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Noncommutative index theory for mirror quantum spheres

We introduce and analyse a new type of quantum 2-spheres. Then we apply index theory for noncommutative line bundles over these spheres to conclude that quantum lens spaces are non-crossed-product examples of principal extensions of C*-algebras.


Introduction
The purpose of this paper is twofold. First, we present and study quantum two-spheres of a new topological type (cf. [5]). They are like twin siblings of the generic Podleś quantum spheres [18], but of opposite gender.
These new spheres were encountered while trying to understand why the locally trivial quantum Hopf fibration built over a generic Podleś sphere comes from an action of U (1) on S 3 that is equivalent to the standard Hopf-fibration action by an orientation-reversing homeomorphism [11]. Taking the natural action leads to a new fixed-point subalgebra that coincides with the fibre product of Toeplitz algebras over C(S 1 ) given by the symbol map and its composition with the automorphism inverting the unitary generator of C(S 1 ). In contrast, the C * -algebra of a generic Podleś sphere is a fibre product of two Toeplitz algebras via two symbol maps (no inversion) [19]. Geometrically, this means that the new sphere arises by gluing over the boundary a quantum disc with its mirror image. It is precisely this type of gluing of oriented discs that yields an oriented sphere. While in this classical case the orientation issue does not show up in the C * -algebra of continuous functions on S 2 , in the noncommutative setting it leads to two C * -algebras that are not even Morita equivalent. Since orientation is vital for the Dirac operator, our new example may be a useful testing ground for spectral triples (cf. [6]). Another curious aspect of the new spheres is that, unlike the aforementioned Podleś C * -algebra, their C * -algebra is not a graph algebra. However, it is very close to graph algebras [12] and points at a more general concept of a graph algebra.
Our other main goal is to construct non-crossed-product examples of principal extensions of C * -algebras [7]. Much as there is more to principal bundles than Cartesian products of groups with spaces, we argue that there is more to principal extensions of C * -algebras than crossed products. In the algebraic context of Hopf-Galois extensions, one might prove that a given extension is not a crossed product by arguing that algebraic crossed products have many nontrivial invertible elements, whereas a given polynomial algebra might lack such elements (e.g., see [9,Appendix]). This argument does not apply on the C * -level because it is easy to construct nontrivial invertible elements in a C * -algebra. It is difficult to decide whether a given C * -algebra is a crossed product with its fixed-point subalgebra because, contrary to the more restricted setting of von Neumann algebras, there appears to be no theory to manage such problems. Von Neumann algebras are at an opposite extreme to polynomial algebras. For them theory tells us when an extension of algebras has a crossed-product structure. (Think of measurable global sections of nontrivial topological bundles.) Therefore, we find the C * -context most interesting to analyse this crossed-product problem. Our method follows the classical pattern of algebraic topology: to prove that a principal bundle is nontrivial, one computes an invariant of an associated vector bundle. Herein we use the result of an index computation for finitely generated projective modules over the C * -algebra of a mirror quantum sphere to prove that the C * -algebras of noncommutative lens spaces cannot be Z-crossed products with the sphere C * -algebra. This makes them essentially different from the standard example of the noncommutative torus:

The comparison of Podleś and mirror quantum spheres
We call the universal unital * -algebra generated by C, E and F satisfying the coordinate algebra of a mirror quantum 2-sphere, and denote it by O(S 2 pq− ).
The first two relations play the role of the sphere equation. Here E should be thought of as the coordinate measuring the square of the distance of a point on the upper hemisphere from the equator plane, while F measures the square of the same distance but of a point on the lower hemisphere. Indeed, if p = 1 = q, then we have the classical sphere, E and F are functions with disjoint support, and the relationship with the Cartesian coordinates is C = x + iy, √ E − √ F = z. These new spheres were found when examining a natural U (1)-action on Heegaard-type quantum 3-spheres S 3 pqθ [1]. Recall that the polynomial algebra O(S 3 pqθ ) is the universal unital * -algebra generated by elements a and b satisfying the relations: (1 − aa * )(1 − bb * ) = 0, a * a = paa * + 1 − p, b * b = qbb * + 1 − q, ab = e 2πiθ ba, ab * = e −2πiθ b * a. Here p, q and θ are parameters with values in [0, 1). Rescaling the generators by a unitary complex number, i.e. a → e iϕ a, b → e iϕ b, gives an action α : U (1) → Aut O(S 3 pqθ ). Moreover, one can show that the subalgebra of all α-invariants O(S 3 pqθ ) U(1) is independent of θ and isomorphic with the unital * -algebra O(S 2 pq− ).
The above defined U (1)-action extends by continuity to the C * -completion C(S 3 pqθ ) of O(S 3 pqθ ). The fixed-point C * -algebra C(S 3 pqθ ) U(1) turns out to be the universal (enveloping) C * -algebra C(S 2 pq− ) of the coordinate algebra of S 2 pq− . Since C(S 3 pqθ ) and C(S 3 00θ ) are isomorphic via a U (1)-equivariant map [1, Theorem 2.8], we can conclude that C(S 2 pq− ) ∼ = C(S 2 00− ) as C * -algebras. Another way to think of this C * -algebra is in terms of the fibre product of two Toeplitz algebras given by the symbol map T σ → C(S 1 ) and its composition with the automorphism of C(S 1 ) sending the generating unitary to its inverse: We use the same symbols ρ + , ρ − and ρ λ to denote the extensions of these representations to the enveloping C * -algebra C(S 2 pq− ). It turns out that the kernels of ρ + and ρ − have zero intersection and are both isomorphic to the algebra K of compact operators on a separable Hilbert space. The quotient of C(S 2 pq− ) by Ker ρ + +Ker ρ − is isomorphic to the algebra of continuous functions on the circle. Thus, the C * -algebra C(S 2 pq− ) admits an exact sequence 0 → K ⊕ K → C(S 2 pq− ) → C(S 1 ) → 0. Using this sequence or the fibre-product formula, one can compute: The properties of C(S 2 pq− ) described so far match the respective properties of the C * -algebra of a generic Podleś sphere. One can show that the latter is isomorphic with C(S 2 pq+ ), that is the universal unital C * -algebra for the relations D * D = 1 − pP − qQ, DD * = 1 − P − Q, P D = pDP, QD = qDQ, P Q = 0, P = P * , Q = Q * , where p, q ∈ [0, 1). These relations appear almost identical to the defining relations for C(S 2 pq− ). Furthermore, for the action given by a → e iϕ a, b → e −iϕ b, we have C(S 2 pq+ ) ∼ = C(S 3 pqθ ) U(1) . The fibre-product formula for C(S 2 pq+ ) also differs only slightly from its counterpart for C(S 2 pq− ), namely C(S 2 pq+ ) ∼ = {(x, y) ∈ T × T | σ(x) = σ(y)}. On the other hand, consider two inequivalent minimal projections e ± 1 and e ± 2 in C(S 2 pq± ): it can be shown that their classes are non-zero in the respective K 0 -groups. However, [e + 1 ] = −[e + 2 ] in K 0 (C(S 2 pq+ )), while [e − 1 ] = [e − 2 ] in K 0 (C(S 2 pq− )). This leads to: Proposition 2.3 The C * -algebras C(S 2 pq+ ) and C(S 2 pq− ) are not stably isomorphic.
Finally, recall that C(S 2 pq+ ) is a graph C * -algebra [12], whose primitive ideal space consists of a pair of points (which is dense in the entire space) and a circle. The primitive ideal space of C(S 2 pq− ) turns out to be homeomorphic to that of C(S 2 pq+ ), but a careful analysis of all graph algebras whose primitive ideal space is a circle plus two points (cf. [14]) shows that no such an algebra can be isomorphic to C(S 2 pq− ).

Index computation for associated noncommutative line bundles
A first step in an index computation for finitely generated projective modules is to find appropriate summable Fredholm modules. Although the index computations for noncommutative line bundles over the generic Podleś and mirror quantum spheres proceed along the same lines (cf. [10]), it is in the structure of Fredholm modules where differences between these spheres are crucial. Indeed, one can take appropriate infinite-dimensional irreducible representations ρ 1 and ρ 2 of C(S 2 pq+ ), and they yield a 1summable Fredholm module over O(S 2 pq+ ), the universal * -algebra for the defining relations of C(S 2 pq+ ) [16,10]. This is in contrast with the situation for mirror quantum spheres, where (ρ + , ρ − ) is not a Fredholm module. Instead, making an appropriate identification of Hilbert spaces 1 , we obtain: Applying [2, Theorem 3.1], one can prove that the modules O(S 3 pqθ ) µ are finitely generated projective and can be explicitly given by the idempotents E µ = R T µ L µ . Here, respectively for µ < 0 and µ > 0, One can argue that these are isomorphic to the finitely generated projective modules C(S 2 pq− ) |µ|+1 E µ (row times matrix), with the E µ 's given above. Now, any summable Fredholm module yields a cyclic cocycle that can be paired with a K-group [3]. In particular, Lemma 3.1 gives us a trace on O(S 2 pq− ) defined by tr(x) = Tr((ρ + ⊕ ρ − )(x) − S 1 ρ λ (x)dλ). The pairing of the cyclic 0-cocycle tr and the K 0 -class [E µ ], µ ∈ Z, yields complicated-looking rational functions that have already been encountered in [8]. Therein, with the help of the noncommutative index formula [3], they were proved to be the desired integers. This brings us to the main technical result:

Quantum lens spaces as non-crossed-product examples of extensions of C * -algebras
Since Z n := Z/nZ is a subgroup of U (1), we can define new fixed-point subalgebras C(L n pqθ ) := C(S 3 pqθ ) Zn . They play the role of the algebras of continuous functions on quantum lens spaces (cf. [17,13]). Much as in the classical case, there is still a natural U (1)-action on C(L n pqθ ). It can be checked that, for any n ∈ N\{0}, this action is principal (free) in the sense of [7], and that the fixed-point subalgebras C(L n pqθ ) U(1) always coincide with C(S 2 pq− ) (cf. [20]). Furthermore, we can replace O(S 3 pqθ ) by C(L n pqθ ) in (3.1) to obtain the C(S 2 pq− )-modules C(L n pqθ ) µ and verify that they are isomorphic with the previously defined modules: ∀ n ∈ N \ {0} : C(L n pqθ ) µ ∼ = C(S 3 pqθ ) nµ as left C(S 2 pq− )-modules. (4.1) Thus it follows from Theorem 3.2 that the modules C(L n pqθ ) µ are not free unless µ = 0. On the other hand, using Fourier analysis with coefficients in a C * -algebra A [4, p.222-3] and plugging A⋊ Z into (3.1) in place of O(S 3 pqθ ), one can prove: Lemma 4.1 Let A be a unital C * -algebra and A⋊ Z an arbitrary crossed product of A with Z. Then, for the natural action of U (1) on A⋊ Z and any µ ∈ Z, we have (A⋊ Z) µ ∼ = A as left A-modules.