately stated cancellation theorem. Its scope includes the positive cone and the unit, rather than only the abstract group or the trace range. Rosenberg, Theorem 4.1, p.117, credits the classical computation. The underlying abstract group is \(\mathbb Z^2\) for every irrational parameter; its placement and order in \(\mathbb R\), rather than that abstract group alone, distinguish Morita classes. ## 2. Two commuting actions on the Schwartz space For the construction assume \(\theta>0\). Write \(\mathcal S=\mathcal S(\mathbb R)\), and denote the generators of \(A_{1/\theta}\) by \(u',v'\). Define \[ \begin{aligned} (\xi u)(t)&=e^{2\pi it}\xi(t),\\ (\xi v)(t)&=\xi(t-\theta),\\ (u'\xi)(t)&=e^{2\pi it/\theta}\xi(t),\\ (v'\xi)(t)&=\xi(t+1). \end{aligned} \tag{2.1} \] For example \(R_uR_v=e^{2\pi i\theta}R_vR_u\), exactly the operator form of the right relation (1.1). On the left, \(L_{v'}L_{u'}=e^{2\pi i/\theta}L_{u'}L_{v'}\). Translation by one commutes with modulation by an integer frequency; modulation by \(1/\theta\) commutes with translation by \(-\theta\). The other pairs are both translations or both modulations. Thus all left and right operators commute. Every monomial acts continuously on Schwartz space. Its weighted derivative seminorms grow at most polynomially in its two indices: translation enlarges weights polynomially, and differentiating a modulation introduces powers of its frequency. Rapid coefficients dominate these powers. Consequently both smooth algebras act, and their actions still commute. Let \((\xi,\eta)_2=\int\overline\xi\eta\), linear in the second variable. Define the right inner product by \[ \begin{gathered} \langle\xi,\eta\rangle_\theta =\sum_{m,n}c_{m,n}(\xi,\eta)b_{m,n},\\ \begin{aligned} c_{m,n}(\xi,\eta)&=\int_{\mathbb R}\overline{\xi(s)}\eta(s+n\theta)\\ &\hspace{2em}\cdot e^{-2\pi ims}\,ds. \end{aligned} \end{gathered} \tag{2.2} \] Indeed \(c_{m,n}=(R_{b_{m,n}}\xi,\eta)_2\), since \(R_{b_{m,n}}\xi(t)=e^{2\pi im(t-n\theta)}\xi(t-n\theta)\). This is precisely the case \(p=0,q=1\) of the finite-frame construction in *Connections and curvature from symmetries of an algebra*, Lemma 7.1 and Theorem 7.2, after the generator identification above. We therefore use its right-linearity, Hermitian symmetry, positivity, definiteness and finite-frame conclusion. Its completion is a nonzero finite projective Hilbert \(A_\theta\)-module \(E_\theta\). Define a proposed left inner product by \[ \begin{gathered} {}_{1/\theta}\langle\xi,\eta\rangle =\sum_{k,l}d_{k,l}(\xi,\eta)u'^kv'^l,\\ \begin{aligned} d_{k,l}(\xi,\eta)&=\frac1\theta\int_{\mathbb R}\xi(s)\overline{\eta(s+l)}\\ &\hspace{2em}\cdot e^{-2\pi iks/\theta}\,ds. \end{aligned} \end{gathered} \tag{2.3} \] This is linear in \(\xi\). The factor \(1/\theta\) is necessary for compatibility. Both coefficient families are rapidly decreasing. For decay in the translation index, use the estimate that two rapidly decreasing functions with arguments differing by \(a\) have product integral bounded by \(C_N(1+|a|)^{-N}\). It follows by splitting the line according to which argument has absolute value at least \(|a|/2\). Derivatives satisfy the same estimate. Integration by parts in the modulation index, with these derivative estimates, then proves joint rapid decay of arbitrary order. Each series thus converges absolutely in C*-norm, because its monomials are unitaries, and belongs to the appropriate smooth algebra. ## 3. Compatibility and completion We first prove the identity on \(\mathcal S\), rather than infer it from commuting actions alone. **Lemma 3.1.** For Schwartz vectors, \[ {}_{1/\theta}\langle\xi,\eta\rangle\zeta =\xi\langle\eta,\zeta\rangle_\theta. \tag{3.1} \] *Proof.* The Poisson formula we use, for Schwartz \(f\) and \(a>0\), is \[ \begin{aligned} &\frac1a\sum_k\widehat f(k/a)e^{2\pi ikt/a}\\ &\quad=\sum_j f(t+ja). \end{aligned} \tag{3.2} \] To justify it, periodize \(f\) on the right. The series and all derivatives converge uniformly on a period. Its \(k\)-th Fourier coefficient is \(\widehat f(k/a)/a\), by changing variables across the period translates. Those coefficients are rapidly decreasing, so its Fourier series converges back to the periodization. This proves (3.2) without formal delta functions. Apply (3.2), with \(a=\theta\), to \(f_l(s)=\xi(s)\overline{\eta(s+l)}\). The left side of (3.1), evaluated at \(t\), becomes \[ \sum_{j,l}\xi(t+j\theta) \overline{\eta(t+j\theta+l)}\zeta(t+l). \tag{3.3} \] For the right side use (2.2) for \(\eta,\zeta\) and the right monomial action. Poisson summation with period one, applied to \(\overline{\eta(s)}\zeta(s+n\theta)\), gives \[ \begin{aligned} &\sum_{n,l}\xi(t-n\theta)\overline{\eta(t-n\theta+l)}\\ &\hspace{2em}\cdot\zeta(t+l). \end{aligned} \tag{3.4} \] Replace \(n\) by \(-j\). These are equal. The sums converge absolutely: the first and third Schwartz factors control the independent lattice indices, and the middle factor is bounded. To control weights in \(t\), use \(t=(t+j\theta)+(t+l)-(t+j\theta+l)\); powers of \(t\) can be assigned to the three Schwartz factors. Differentiating only gives finite sums of the same kind. These bounds and the rapid coefficient estimates justify all rearrangements and convergence in Schwartz seminorms. ∎ **Theorem 3.2 (The Heisenberg equivalence).** The completion \(E_\theta\) is an \(A_{1/\theta}\)–\(A_\theta\) imprimitivity bimodule. Its conjugate is an \(A_\theta\)–\(A_{1/\theta}\) imprimitivity bimodule. *Proof.* Each left generator is unitary on \(L^2(\mathbb R)\) and commutes with every right monomial. Formula \(c_{m,n}=(R_b\xi,\eta)_2\) therefore gives \[ \begin{gathered} \langle L\xi,L\eta\rangle_\theta =\langle\xi,\eta\rangle_\theta,\\ \langle L\xi,\eta\rangle_\theta =\langle\xi,L^*\eta\rangle_\theta. \end{gathered} \tag{3.5} \] Thus it extends to an adjointable unitary on \(E_\theta\). The universal relation gives a unital homomorphism \[ \rho:A_{1/\theta}\longrightarrow\mathcal L(E_\theta). \tag{3.6} \] It is nonzero, and its domain is simple by Lemma 1.1, so it is injective and isometric. Compatibility says that \[ \rho({}_{1/\theta}\langle\xi,\eta\rangle) =\theta_{\xi,\eta} \tag{3.7} \] on the dense Schwartz module. Hence the range contains all its rank-one operators, and by norm closure all of \(\mathcal K(E_\theta)\). Conversely the right finite frame makes \(1_{E_\theta}\) compact; therefore every adjointable operator is compact, including every element of the range. We obtain \[ \rho:A_{1/\theta}\cong\mathcal K(E_\theta). \tag{3.8} \] This also proves positivity of (2.3): its image for \(\xi=\eta\) is a positive rank-one operator, and an injective C*-homomorphism reflects positivity. Its norm is \(\|\theta_{\xi,\xi}\|=\|\xi\|^2\). For all completed vectors define the left product as \(\rho^{-1}(\theta_{\xi,\eta})\); norm continuity extends the Schwartz formula. The rank-one identities now prove left linearity, Hermitian symmetry and compatibility on the completion. For instance \(\rho(a)\theta_{\xi,\eta}=\theta_{a\xi,\eta}\) proves left linearity, and \(\theta_{\xi b,\eta}=\theta_{\xi,\eta b^*}\) proves the coefficient adjoint identity. Both left and right module norms agree, so completeness holds on both sides. The left products span the compact algebra and hence all of \(A_{1/\theta}\). The closed right coefficient ideal is nonzero, since \[ \tau_\theta(\langle\xi,\xi\rangle_\theta) =\|\xi\|_2^2>0\quad(\xi\ne0). \tag{3.9} \] Simplicity of \(A_\theta\) makes that ideal all of it. The module is full on both sides and satisfies every imprimitivity axiom. The conjugate-module theorem from the earlier Morita lesson gives the reverse equivalence, with its specified reversed actions and inner products. ∎ The reverse equivalence also has a literal Schwartz-space model. Identify \(\overline\xi\) in the conjugate module with the function \(f(t)=\overline{\xi(t)}\). The conjugate rules \(b\overline\xi=\overline{\xi b^*}\) and \(\overline\xi a=\overline{a^*\xi}\) give \[ \begin{aligned} (u f)(t)&=e^{2\pi it}f(t),\\ (v f)(t)&=f(t+\theta),\\ (f u')(t)&=e^{2\pi it/\theta}f(t),\\ (f v')(t)&=f(t-1). \end{aligned} \tag{3.10} \] Its right product is the old left product, and its left product is the old right product, with the vectors transported by this identification. This gives exactly a completion of \(\mathcal S(\mathbb R)\) as an \(A_\theta\)–\(A_{1/\theta}\) imprimitivity module; the different direction is not hidden in a generator sign. ## 4. Finite frames, projections and trace direction The frame prerequisite permits a concrete normalization. Choose finitely many compact smooth windows \(g_i\), each with support diameter less than one, such that \[ \sum_{i,n}|g_i(t+n\theta)|^2=1. \tag{4.1} \] To obtain them, cover the circle \(\mathbb R/\theta\mathbb Z\) by small intervals, lift smooth bumps \(\beta_i\), and divide them by the square root of the positive periodic sum \(\sum_{i,n}|\beta_i(t+n\theta)|^2\). Theorem 7.2 of the frame prerequisite gives \[ \begin{gathered} \xi=\sum_i g_i\langle g_i,\xi\rangle_\theta,\\ P=(\langle g_i,g_j\rangle_\theta)_{i,j},\\ E_\theta\cong P A_\theta^N. \end{gathered} \tag{4.2} \] The trace of this projection is \[ \begin{aligned} \tau_\theta^{(N)}(P) &=\sum_i\int_{\mathbb R}|g_i(t)|^2\,dt\\ &=\theta. \end{aligned} \tag{4.3} \] Integrate (4.1) over one period; its translates partition the real line. This proves the second equality. The trace induced on the left algebra by this frame is a positive trace of value \(\theta\) at identity. Lemma 1.1 makes it \(\theta\tau_{1/\theta}\). By the previous lesson’s Proposition 7.5, \[ \begin{gathered} T_{E_\theta}:K_0(A_{1/\theta})\to K_0(A_\theta),\\ \tau_{\theta*}(T_{E_\theta}x) =\theta\tau_{1/\theta,*}(x). \end{gathered} \tag{4.4} \] The conjugate map in the reverse direction scales by \(1/\theta\). Thus the trace multiplier must always be stated with the chosen direction. When \(0<\theta<1\), one window suffices. Choose a support interval of length \(L\) with \(\theta1\) this module needs a matrix projection: a projection in \(A_\theta\) itself has trace at most one, whereas the module dimension is \(\theta\). The finite-window construction works for every positive irrational parameter and supplies that matrix projection. **Example 4.1 (The modules \(E_{p,q}\)).** For \(p\in\mathbb Z\), \(q\geq1\), set \(\varepsilon=p/q-\theta\). The same prerequisite constructs \(\mathcal S(\mathbb R\times\mathbb Z/q)\) with our renamed generators acting by \[ \begin{aligned} (\xi u)(s,h)&=e^{2\pi i(s-h/q)}\xi(s,h),\\ (\xi v)(s,h)&=\xi(s+\varepsilon,h+p). \end{aligned} \tag{4.6} \] Its completed right module is finite projective and has trace dimension \(q|\varepsilon|=|p-q\theta|\), by that theorem. In the positive branch \(p-q\theta>0\), this is \(p-q\theta\). A negative signed expression is not a negative module dimension. This statement permits nonprimitive \((p,q)\); it does not identify every such module’s left endomorphism algebra with a single rotation algebra. ## 5. Necessity of the fractional-linear orbit **Lemma 5.1.** An ordered-group isomorphism \(f:G_\theta\to G_{\theta'}\) is multiplication by a positive scalar. *Proof.* Put \(\lambda=f(1)>0\). If integers \(n,m\), with \(m>0\), satisfy \(n0),\\ \theta'=\frac{a\theta+b}{c\theta+d},\\ \begin{pmatrix}a&b\\c&d\end{pmatrix}\in GL_2(\mathbb Z). \end{gathered} \tag{5.2} \] *Proof.* Both algebras are unital, hence σ-unital. The previous lesson’s Theorem 5.1 gives the K-isomorphism by tensoring projective modules, and its conjugate inverse has the same property. Thus it preserves the positive cone in both directions. Under (1.2) it becomes an order isomorphism, so Lemma 5.1 supplies the positive scale. Equivalently the unique traces and the preceding lesson’s trace theorem give this scale directly. Write the two target basis elements in the scaled source basis: \[ \begin{aligned} 1&=\lambda(c\theta+d),\\ \theta'&=\lambda(a\theta+b). \end{aligned} \tag{5.3} \] The pair \((1,\theta')\) is a \(\mathbb Z\)-basis of \(G_{\theta'}\), since \(\theta'\) is irrational. The pair \((\lambda,\lambda\theta)\) is another basis. Their change-of-basis integer matrix is invertible over \(\mathbb Z\), giving \(ad-bc=\pm1\). The first equality says \(c\theta+d=1/\lambda>0\); divide the second by the first to obtain (5.2). ∎ ## 6. Sufficiency, signs and the groupoid picture **Theorem 6.1 (Classification).** For irrational real parameters, \[ \begin{aligned} &A_\theta\sim_M A_{\theta'}\\ &\quad\Longleftrightarrow\quad \theta'\in GL_2(\mathbb Z)\cdot\theta. \end{aligned} \tag{6.1} \] *Proof.* Necessity is Theorem 5.2. For sufficiency, the transformation \(T:\theta\mapsto\theta+1\) preserves the defining phase, so the universal algebras are isomorphic by the same generators. The transformation \(R(\theta)=-\theta\) is implemented by replacing \(v\) with \(v^*\). The transformation \(S(\theta)=1/\theta\) is the equivalence of Theorem 3.2 for positive parameters. For negative parameters, apply the sign isomorphisms before and after that positive construction. These three transformations generate \(GL_2(\mathbb Z)\). Here is the integer-matrix argument: powers of \(T\) add integer multiples of the second row to the first; \(S\) interchanges rows; \(R\) changes a row sign. Their conjugates allow the corresponding operations on the second row. The first column of a unimodular matrix has relatively prime entries. The Euclidean algorithm reduces it to \((1,0)\). The determinant condition makes the other diagonal entry \(\pm1\); sign change and a final row addition reduce the matrix to identity. Reversing these operations proves generation. Every intermediate parameter remains irrational, since an inverse fractional-linear transformation would otherwise make the original parameter rational. Its denominator never vanishes: an integer equation \(c\theta+d=0\) would make an irrational \(\theta\) rational, unless \(c=d=0\), which no unimodular matrix has. Tensor the equivalence modules along a generator word, using the transitivity theorem from the imprimitivity lesson. The resulting module proves sufficiency. ∎ The determinant \(-1\) transformations matter. Bare \(SL_2(\mathbb Z)\)-orbits are generally smaller. For example, take transcendental \(\theta\in(0,1/2)\) and \(\theta'=1/(\theta+3)\). Theorems above give Morita equivalence. If a determinant-one matrix gave that same fraction, cross multiplication would give \[ a\theta^2+(3a+b-c)\theta+3b-d=0. \tag{6.2} \] Transcendence forces \(a=0,c=b,d=3b\), whose determinant is \(-b^2\), never one. Both parameters lie in \((0,1/2)\). One may use \(SL_2\) transformations together with the sign isomorphism, but that is a larger equivalence relation than the bare \(SL_2\)-orbit. **Example 6.2 (Two transversals).** On \(\mathbb T^2=\mathbb R^2/\mathbb Z^2\), consider \[ \beta_t(x,y)=(x+\theta t,y+t). \tag{6.3} \] The horizontal circle \(y=0\) is a complete transversal: every orbit reaches it, and its return times are the integers. The return map rotates its coordinate by \(\theta\). The vertical circle \(x=0\) is also complete; its return times are \(n/\theta\), and its return map rotates by \(1/\theta\). Thus the two reduced groupoids are circle rotation groupoids for the two parameters. The analytical passage from complete transversal reduction to Morita equivalence is proved in Equivalence of groupoids and Morita equivalence of their C*-algebras, Theorem 15.3 and the suspension construction (15.31)–(15.37), and C*-algebras of foliations and their Morita equivalences, Theorem 16.2. Those written providers cover both full and reduced algebras for these second-countable Hausdorff manifolds, using the Haar and counting systems on the full and transverse groupoids. Li, §5.1.1, p.23, credits the geometric example. Applied to each circle, it identifies both rotation algebras as Morita equivalent to the reduced Kronecker-flow algebra. The equivalence space for a transversal \(Y\) consists of arrows with source in \(Y\), with range in the full object space; it is not just the arrows whose two endpoints are in \(Y\). Its range and source anchors give the commuting actions of the full and reduced groupoids. We verified the return parameters here; the linking-groupoid positivity and both completion norms are proved in the stated provider. This is a geometric explanation of the equivalence, whereas Theorem 3.2 supplies its Hilbert-module proof. **Example 6.3 (A self-equivalence).** Let \(\theta=(\sqrt5-1)/2\). Then \(1/\theta=\theta+1\). Identify the left algebra of \(E_\theta\) with \(A_\theta\) by the shift isomorphism. Equation (4.4) gives a positive automorphism of the ordered K-group multiplying trace coordinates by \(\theta\). It sends the unit class to the class of the projection in (4.5), of trace \(\theta\). A Morita self-equivalence need not preserve the distinguished unit class. ## 7. Exercises with complete solutions **Exercise 15.1.** Prove \(A_\theta\cong A_{\theta+1}\), and derive the sufficiency direction of the orbit classification. *Solution.* The phases agree, so each generator pair satisfies the other algebra’s universal relation. The two resulting homomorphisms are inverse on generators and hence on the dense polynomials and their completions. The sign replacement \(v\mapsto v^*\) likewise gives the sign isomorphism. Theorem 3.2 gives inversion, using sign isomorphisms for a negative parameter. Euclidean row reduction in Theorem 6.1 writes any unimodular fractional-linear transformation as a word in these three operations. Tensoring the corresponding equivalences, in their successive domain and codomain order, realizes that word. Thus each complete \(GL_2(\mathbb Z)\)-orbit belongs to one Morita class; Theorem 5.2 gives the converse. The shift isomorphism alone would not establish the other generators. **Exercise 15.2.** Verify the compatibility identity for Gaussian functions. *Solution.* Put \(g_\alpha(t)=e^{-\pi\alpha t^2}\), with \(\alpha>0\), and take all three vectors equal to it. Completing the square in (2.2) gives \[ \begin{aligned} c_{m,n}(g_\alpha,g_\alpha) &=(2\alpha)^{-1/2}\\ &\quad\cdot e^{-\pi\alpha n^2\theta^2/2}\\ &\quad\cdot e^{-\pi m^2/(2\alpha)}e^{\pi imn\theta}. \end{aligned} \tag{7.1} \] The Gaussian Fourier integral is \(\int e^{-2\pi\alpha s^2}e^{-2\pi ims}\,ds=(2\alpha)^{-1/2}e^{-\pi m^2/(2\alpha)}\). For completeness, integration by parts gives the transform equation \(H'(\omega)=-(\pi\omega/\alpha)H(\omega)\). Squaring the real Gaussian integral and using polar coordinates gives \(H(0)=(2\alpha)^{-1/2}\). Solving the equation gives the stated transform. The left coefficients are similarly \[ \begin{aligned} d_{k,l}(g_\alpha,g_\alpha) &=(\theta\sqrt{2\alpha})^{-1}\\ &\quad\cdot e^{-\pi\alpha l^2/2}\\ &\quad\cdot e^{-\pi k^2/(2\alpha\theta^2)}e^{\pi ikl/\theta}. \end{aligned} \tag{7.2} \] The phases come from shifting the integration variable by half the translation. Multiply these coefficient series by their corresponding actions from (2.1). Poisson summation, with periods \(\theta\) and one respectively, gives in both cases \[ \begin{aligned} Q_{j,l}(t)&=(t+j\theta)^2\\ &\quad+(t+j\theta+l)^2\\ &\quad+(t+l)^2,\\ \text{common value}&=\sum_{j,l}e^{-\pi\alpha Q_{j,l}(t)}. \end{aligned} \tag{7.3} \] This double series is absolutely convergent, as its first and third squared terms control the independent indices. It proves the equality pointwise and in Schwartz space as in Lemma 3.1. The unnormalized Gaussian’s inner product is not thereby a projection; that would need an additional normalization. **Exercise 15.3.** Prove necessity of the \(GL_2(\mathbb Z)\)-orbit using trace ranges. *Solution.* Let \(X\) give an equivalence from \(A_\theta\) to \(A_{\theta'}\). Its right module is finite projective, and the frame-induced trace on the left is \(c\tau_\theta\), where \(c>0\) is its right trace dimension. Positivity of \(c\) follows from the faithful trace of its nonzero frame projection. The preceding lesson’s trace identity says \(\tau_{\theta'*}T_X=c\tau_{\theta*}\). Since \(T_X\) is onto, the ranges satisfy \(G_{\theta'}=cG_\theta\). Write \(1=c(c_0\theta+d)\) and \(\theta'=c(a\theta+b)\). These are two integer bases of the same free rank-two group, so \(ad-bc_0=\pm1\). Division gives the fractional-linear formula, with denominator \(1/c>0\). This is the trace version of Theorem 5.2’s order argument; it tracks the scale rather than assuming the traces are preserved without normalization. **Exercise 15.4.** Compute the trace of the projection representing the basic Heisenberg module in the one-window case, and give the matrix version for general positive \(\theta\). *Solution.* Assume first \(0<\theta<1\). Choose the normalized compact window \(\eta\) of (4.5). Compatibility makes \(p=\langle\eta,\eta\rangle_\theta\) a projection representing \(E_\theta=pA_\theta\). Its constant coefficient is \(\int|\eta(t)|^2dt\). Integrating \(\sum_n|\eta(t+n\theta)|^2=1\) over \([0,\theta]\) gives that integral exactly \(\theta\). Thus \(\tau_\theta(p)=\theta\); it is not an integer rank. For any positive irrational \(\theta\), use the finite windows and the projection \(P\) in (4.2). Summing their squared integrals and integrating (4.1) gives \(\tau_\theta^{(N)}(P)=\theta\). If \(\theta>1\), a single projection inside \(A_\theta\) cannot have that trace, which explains the need for a matrix. The reverse equivalence has right dimension \(1/\theta\), consistently with (4.4). ## What this lesson does not prove The ordered K-group, trace identification, positive cone and distinguished unit have the written provider *Irrational rotation algebras*, Theorems 2.1, 3.1 and 5.1, in *K-theory of C*-algebras*. The order proof is independent of the separately stated cancellation theorem, which this classification does not require. Rosenberg, Theorem 4.1, remains its historical credit. The smooth algebra and finite-projective right Schwartz module, including coefficient positivity and its explicit frame, are *Connections and curvature from symmetries of an algebra*, Theorem 6.1, Lemma 7.1 and Theorem 7.2. We prove the additional commuting left action, compatibility, compact left algebra and fullness here. Morita K-maps and trace transport are the preceding lesson’s Theorem 5.1 and Proposition 7.5; conjugates and tensor transitivity are the earlier imprimitivity lesson’s Theorem 3.2 and Corollary 4.2. Equivalence of groupoids and Morita equivalence of their C*-algebras, Theorem 15.3, and C*-algebras of foliations and their Morita equivalences, Theorem 16.2, supply the written analytical groupoid-reduction proof in Example 6.2. General higher-dimensional torus classification, constant-curvature connection classification and Gaussian projection normalization are not developed here. ## References - H. Emerson, *An Introduction to C*-Algebras and Noncommutative Geometry*, §6.6, especially Corollaries 6.6.2–3, Exercise 6.6.5 and Corollary 6.6.7. - B. Blackadar, *Operator Algebras*, II.10.4.12(i), for the rotation crossed-product convention and classification context. - J. Rosenberg, *Examples and applications of noncommutative geometry and K-theory*, §4, especially Theorems 4.1–3, pp.117–118. - Y. Li, *Groupoid C*-algebras*, §5.1.1, “Kronecker flow of irrational angle,” p.23. - *Connections and curvature from symmetries of an algebra*, §§6–7, with the generator identification specified in Section 1. - *Morita invariance of K-theory and maps induced by correspondences*, Theorem 5.1 and Proposition 7.5.