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DOI: 10.17516/1997-1397-2022-15-5-598-609 УДК 512
Tutorial on Rational Rotation C*-Algebras
Wayne M. Lawton*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 10.11.2021, received in revised form 23.04.2022, accepted 30.06.2022 Abstract. The rotation algebra Ae is the universal C*-algebra generated by unitary operators U, V satisfying the commutation relation UV = &VU where ш = в2жге. They are rational if в = 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.
Keywords: bundle topology, Gelfand-Naimark-Segal construction, irreducible representation, spectral decomposition.
Citation: W.M. Lawtonl, Tutorial on Rational Rotation C*-Algebras, J. Sib. Fed. Univ. Math. Phys., 2022, 15(5), 598-609. DOI: 10.17516/1997-1397-2022-15-5-598-609.
1. Uniqueness of universal rational rotation C^-algebras
N, Z, Q, R, C and T c 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 a := e2nl/q and w := ap, and let Cp/q be the set of all C*-algebras generated by a set {U,V} c B(H) satisfying UV = wVU. Since {U, V} = {V, U}, C(q-p)/q = C(q-p)/q. Mq and the circle subalgebra of L2(T) generated by (Uf)(z) := zf (z) and (Vf)(z) := f (wz) 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} c B(H) satisfying UV = wVU is called universal if for every Ai £ Cp/q generated by {Ui,V\} c B(H\) satisfying U1V1 = wV\U\, there exists a *-homomorphism ^ : A^ A1 satisfying ^(U) = U1 and ^(V) = V1.
Lemma 1. If A, A1 £ Cp/q are both universal, then they are isomorphic.
* wlawton50@gmail.com © Siberian Federal University. All rights reserved
Proof. Let U,V,Ui,Vi be as in Definition 1. There exists *-homomorphisms $ : A ^ Ai and $i : Ai ^ A with $i o $(U) = U, $i o $(V) = V, $ o $i(Ui) = Ui, $ o $i(Vi) = Vi. Since {U, V} generates A, $i o $ is the identity map on A. Similarly, $ o $i is the identity map on Ai. Therefore $ is a ^-isomorphism of A onto Ai and A is ^-isomorphic to Ai. □
2. Construction of universal rational rotation C^-algebras
Define the Hilbert space Hq := L2(R2, Cq) consisting of Lebesgue measurable v : R2 ^ Cq satisfying f v*v < x>, equipped with the scalar product <v,w> := J w*v. Define Pq to be the
R2 R2
subset of continuous a : R2 ^ Mq satisfying
a(xi,x2) = a(xi + q,X2) = a(xi,x2 + q), (xi, X2) G R2 (1)
and regarded as a C*-subalgebra of B(Hq) acting by (av)(x) := a(x)v(x), a G Pq, v G Hq. The operator norm of a GPq satisfies
\\a\\ = max2 ||a(x)||. (2)
xe[0,q\2
Define U,V GPq by
U(xi,x2) := e2nixi/qUo, V(xi,x2) := e2nix2/qVo, (3)
where U0, V0 G Mq are defined by (7), and define Ap/q to be the C*-subalgebra of Pq generated by {U, V}. Choose r G {1,... ,q — 1} such that pr = 1 mod q. Then r is unique, gcd(r, q) = 1. Define a := e2ni/q and w := wp. Then wr = a.
Theorem 1. If a G Ap/q then
a(xi + 1,x2) = V—ra(xi,x2)V0r and a(xi,x2 + 1) = U0a(xi, x2)U-i. (4)
Conversely, if a GPq satisfies (4), then a G Ap/q.
Proof. (3) and (8) give V-rUVr = aU and UrVU-r = aV. If a = UmVn, then
a(xi + 1, x2) = ama(xi, x2) = V—ra(xi, x2)V0r; a(xi, x2 + 1) = ana(xi, x2) = a(xi, x2)U-r.
The first assertion follows since span{UmVn : m,n G Z} is dense in Ap/q. Conversely, if a G Pq, then (1), Lemma 3, and Weierstrass' approximation theorem implies that there exist unique c(m, n, j, k) G C with
q-i
a(xi,x2) — Y, Y,c(m,n,j,k) e2ni(mxi+nx2)/q UjV0k (m,n)ez2 j,k=0
where — denotes Fourier series. Then (4) gives c (m,n,j,k) am = c (m,n,j,k) aj and c(m,n,j,k)an = c(m,n,j,k)ak. Since aq = 1, c(m,n,j,k) = 0 unless j = m mod qand k = n mod q. Define c(m, n) := c(m, n, m mod q, n mod q). Then a G Ap/q since
a — Y, c(m,n)Um Vn.
(m,n)ez2
Representations pi, pi : A ^ B(H) of a C*-algebra A are unitarily equivalent if there exists U G U(H) such that p2(a) = Upi(a)U-i, a G A. □
Theorem 2. If A £ Cp/q is generated by {U,V} with UV = wVU and p : A ^ B(H) is an irreducible representation then:
1. dim H = q so B(H) = Mq,
2. there exist z1, z2 £ T such that p = Pzt,z2 where Pzt,z2 (UjVk) :=
z1 z2 U0 Vo.
3. pz[,z2 is unitarity equivalent to pZl,z2 iff (z'1 /z1)q = (z2/z2)q = 1.
Proof. Boca gives a proof in ([1], p. 5, Lemma 1.8, p. 7, Theorem 1.9). We give a proof based on Schur's lemma. Let C c A be the C*-subalgebra generated by {Uq, Vq}. Since p is irreducible and p(C) commmutes with p(A), there exists a *-homomorphism y : C ^ C such that p(c) = j(c)I, c £ C. Choose h £ H\{o} and define H1 := span {p(UjVk)h;0 < j,k < q — l}. Since H1 is closed, p-invariant, H1 = {0}, and p is irreducible, H = H1. Since dim H < q2, p(V) has an eigenvector b with eigenvalue A £ T and \\b\\ = 1. Define z2 := Aw. Choose z1 £ T so zq = Y(Uq) and define bj := zjp(U-j)b, 1 < j < q. Then p(V)bj = z2wj-1bj, j = 1,...,q, and p(U)b1 = z1bq, and p(U)bj = z1bj-1, 2 < j < q. Therefore {b1,..., bq} is a basis for H, and (7) implies that p(U) = z1U0, and p(V) = z2V0 with respect to this basis. This proves assertions 1 and 2. Assertion 3 follows since the set of eigenvalues of p(U) is {z1wj, 0 < j < q — 1}, the set of eigenvalues of p(V) is {z2wj, 0 < j < q — 1}, and the set of eigenvalues determines unitary equivalence. □
Theorem 3. Ap/q c B(H) is the universal C* -algebra in Cp/q.
Proof. Assume that B £ Cp/q. Then there exists a Hilbert space H1 and U1,V1 £ B(H1) with U1V1 = wV1U1 and B is generated by {U1,V1}. It suffices to construct a continuous *-homomorphism p : Ap/q ^ B satisfying ^(U) = U1 and V) = V1. Define dense *-subalgebras
A/q := span {UjVk : j,k £ Z} c Ap/q, B := span {UjV? : j,k £ Z} c B,
and a *-homomorphism p : Ap/q ^ B by ip(UjVk) := UjVk. To extend to *-homomorphism p : Ap/q ^ B it suffices to show that for every Laurent polynomial of two variables p(u,v) the following inequality is satisfied \\p(UuV1)\ \ < \\p(U,V)\\ sincep(UV1) = p(p(U,V)). Then ([13], Corollary I.9.11), which follows directly from the Gelfand-Naimark-Segal construction, implies that there exists an irreducible representation p1 : B ^ Mq and v £ H1 with \\v\\ = 1 such that \\p(U1,V1)\\ = \\p1(p(U1,V1))v\\. Theorem 2 implies that p1(U1) = zU and p^) = z2Vo for some z1, z2 £ T. Let p : Ap/q ^ Mq be the irreducible representation defined by Theorem 2 so p(U) = z1U0 and p(V) = z2V0. Since p1 op = the restriction of p to Ap/q, (2) and (3) imply that
\\p(U1,V1)\\ = \\p1(p(U1,V1))v\\ <\\p(p(U,V))\\ < \\p(U,V)\\
which concludes the proof. □
3. Bundle topology and isomorphism classes
Define E1 to be the Cartesian product [0,1]2 x Mq with the identification
(1,X2,M) = (0,X2,V0-rMV0), X2 £ [0,1], M £Mq
and
(x1,1, M) = (x1, 0, UrMU-r), x1 £ [0,1], M £Mq
and define the algebra bundle ni : Ei ^ T2 by
ni(xi,x2,M) = (e2nixi ,e2nix2), (xi,x2,M) G Ei.
A map s : T2 ^ Ei is called a section if it is continuous and ni o s = I where I denotes the identity map on T2. Since for every p G T2, the fiber n- (p) — ^^q, the set of sections under pointwise operations is a C*-algebra. The theorems above show that this algebra is isomorphic to Ap/q. Furthermore, since points in T2 correspond to unitary equivalence classes of irreducible representations, isomorphism of algebras induces homeomorphisms of T2. In order to compute isomorphism classes of universal rational rotation C*-algebras it is convenient to use a slightly different bundle representation of Ap/q. Define W GPq
W(xi,x2) :--
10
0 e2nixi/q
0 0
0
e2ni(q-i)xi/q
and A'/q := WAp/qW-i, which is ^-isomorphic to Ap/q. Then A'/q is represented as the algebra of sections of the algebra bundle n2 : E2 ^ T2 where E2 is the Cartesian product T x [0,1] x Mq with the identification
(zi, 1, M) = (zi, 0, GrMG-r), zi G T, M G Mq
and
G(zi) :--
1 0 0 0 0
0 1 0 0 0
0 0 0 0
0 0 0 1 0
0 0 0 0 zi
U .
Gr is the clutching function of the bundle. Let Gin : T ^ Aut* be the map defined by conjugation by G. Using the arguments for vector bundles in [18], it can be shown that the isomorphism classe of Ap/q is determined the homotopy class of Grin : T ^ Aut*. Since ni(Gin) = —1, ni(Grin) = —r which gives:
Theorem 4. Ap/q is isomorphic to Ap'/q' iff q' = q and either p' = p or p' = q — p.
0
4. Requisite results
4.1. Hilbert spaces and adjoints
H is a Hilbert space with inner product < •, • >: H x H ^ C, norm WW := yj< v, v >, and metric d : H x H ^ [0, x) defined by d(v,w) := ||v — w||. B(H) is the Banach algebra of bounded operators on H (continuous linear maps from H to H) with operator norm
HaH := sup{ WavW : v G H, WW = 1 }.
The dual space H* is the set of continuous linear functions L : H ^ C. For w G H define
Lw G H* by Lwv := <v,w>, v G H.
Lemma 2. If L G H * then there exists a unique w G H such that L = Lw.
Proof. Rudin gives a direct proof ( [28], Theorem 4.12). If B is an orthonormal basis for H and w := J2 Lbb, then since for every v G H, v = J2 < v,b > b, it follows that
beB
beB
Lv < v,b > Lb
be B
L E Lbb)
be B
< v,w > = Lw v.
Lemma 2 ensures the existence of adjoints. For a G B(H) define its adjoint a* G B(H) by La»w := Lw o a, w G H where o denotes composition of functions. Therefore
Clearly a*
and
< a v,w >= < v, a*w >, v,w G H. a, (ab)* = b*a*, and the Cauchy-Schwarz inequality gives
||a*|| = sup{ | < a*v,w > | : v,w G H, ||v|| = ||w|| = 1} = = sup{ | < v, aw > | : v,w G H, ||v|| = ||w|| = 1} = ||a||
||a*a|| = sup{ | < a*av, w > | : v,w G H, ||v|| = ||w|| = 1} = = sup{ | < av,aw > | : v,w G H, ||v|| = ||w|| = 1} = ||a||2.
(5)
(5) is called the C*-identity. It makes B(H) equipped with the adjoint a C*-algebra. The identity operator I £ B(H) is defined by Iv := v for all v £ H.
U(H) := {U £ B(H) : UU* = U*U = I},
the set of unitary operators, is a group under multiplication. A subalgebra A c B(H) is a C*-algebra if it is closed in the metric space topology on B(H) and a* £ A whenever a £ A. The intersection of any nonempty collection of C*-subalegras of B(H) is a C *-algebra. If S c B(H)
the intersection of all C*-subalgebras of B(H) that contain S is the C*-algebra generated by S.
□
4.2. Matrix algebras
For m,n £ N, Cmxn denotes the set of m by n matrices with complex entries and Cn := Cnx1. The adjoint of a £ Cmxn is the matrix a* £ Cnxm defined by a* k := akj. Cn is a Hilbert space with scalar product <v,w> := w*v, v,w £ Cn. Clearly
B(Cn) = Mn
where for a £ Mn the adjoint of a as an operator corresponds to the adjoint of a as a matrix. In denotes the n by n identity matrix whose diagonal entries equal 1 and other entries equal 0. The operator norm of a £ Mn is \\a\\ = \Jspectral radius a*a where the spectral radius is the largest moduli of the eigenvalues of a matrix. Thus Mn is a C*-algebra. It is also a Hilbert space a Hilbert space of dimension n2 with inner product
< a,b >:= Trace b*a
(6)
and orthonormal basis Cjtk := matrix with 1 in row j and column j with all other enties = 0. Fix p,q G N with p < q — 1 and gcd(p, q) = 1. Define Uo, Vo G Mq by
Uo :=
0 1 0 0
: 0 00 10
Vo :=
1 0 0 0
0 w 0 0
0 0 0
0 0 0 wq-1
(7)
Lemma 3. {(1/^q)UjV£ : 0 ^ j,k ^ q — 1} is an orthonormal basis for Mq with the scalar
product defined by (6). Furthermore,
UoV0 = uVoUo. Proof. (8) is obvious. The first assertion follows since
(8)
< Uj Vk, U^V" > Trace V-nU-mUjV£ = Trace Uj
mV V
k—n
q if j = m and k = n, 0 otherwise.
Define the groups of unitary matrices Un := U(Cn) and special unitary matrices Sn := {a G Un : det a = 1}. Clearly Uo and Vo are unitary. Since det Uo = det Vo = ( — 1)q-1, they are special unitary iff q is odd. A map ^ : Mn ^ Mn is a homomorphism if it is linear and satisfies ^(ab) = ^(a)^(b) for all a,b G Mn and an automorphism if is also a bijective. An automorphism ^ is a *-automorpism if ^(b*) = ^(b)* for all b G Mn. Autn, Aut" denote the group of all automorpisms, *-automorphisms of Mn. ^ G Autn is called inner if there exists an invertible a G Mn such that ^(b) = aba-1 for every b G Mn.
Theorem 5 (Skolem-Noether). Every ^ G Autn is inner.
Proof. The algebra Mn is simple, meaning it has no two-sided ideals othe that itself ( [29], 11.41), so the result follows from the classic Skolem-Noether theorem. An elementary constructive proof is given in [32]. □
Theorem 6. If ^ G Aut*n then there exists a GUn such that b) = aba* for every b G Mn.
Proof. Every ^ G Aut" induces an irreducible representation ^ : Mn ^ B(Cn) so Theorem 2 implies that there exists a basis {bl7... ,bn} with respect to which ^(Uo) has the matrix representation z1Uo and ^(Vo) has the representation z2Vo. Since U" = V" = I, z" = z" = 1 so without loss of generality this basis can be chosen to make z1 = z2 = 1 and then a) = aba-1—1 where aej = bj and {e1,..., en} is the standard basis for Cn. This theorem can also be derived as a corollary of of Theorem 5. Clearly ^(In) = In. Theorem 5 implies that there exists an invertible a G Mn such that ^(b) = aba-1 for all b G Mn. Since ^ is a *-homomorphism ab*a-1 = (aba-1)* = (a-1)*b*a* hence a*ab* = b*a*a for every b G Mn which implies that a*a = cIn for some c > 0. Replacing a by a/y/c gives the conclusion.
Corollary 1. Let Tn C T be the subgroup of n-th roots of unity. Tn In C Sn is isomorphic to Z/nZ. Aut" is isomorphic to the quotient group Sn/Tn In. The fundamental group n1(Aut*n) is isomorphic to Z/nZ.
Proof. Assertion one is obvious. Define Z : Un ^ Aut" by Z(a)(b) := aba*. Z is a *-homomorphism, kernel Z = T In, and Corollary 1 implies that Z is onto. The first homomorphism theorem of group theory ([33], 7.2) implies that Aut" is isomoprhic to Un/TIn. Since
Sn = (TIn)((TnIn) and Tn In = Sn n (TIn), the second isomorphism theorem of group theory ( [33], 7.3) implies that Autn is isomporphic to Sn/TnIn. Sn is simply connected ( [17], Proposition 13.11) hence since TnIn is discrete Sn is the univeral cover of Sn n (TIn) hence the discussion in ( [18], 1.3) implies the last assertion.
4.3. Spectral Decomposition Theorem for Unitary Operators
E G B(H) is called a projection if E* = E and E2 = E. Then P : H ^ PH is orthogonal projection. A collection of projections {Ev : p G [0,2n] is called a spectral family if
EV1EV2 = EV2 EV1 = EV1 whenver < EV2.
Let A,P,N G B(H). A is self-adjoint if A* = A. P G B(H) is positive if < Pv,v > > 0 for all v G H.N G B(H ) is called normal if AA* = A* A. Clearly self-adjoint and unitary operators (or transformations) are normal. Furthermore eigenvalues of self-adjoint operators are real and eigenvalues of unitary operators have modulus 1. If dim H < œ, then H admits an orthonormal basis of eigenvectors ([29], Theorem 9.33). Therefore every unitary matrix in Mn can be diagonalized and its diagonal entries have modulus 1. The following result, copied verbatim from the classic textbook by F.Riesz and B.Sz.-Nagy ([27], p. 281), extends this diagonalization to unitary operators on arbitrary Hilbert spaces.
Theorem 7. Every unitary transformation U has a spectral decomposition
-0
U = Í eipdEp,
where [Ev] is a spectral family over the segmen 0 ^ p ^ 2n. We can require that Ev be continuous at the point p = 0, that is, E0 = 0; [Ev] will then be determined uniquely by U. Moreover, Ev is the limit of a sequence of polynomials in U and U— 1.
Proof. The authors of [27] reference 1929 papers by von Neumann [25] and Wintner [37], 1935 papers by Friedricks and Wecken, and a 1932 book by Stone. They observe that the theorem can be deduced from the one on symmetric transformation ( [27], p. 280) (since U = A + iB where A := (U + U*)/2 and B := -i(U — U*)/2 are symmetric) or from the theorem on trigonometric moments ( [27], Section 53), but they give a direct three page
n
proof. We sketch their proof. For every trigomometric polynomial p(ellp) = ^ ckelklp we as-
—n
n
sociate the transformation p(U) := J2 ckUk. This gives a *-homomorphism of the algebra of
—n
trigonometric polynomials (where * means complex conjugation) into the subalgebra of B(H) generated by U and U* = U—1. Clearly if p(elv) is real-valued then p(U) is self-adjoint. If p(elv) > 0 the Riesz-Fejer factorization Lemma ( [27], Section 53) implies that there exists a trigonometric polynomial q(ellp) with p(ellp) = q(eltp)q(elv) hence p(U) = q(U)q(U)*. Therefore < p(U)v,v >=< q(U)v,q(U) > > 0, v G H, hence p(U) is a positive operator. For 0 < < 2n let e^ be the characteristic function of (0, extended to a 2n periodic function on R. Let pn be a monotonically sequence of positive trigonometric functions with pn(U)v = E^v, v G H (pn(U) converges to E^ G B(H) in the strong operator topology). E^ is a projection since E^ = E^ and E= E^, so , and the set {Ev : p G [0,2n] is a spectral family. Since the functions e^ are upper semi-continuous Ex = E^.
Given e > 0 choose 0 < ^o < < • • • < = 2n with max (^k+1 — 4>k) < e and choose Pk G [^k-1,^k], k =1,...,n. Then for p G (^k-1,^k]
n
3 V e iPk [e^k — ] k=1
= — eifk)| < \p — pk| < e
with a similar inequality for p = 0. Since this inequality holds for all p G [0,2n]
U — ^ eifk (E^k — )
k=1
^ e
A subspace H1 C H is called proper if H1 = {0} and H1 = H. The following is an immediate consequence of Theorem 7
Corollary 2. If U G U(H) then either U = jI for some y G T or there exists a projection operator E : H ^ H satisfying
1. E is the limit in the strong operator topology on B(H) of polynomials p(U,U-1).
2. EH is a proper closed U-invariant subspace of H.
4.4. Irreducible representations and Schur's lemma
A representation of a C*-algebra A on a Hilbert space H is a *-homomorphism p : A ^ B(H). A subspace H1 C H is called p-invariant if p(a)H1 C H1 for every a G A. p is irreducible iff it H has no closed proper p-invariant subspaces. The following result extends Shur's lemma for finite dimensional representations ([29], 11.33) for unitary operators.
Theorem 8 (Schur's Lemma). If p : A ^ B(H) is an irreducible representation and U G U(H) commutes with p(a) for every a G A, then there exists y G T with U = yI.
Proof. If the conclusion does not hold then Corollary 2 implies that there exists a projection E satisfying conditions (1) and (2). Condition (1) implies that Up(a) = p(a)U for every a G A. The EH C H is closed and p-invariant since for every a G A, p(a)EH = Ep(a)H. Condition (2) asserts that EH is a proper subspace thus contradicting the hypothesis that p is irreducible, and concluding the proof.
Corollary 3. If A is an commutative C* -algebra generated by a set of unitary operators and p : A^ B(H) is irreducible then dim H = 1 and there exists a *-homomorphism r : A^ C with p(a) = r(a)I for every a G A.
Proof. Follows from from Theorem 8 since if u G A is unitary the p(u) G U(H) and p(u) commutes with p(a) for every a G A.
5. Remarks
We relate concepts introduced to explain rational rotation algebras to general C*-algebra theory, especially two breakthrough results obtained by teams of computer scientists.
Remark 1. Dauns [14] initiated a program to represent C*-algebras by continuous sections over bundles over their primitive ideal spaces (kernels of irreducible representations equipped with
the hull-kernel topology). The primitive ideal space of rational rotation algebras is homemorpic to the torus T2.
Remark 2. Bratteli, Elliot, Evans, and Kishimoto [5] represent fixed point C*-subalgebras of Ap/q by algebras of sections of Mq-algebra bundles over the sphere S2, which is the space of orbits of T2 under the map g ^ g—1.
Remark 3. Elliot and Evans [15] derived the structure of irrational rotation algebras. They proved that if p/q < 0 < p'/q', then Ae can be approximated by a C*-subalgebra isomorphic to C(T) ® Mq © C(T) ® Mq'. This approximation, combined with the continued fraction expansion of 0, represents Ae as an inductive limit of these subalgebras.
Remark 4. Williams [36] gives an extensive explanation of crossed product C*-algebras, which include rotation algebras.
Remark 5. Kadison and Singer [21] formulated a problem about extending pure states. Such an extension is used in the Gelfand-Naimark-Segal construction which we used to prove Theorem 3. This problem was shown to be equivalent to numerous problems in functional analysis and signal processing [6], dynamical systems [23,24], and other fields [3]. Weaver [35] gave a discrepancy-theoretic formulation that was proved in a seminal paper by three computer scientists: Marcus, Spielman, and Shrivastava [19].
Remark 6. Courtney [8,9] proved that the class of residually finite dimensional C*-algebras, those whose structure can be recovered from their finite dimensional representations, coincides with the class of algebras containing a dense set of elements that attain their norm under a finite dimensional representation, this set is the full algebra iff every irreducible representation is finite dimensional (as for rational rotation algebras), and related these concepts to Conne's embedding conjecture [7]. Her publications [10-12] cite many references that discuss equivalent formulations of this conjecture.
Remark 7. In January 2020 five computer scientists: Ji, Natarajan, Vidick, Wright and Yuen submitted a proof that the Conne's embedding conjecture is false. As of November 2021 their paper is still under peer review. However, the editors of the ACM decided, based on the enormous interest that their paper attracted, to publish it [20].
Acknowledgments. We thank Paolo Bertozzini, Kristin Courtney, S^ren Eilers, Vern Paulsen, Beth Ruskai, David Yost, and Saeid Zahmatkesh for sharing their insights.
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. 075-02-2020-1534/1).
References
[1] A.Avila, S.Jitomirskaya, Solution of the ten martini problem, Annals of Mathematics, 170, no. 1, (2000), 303-341. DOI 10.4007/annals.2009.170.303
[2] F.Boca, Rotation C*-algebras and Almost Mathieu Operators, The Theta Foundation, Bucharest, 2001.
[3] M.Bownik, The Kadison-Singer problem. arXiv:1702.04578
[4] M.De Brabanter, The structure of rational rotation CD-algebras, Arch. Math, 43(1984), 79-83.
[5] O.Bratteli, G.A.Elliot, D.E.Evans, A.Kishimoto, Non-commutative spheres. II: rational rotations, J. Operator Theory, 27(1992), 53-85.
[6] P.G.Casazza, J.C.Tremain, The Kadison-Singer problem in mathematics and engineering, Proc. Nat. Acad. Sci., 103(2006), no. 7, 2032-2039.
[7] A.Connes, Classification of injective factors, Annals of Mathematics, 104(1976), 73-115.
[8] K.E.Courtney, C*-algebras and their finite-dimensional representations, PhD Dissertation, Department of Mathematics, University of Virginia, 2018.
[9] K.E.Courtney, Kirchberg's QWEP conjecture: between Connes' and Tsirelson's problems, UK Operator Algebra Seminar, 2020.
10] K.E.Courtney, T.Schulman, Elements of C*-algebras attaining their norm in a finite-dimensional representation, Canadian J. Math., 71(2019), no. 1. arXiv:1707.01949
11] K.E.Courtney, D.Sherman, The universal C*-algebra of a contraction, J. Operator Theory, 84(2020), no. 1, 153-184. arXiv:1811.04043
12] K.E.Courtney, Universal C*-algebras with the local lifting property, Math. Scand., 127(2021), 361-381. arXiv:2002.02365
13] K.R.Davidson, C*-Algebras by Examples, American Math. Soc., Providence, Rhode Island, 1991.
14] J.Dauns, The primitive ideal space of a C*-algebra, Canadian J. Math., XXVI(1974), no. 1, 42-49.
15] G.A.Elliot, D.E.Evans, The structure of irrational rotation C*-algebras, Annals of Mathematics, 138(1993), no. 3, 477-501.
16] I.M.Gelfand, M.A.Naimark, On the embedding of normed rings into the ring of operators on a Hilbert space, Mat. Sbornik, 12(1943), no. 2, 197-217.
17] B.Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Springer, Switzerland, 2003.
18] A.Hatcher, Vector Bundles and K-Theory. https://openlibra.com/en/book/vector-bundles-and-k-theory
19] A.W.Marcus, D.A.Spielman, N.Shrivastava, Interlacing Famlies II: mixed characteristic polynomials and the Kadison-Singer problem, Annals of Mathematics, 182(2015), no. 1, 327-350. DOI: 10.4007/ANNALS.2015.182.1.8
20] Z.Ji, A.Natarajan, T.Vidick, J.Wright, H.Yuen, MIP* = RE, Communications of the ACM, 64(2021), no. 11, 131-138. DOI: 10.48550/arXiv.2001.04383
[21] R.V.Kadison, I.M.Singer, Extensions of pure states, American J. Math., 81(1959), no. 2, 383-400.
[22] W.Lawton, A.Mouritzen, J.Wang, J.Gong, Spectral relationships between kicked Harper and on-resonance double kicked rotor operators, J. Math. Phys., 50(2009), no. 3, 032103. DOI: 10.1063/1.3085756
[23] W.Lawton, Minimal sequences and the Kadison-Singer problem, Bull. Malaysian Math. Science Society, 33(2010), 169-176.
[24] V.Paulsen, A dynamical system approach to the Kadison-Singer problem, J. Functional Analysis, 255(2008), no. 1, 120-132.
[25] J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Annalen, 102(1929), 49-131.
[26] R.Rammal, J.Bellisard, An algebraic semi-classical approach to Bloch electrons in a magnetic field, J. Physics France, 51(1990), 1803-1830.
[27] F.Riesz, B.Sz.-Nagy, Functional Analysis, Frederick Ungar Publishing Company, New York, 1955.
[28] W.Rudin, Real and Complex Analysis, McGraw-Hill, Singapore, 1987.
[29] G.Shilov, Linear Algebra, Dover, New York, 1977.
[30] B.Simon, Almost periodic Schrödinger operators: a review, Advances in Applied Math., 3(1982), 463-490.
[31] P.J.Stacey, The automorphism groups of rational rotation algebras, J. Operator Theory, 39(1998), 395-400.
[32] J.Szigeti, L.Wyk, A constructive elementary proof of the Skolem-Noether theorem for matrix algebras, The American Mathematical Monthly, 124(2017), no. 16, 966-968. arXiv:1810.08368
[33] B.van der Waerden, Algebra I, Springer, New York, 1991.
[34] H.Wang, D.Ho, W.Lawton, J.Wang, J.Gong, Kicked-Harper model versus on-resonance double-kicked rotor model: From spectral difference to topological equivalence, Physical Review E, 88(2013), 1-15. 052920 arXiv:1306.6128
[35] N.Weaver, he Kadison-Singer problem in discrepancy theory, TDiscrete Math., 278(2004), no. 1-3, 227-239.
[36] D.Williams, Crossed Products of C*-Algebras, American Math. Society, 2007.
[37] A.Wintner, Zur Theorie der beschrönkten Bilinearformen, Math. Zeitschr., 30(1929), 228-289.
[38] H.S.Yin, A simple proof of the classification of rational rotation C*-algebras, Proceedings of the American Math. Society, 98(1986), no. 3, 469-470.
Учебник по рациональному вращению С*-Алгебры
Уэйн М. Лоутон
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Алгебра вращений Лв — это универсальная С*-алгебра, порожденная унитарными операторами и,У, удовлетворяющими коммутационному соотношению иУ = шУи, где ш = е2пгв. Они рациональны, если 0 = р/д с 1 < р < д — 1, в противном случае иррациональны. Операторы в этих алгебрах связаны с квантовым эффектом Холла [2,26,30], квантовыми системами [22,34] и эффектным решением проблемы Тена Мартини [1]. Брабантер [4] и Инь [38] классифицировали С*-алгебры рационального вращения с точностью до ^-изоморфизма. Стейси [31] построила свои группы автоморфизмов. Они использовали известные специалистам методы: коциклы, скрещенные произведения, классы Диксмье-Дуади, эргодические действия, К-теорию и эквивалентность Мориты. Эта пояснительная статья определяет Лр/д как С*-алгебру, порожденную двумя операторами в гильбертовом пространстве, и использует линейную алгебру, ряды Фурье и конструкцию Гельфанда-Наймарка-Сигала [16] для доказательства его универсальности. Затем он представляет его как алгебру сечений расслоения матричной алгебры над тором для вычисления его класса изоморфизма. Раздел примечаний связывает эти концепции с общей теорией операторной алгебры. Мы пишем для математиков, не являющихся экспертами в С*-алгебре.
Ключевые слова: топология расслоения, конструкция Гельфанда-Наймарка-Сигала, неприводимое представление, спектральное разложение.
