Distinguished bases in the K-theory of multipullback quantum complex projective spaces
Francesco D'Andrea, Piotr M. Hajac, Tomasz Maszczyk, Albert Sheu, Bartosz Zielinski
DDISTINGUISHED BASES IN THE K-THEORY OFMULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES
F. D’ANDREA, P.M. HAJAC, T. MASZCZYK, A. SHEU, AND B. ZIELI ´NSKI
Abstract.
We construct distinguished free generators of the K -group of the C*-algebra C ( CP n T ) of the multipullback quantum complex projective space. To this end, first weprove a quantum-tubular-neighborhood lemma to overcome the difficulty of the lack of anembedding of CP n − T in CP n T . This allows us to compute K ( C ( CP n T )) using the Mayer-Vietoris six-term exact sequence in K-theory. The same lemma also helps us to prove acomparison theorem identifying the K -group of the C*-algebra C ( CP nq ) of the Vaksman-Soibelman quantum complex projective space with K ( C ( CP n T )). Since this identificationis induced by the restriction-corestriction of a U (1)-equivariant *-homomorphism fromthe C*-algebra C ( S n +1 q ) of the (2 n +1)-dimensional Vaksman-Soibelman quantum sphereto the C*-algebra C ( S n +1 H ) of the (2 n +1)-dimensional Heegaard quantum sphere, weconclude that there is a basis of K ( C ( CP n T )) given by associated noncommutative vectorbundles coming from the same representations that yield an associated-noncommutative-vector-bundle basis of the K ( C ( CP nq )). Finally, using identities in K-theory afforded byToeplitz projections in C ( CP n T ), we prove noncommutative Atiyah-Todd identities. Contents
1. Introduction 22. Preliminaries 32.1. Pullbacks 32.2. Multipullback quantum spheres and projective spaces 53. The Milnor connecting homomorphism 63.1. A tubular neighbourhood lemma 63.2. K-groups 104. Associated noncommutative vector bundles 114.1. Vaksman-Soibelman quantum spheres 114.2. A pullback structure of the Hong-Szyma´nski quantum even balls 134.3. A comparison theorem 165. The Atiyah-Todd picture 195.1. The classical case revisited 195.2. The multipullback noncommutative deformation 21Acknowledgements 26References 26
Date : February 5, 2021. a r X i v : . [ m a t h . K T ] F e b F. D’ANDREA, P.M. HAJAC, T. MASZCZYK, A. SHEU, AND B. ZIELI ´NSKI Introduction
The goal of this paper is to unravel new noncommutative-topological origins of distin-guished bases of the K -groups of the multipullback quantum complex projective spaces.To this end, we use the Milnor connecting homomorphism in the Mayer-Vietoris six-term exact sequence coming from realizing the multipullback quantum complex projec-tive spaces CP n T , introduced in [8], as a gluing of a quantum tubular neighbourhood of ahyperplane T N ( CP n − T ) and a quantum polydisc (given by the n -th tensor power of theToeplitz algebra T ) in the complement of the hyperplane. This leads to the inclusion T N ( CP n − T ) ⊂ CP n T (1.1)replacing the classical inclusion CP n − ⊂ CP n , whose direct analog is lacking for the multi-pullback quantum complex projective spaces despite existing for the Vaksman-Soibelmanquantum complex projective spaces: CP n − q ⊂ CP nq .Now, a key obstacle to overcome is to prove that the natural quotient map T N ( CP n − T ) −→ CP n − T (1.2)induces an isomorphism on K-theory K ( CP n − T ) → K ( T N ( CP n − T )) in spite of the quan-tum disc not being contractible [6] (see below Corollary 2.7 therein). Inverting this isomor-phism and combining it with a homomorphism induced by (1.1) yields the desired grouphomomorphism K ( CP n T ) → K ( CP n − T ). Furthermore, the Mayer-Vietoris six-term exactsequence extends this homomorphism to the short exact seuence0 −→ Z −→ K ( CP n T ) −→ K ( CP n − T ) −→ . (1.3)By induction, we can conclude now that we have a non-canonical isomorphism of abeliangroups: K ( CP n T ) ∼ = Z n +1 . (1.4)This gives an alternative proof of (1.4) shown already in [10, Theorem 5.6].However, the associated noncommutative complex line bundle construction for thequantum U (1)-principal bundles introduced in [10] makes (1.3) a short exact sequencesof cyclic modules over the representation ring R ( U (1)), which provides a splittings of thissequences coming from the standard splitting of the filtration of R ( U (1)) by powers ofthe augmentation ideal. For this respect the isomorphism (1.4) can be made canonical,what provides a distinguished set of generators of the abelian groups K ( CP n T ).Next, we construct a K-equivalence between our quantum complex projective spaces CP n T and Vaksman-Soibelman quantum projective spaces CP nq obtained in the same wayfrom the corresponding Vaksman-Soibelman spheres S n +1 q introduced in [17]. In par-ticular, this reproves and strenghtens the results of [1]. To this end, we construct a U (1)-equivariant map between the Toeplitz quantum polydisc and the Hong-Szyma´nskinoncommutative ball B nq [12] that is compatible with an appropriate pushout structureof both of them [2]. In K-theoretical computations we apply the graph C*-algebra modelsof Vaksman-Soibelman quantum spheres [12]. To compare the generators of K-groups of HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 3 both models of quantum complex projective spaces, coming from noncommutative associ-ated vector bundles, we apply the quantum version of the principle that under equivariantmaps of principal bundles association commutes with pullbacks [9].Finally, based upon the results of [16], we show that the set of all K -classes of thenoncommutative associated line bundles satisfy some relations satisfied by powers of theclass of the Hopf line bundle in the classical Atiyah-Todd description of the ring structureof K ( CP n ). This is the best one can expect in the noncommutative setting since thenK-groups do not posses an intrinsic ring structure. It is worthwhile to emphasize that theclassical Atiyah-Todd identities come from the intrinsic ring structure enjoyed by the K -group of any commutative C*-algebra and lacking in the K -group of a noncommutativeC*-algebra. Thus Toeplitz projections, which do not exist in the classical case, bring aboutAtiyah-Todd identities in the noncommutative case, while the ring structure, which doesnot exist in the noncommutative case, brings about Atiyah-Todd identities in the classicalcase. 2. Preliminaries
Pullbacks.
Given two morphisms of C*-algebras A π −−→ A π ←−− A , a pullback ofsuch a diagram is a pair of morphisms A p ←−− A p −−→ A making the diagram A p (cid:15) (cid:15) p (cid:47) (cid:47) A π (cid:15) (cid:15) A π (cid:47) (cid:47) A (2.1)commutative, and universal in the sense that if A q ←− B q −→ A is any other pair ofmorphisms such that π ◦ q = π ◦ q , then there is a unique morphism f : B → A suchthat p i ◦ f = q i for all i = 1 ,
2. For any diagram A π −−→ A π ←−− A , a canonical pullbackis provided by the algebra A × A A := (cid:8) ( a , a ) ∈ A × A (cid:12)(cid:12) π ( a ) = π ( a ) (cid:9) (2.2)with maps to A and A given by the projections on the two factors. If A p ←−− A p −−→ A is any other pullback, the unique isomorphism to the canonical pullback C*-algebra isgiven by A (cid:51) a (cid:55)→ (cid:0) p ( a ) , p ( a ) (cid:1) ∈ A × A A . If π is injective, A × A A is isomorphic to the C*-subalgebra π − ( π ( A )) of A , theisomorphism being realized by the map A × A A (cid:51) ( a , a ) (cid:55)→ a ∈ π − ( π ( A )) . F. D’ANDREA, P.M. HAJAC, T. MASZCZYK, A. SHEU, AND B. ZIELI ´NSKI
Remark . Given a commutative diagram of C*-algebras and morphisms: A (cid:15) (cid:15) (cid:47) (cid:47) B (cid:15) (cid:15) (cid:47) (cid:47) C (cid:15) (cid:15) E (cid:47) (cid:47) F (cid:47) (cid:47) G (2.3)one proves by diagram chasing that if the two squares are pullbacks, then so is the outerrectangle; if the right square and the outer rectangles are pullbacks, so is the left square.If all algebras in (2.1) carry a U (1)-action and all arrows are equivariant we will talkabout U (1) -equivariant pullback diagram. Given a U (1)-equivariant pullback diagram,restricting/corestricting all maps to the corresponding U (1)-fixed point subalgebras weget a new pullback diagram.The following observation will be useful to “gauge” U (1)-actions. Remark . Suppose we have a commutative diagram:
AB CD E Fp p p (cid:48) π π (cid:48) φ φ π π (cid:48) and the maps φ and φ are isomorphisms. Using Remark 2.1 twice one proves that theupper-left square is a pullback diagram if and only if the outer rectangle is a pullback.To every pullback diagram (2.1), with π surjective, is associated a six-term exactsequence in K -theory: K ( A ) (cid:47) (cid:47) K ( A ) ⊕ K ( A ) (cid:47) (cid:47) K ( A ) (cid:15) (cid:15) K ( A ) (cid:79) (cid:79) K ( A ) ⊕ K ( A ) (cid:111) (cid:111) K ( A ) (cid:111) (cid:111) An important tool to compute K-theory is the next theorem, that we state for future use.
Theorem 2.3 ([7, Thm. 3.1]) . Suppose we have a commutative diagram (of C*-algebrasand morphisms):
HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 5 A p (cid:125) (cid:125) p (cid:33) (cid:33) φ (cid:45) (cid:45) B q (cid:125) (cid:125) q (cid:33) (cid:33) A π (cid:33) (cid:33) φ (cid:51) (cid:51) A π (cid:125) (cid:125) φ (cid:43) (cid:43) B ρ (cid:33) (cid:33) B ρ (cid:125) (cid:125) A φ (cid:49) (cid:49) B (2.4) with π and ρ surjective, and suppose the two squares are pullback diagrams. Assumealso that the morphisms φ , φ and φ induce isomorphisms on K -groups: φ i ∗ : K ∗ ( A i ) ∼ = −→ K ∗ ( B i ) , φ ∗ : K ∗ ( A ) ∼ = −→ K ∗ ( B ) . Then the morphism φ also induces an isomorphim on K -theory: φ ∗ : K ∗ ( A ) ∼ = −→ K ∗ ( B ) . Multipullback quantum spheres and projective spaces.
Given a family of ∗ -homomorphisms of C*-algebras: (cid:8) π ij : A i → A ij = A ji (cid:9) i,j ∈ I, i (cid:54) = j , with I = { , . . . , k } a finite set, one can similarly define the canonical multi-pullback A π as a suitable limit, or concretely as A π := (cid:110) ( a , . . . , a k ) ∈ A × . . . × A k (cid:12)(cid:12)(cid:12) π ij ( a i ) = π ji ( a j ) ∀ i, j ∈ I, i (cid:54) = j (cid:111) . Recall that the C*-algebra T of Toeplitz operators is the C*-subalgebra of B ( (cid:96) ( N ))generated by the unilateral right shift t on (cid:96) ( N ), which is the operator given on thecanonical basis { ξ n : n ∈ N } of (cid:96) ( N ) by tξ n = ξ n +1 . It can be equivalently defined in amore abstract way as universal C*-algebra generated by a partial isometry t . We denoteby σ : T → C ( S ) the symbol map.For n ≥
0, the choice A i := T ⊗ i ⊗ C ( S ) ⊗ T ⊗ n − i , ∀ i = 0 , . . . , n,A ij = A ji := T ⊗ i ⊗ C ( S ) ⊗ T ⊗ j − i − ⊗ C ( S ) ⊗ T ⊗ n − j ∀ ≤ i < j ≤ n,π ij := id ⊗ j ⊗ σ ⊗ id ⊗ ( n − j ) ∀ i, j = 0 , . . . , n : i (cid:54) = j, defines the 2 n + 1-dimensional multipullback quantum sphere [10, § C ( S n +1 H ) defined in termsof generators and relations as follows (this is a special case of Theorem 2.3 of [10]). Definition 2.4.
For n ≥
0, we denote by C ( S n +1 H ) the universal C*-algebra with gener-ators s , . . . , s n satisfying the relations:[ s i , s j ] = [ s i , s ∗ j ] = 0 ∀ ≤ i (cid:54) = j ≤ n F. D’ANDREA, P.M. HAJAC, T. MASZCZYK, A. SHEU, AND B. ZIELI ´NSKI s ∗ i s i = 1 ∀ i = 0 , . . . , n, n (cid:89) i =0 (cid:0) − s i s ∗ i ) = 0 . We think of C ( S ) – the C*-algebra of continuous complex-valued functions on theunit circle – as a compact quantum group with standard coproduct dual to the groupmultiplication. Thus, C ( S ) is the universal C*-algebra generated by a unitary u withcoproduct defined by ∆ u = u ⊗ u .We denote the generators of T ⊗ n +1 by t i := 1 ⊗ . . . ⊗ (cid:124) (cid:123)(cid:122) (cid:125) i times ⊗ t ⊗ ⊗ . . . ⊗ (cid:124) (cid:123)(cid:122) (cid:125) n − i times , i = 0 , . . . , n. Right coactions of C ( S ) on C ( S n +1 H ) and T ⊗ n +1 , all denoted by δ , are defined on gener-ators by δ ( s i ) = s i ⊗ u and δ ( t i ) = t i ⊗ u respectively (for i = 0 , . . . , n ). Lemma 2.5 ([10, Eq. (4.5)]) . For all n ≥ , there is a U (1) -equivariant short exactsequence → K ( (cid:96) ( N ⊗ n +1 )) → T ⊗ n +1 σ n −→ C ( S n +1 H ) → σ n is defined explicitly on generators by σ n ( t i ) := s i , i = 0 , . . . , n . Remark.
Multipullback quantum spheres admit a presentation as higher rank graph C*-algebras [10], while such a presentation for multipullback projective spaces is not known.3.
The Milnor connecting homomorphism
A tubular neighbourhood lemma.
For every n ≥ C ( S n +1 H ) C ( S n − H ) ⊗ T T ⊗ n ⊗ C ( S ) C ( S n − H ) ⊗ C ( S ) p p π π (3.1)which is U (1)-equivariant with respect to the diagonal action on each vertex. Let us givethe explicit definition of the four maps and state the result in the form of a theorem.With a slight abuse of notation, we denote by the same symbol the generators of C ( S n +1 H ) for different values of n , and similar for T n . HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 7
Lemma 3.1.
Define the maps in (3.1) by π = id ⊗ σ , π = σ n − ⊗ id , and the remainingtwo in terms of generators by: p ( s i ) := (cid:26) s i ⊗ ∀ i = 0 , . . . , n − ⊗ t if i = np ( s i ) := (cid:26) t i ⊗ ∀ i = 0 , . . . , n − ⊗ u if i = n Then (3.1) is a pullback diagram, U (1) -equivariant w.r.t. the diagonal U (1) -action oneach vertex. Furthermore, all four maps in the diagram are surjective. We need one more preliminary lemma.
Lemma 3.2.
For all k ≥ and all n ≥ , we have a U (1) -equivariant pullback diagram: C ( S n +1 H ) ⊗ T k C ( S n − H ) ⊗ T k +1 T ⊗ n ⊗ C ( S ) ⊗ T k C ( S n − H ) ⊗ C ( S ) ⊗ T k p k p k π k π k (3.2) where the U (1) -action is diagonal on the top and left vertices, and only on the C ( S ) factor on the bottom and right vertices. The map π k is given by π k = σ n − ⊗ id C ( S ) ⊗ id T k . Proof.
Given a pullback diagram, if we tensor each vertex with a fixed unital C*-algebraand each map with the identity map, the new diagram we get is still a pullback diagram[14, Theo. 3.9]. We get (3.2) from (3.1) first tensoring all algebras with T k and all mapswith the identity. This gives a U (1)-equivariant pullback diagram: C ( S n +1 H ) • ⊗ T k • C ( S n − H ) • ⊗ T k +1 • T ⊗ n • ⊗ C ( S ) • ⊗ T k • C ( S n − H ) • ⊗ C ( S ) • ⊗ T k • p ⊗ id T k p ⊗ id T k π ⊗ id T k π ⊗ id T k where p , p , π , π are the maps in Lemma 3.1 and a dot denotes the factors where U (1)acts. Now we use the gauging automorphism(see e.g., [10, Section 2.3]) and Remark 2.2 F. D’ANDREA, P.M. HAJAC, T. MASZCZYK, A. SHEU, AND B. ZIELI ´NSKI to move the actions on the bottom and right vertices on the C ( S ) factor. More precisely,we apply to the bottom and right vertices in the above diagram which are of the form A ⊗ C ( S ) ⊗ B the automorphism φ : A • ⊗ C ( S ) • ⊗ B • → A ⊗ C ( S ) • ⊗ B, a ⊗ f ⊗ b (cid:55)→ a (0) ⊗ a (1) f b ( − ⊗ b (0) where a (cid:55)→ a (0) ⊗ a (1) and b (cid:55)→ b ( − ⊗ b (0) are the coactions of C ( S ) on A and B dual tothe U (1)-action. This, by the remark 2.2 yields a new U (1)-equivariant pullback diagram: C ( S n +1 H ) • ⊗ T k • C ( S n − H ) • ⊗ T k +1 • T ⊗ n ⊗ C ( S ) • ⊗ T k C ( S n − H ) ⊗ C ( S ) • ⊗ T k p k p k π k π k where with a slight abuse of notation (two different maps are both denoted by φ ): p k := p ⊗ id T k , π k := φ ◦ ( π ⊗ id T k ) ,π k := φ ◦ ( π ⊗ id T k ) ◦ φ − , p k := φ ◦ ( p ⊗ id T k ) . Equivariance of π implies that π k = π ⊗ id T k . (cid:3) For k = 0, the U (1)-equivariant part of (3.2) gives the pullback diagram: C ( CP n T ) (cid:0) C ( S n − H ) ⊗ T (cid:1) U (1) T ⊗ n C ( S n − H ) p φ ◦ p φ ◦ (id ⊗ σ ) σ n − (3.3)In order to compute recursively the K-theory of multipullback quantum projective spaces,we need to relate the K-theory of the algebra of functions C (cid:0) T N ( CP n − T ) (cid:1) := ( C ( S n − H ) ⊗T ) U (1) on a quantum tubular neighbourhood of a hyperplane to that of C ( CP n − T ). Thisis obtained from the following tubular neighbourhood lemma . Lemma 3.3.
For all k ∈ N and all n ≥ , the U (1) -equivariant map (w.r.t. the diagonal U (1) -action): id ⊗ T : C ( S n − H ) ⊗ T ⊗ k → C ( S n − H ) ⊗ T ⊗ k +11 Geometrically, we can think of the former algebra as describing a bundle of closed quantum disksof the normal bundle (isomorphic to the Hopf line bundle) of the quantum hyperplane C P n − T in C P n T .Note that classically, the bundle of normal discs is homeomorphic to a tubular neighbourhood. HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 9 restricted and corestricted to the U (1) -fixed point algebras induces an isomorphism inK-theory: K ∗ (cid:0) ( C ( S n − H ) ⊗ T ⊗ k ) U (1) (cid:1) ∼ = −→ K ∗ (cid:0) ( C ( S n − H ) ⊗ T ⊗ k +1 ) U (1) (cid:1) . (3.4) Proof.
Let us consider then the following commutative diagram: C ( S n +1 H ) ⊗ T k C ( S n − H ) ⊗ T k +1 T n ⊗ C ( S ) ⊗ T k C ( S n − H ) ⊗ C ( S ) ⊗ T k C ( S n +1 H ) ⊗ T k +1 C ( S n − H ) ⊗ T k +2 T n ⊗ C ( S ) ⊗ T k +1 C ( S n − H ) ⊗ C ( S ) ⊗ T k +1 The left diamond is (3.2), the right diamond is (3.2) with k replaced by k +1, the horizontalarrows are all given by id ⊗ T . Passing to fixed point algebras we get the commutativediagram: (cid:0) C ( S n +1 H ) ⊗ T k (cid:1) U (1) (cid:0) C ( S n − H ) ⊗ T k +1 (cid:1) U (1) T n ⊗ T k C ( S n − H ) ⊗ T k (cid:0) C ( S n +1 H ) ⊗ T k +1 (cid:1) U (1) (cid:0) C ( S n − H ) ⊗ T k +2 (cid:1) U (1) T n ⊗ T k +1 C ( S n − H ) ⊗ T k +1 φ n,k φ n − ,k +1 (3.5)where the two diamonds are still pullback diagrams. We now prove by induction on n ≥ k ≥
0, the map φ n,k induces an isomorphism in K-theory.Let us start with n = 0 and consider the U (1)-equivariant commutative diagram: C ( S ) • ⊗ T k • C ( S ) • ⊗ T k +1 • C ( S ) • ⊗ T k C ( S ) • ⊗ T k +1 id ⊗ T id ⊗ T where the vertical arrows are the isomorphisms a ⊗ b (cid:55)→ ab ( − ⊗ b (0) (c.f., [10, Section 2.3]).The U (1)-invariant part gives: (cid:0) C ( S ) • ⊗ T k • (cid:1) U (1) (cid:0) C ( S ) • ⊗ T k +1 • (cid:1) U (1) T k T k +1 φ ,k ⊗ T (3.6)Suppose A and B are two unital C*-algebras both with K = Z [1] and K = 0. Thenany unital *-homomorphism A → B induces an isomorphism in K-theory. In particular, T k ⊗ T −−→ T k +1 induces an isomorphism in K-theory and, since the vertical arrows in (3.6)are isomorphisms, φ ,k induces an isomorphism in K-theory as well.Let us now assume by inductive hypothesis that φ n − ,k induces an isomorphism in K-theory for all k ≥
0. Let us look again at diagram (3.5). The unital *-homomorphisms C ( S n − H ) ⊗ T k → C ( S n − H ) ⊗ T k +1 and T n ⊗ T k → T n ⊗ T k +1 induce isomorphisms in K-theory by the same argument as before (all algebras have K = Z [1] and K = 0); φ n − ,k +1 induces an isomorphism in K-theory by inductive hypothesis. It follows from Thm. 2.3that the top arrow φ n,k induces an isomorphism in K-theory as well, thus completing theproof. (cid:3) K-groups.
Combining the six term exact sequence induced by the pullback diagram(3.3) with Lemma 3.3 we obtain the following description of the K-theory of multipullbackquantum projective spaces.
Proposition 3.4.
For all n ≥ : K ( C ( CP n T )) ∼ = K ( C ( CP n − T )) ⊕ d (cid:0) K ( C ( S n − H )) (cid:1) , (3.7) where d is Milnor connecting homomorphism.Proof. For arbitrary n ≥ k = 0 Lemma 3.3 gives an isomorphism K ∗ (cid:0) C ( CP n − T ) (cid:1) ∼ = −→ K ∗ (cid:0) ( C ( S n − H ) ⊗ T ) U (1) (cid:1) . Recall [10] that K ( C ( S n − H )) = Z [1], K ( C ( S n − H )) ∼ = Z (a generator is given in Corollary4.9), K ( CP n T ) ∼ = Z n +1 and K ( CP n T ) = 0.Let p and p be the maps defined in Lemma 3.1, φ the map a ⊗ b (cid:55)→ a (0) ⊗ a (1) b . Thesix term exact sequence induced by (3.3) becomes K ( C ( CP n T )) K (cid:0)(cid:0) C ( S n − H ) ⊗ T (cid:1) U (1) (cid:1) ⊕ Z K ( C ( S n − H )) ∼ = Z K ( C ( S n − H )) ∼ = Z K ( C ( CP n − T )) ⊕ K ( T ) K ( C ( CP n T )) ( p ∗ , p (cid:48) ∗ ) d HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 11 v e v v e e (a) Graph Σ . (b) Graph Γ . Figure 1. where p (cid:48) = φ ◦ p . We want to prove that we can extract from this a short exact sequence:0 → K ( C ( S n − H )) d −−→ K ( C ( CP n T )) p ∗ −−→ K (cid:0)(cid:0) C ( S n − H ) ⊗ T (cid:1) U (1) (cid:1) → p is surjective (Lemma 3.1), so that p ∗ is surjective as well. Weneed to prove that Im( d ) = ker( p ∗ ). Since the six term sequence is exact, it is enoughto show that p ∗ and ( p ∗ , p (cid:48) ∗ ) have the same kernel. Butker( p ∗ , p (cid:48) ∗ ) = ker( p ∗ ) ∩ ker( p (cid:48) ∗ ) , so that we only have to prove that ker( p ∗ ) ⊂ ker( p (cid:48) ∗ ). This follows from (3.3) andfunctoriality of K-theory:ker( p ∗ ) ⊂ ker( π ∗ ◦ p ∗ ) = ker( σ n − ∗ ◦ p (cid:48) ∗ ) = ker( p (cid:48) ∗ ) , where π = φ ◦ (id ⊗ σ ) and last equality holds because σ n − ∗ : K ( T ⊗ n ) → K ( C ( S n − H ))is an isomorphism (both domain and codomain are Z [1], and σ n − is a unital ∗ -homomorphism).Composing p ∗ with the inverse of the isomorphism (3.4) (for k = 0) we get a shortexact sequence0 → K ( C ( S n − H )) d −−→ K ( C ( CP n T )) → K ( C ( CP n − T )) → . (3.9)Finally, such a sequence splits by the freeness of the Z -module K ( C ( CP n − T )) ∼ = Z n , andwe get (3.7). (cid:3) Associated noncommutative vector bundles In § q -spheres and balls, in § § U (1)-equivariant*-homomorphisms from q -sphere to multipullback quantum spheres that induces an iso-morphism between the K-theory of U (1)-fixed point algebras. This will be used to describegenerators of K ( C ( CP n T )) in terms of associated vector bundles.4.1. Vaksman-Soibelman quantum spheres.
Let E = ( E , E , s, r ) be a directedgraph, with s, r : E → E the source and range maps. Recall that E is row-finite if s − ( v )is a finite set for all v ∈ E . A sink is a vertex v that emits no edges, i.e. s − ( v ) = ∅ . A path is a sequence e e . . . e n of edges with r ( e i − ) = s ( e i ) for all i = 1 , . . . , n ; such a pathis a cycle if r ( e n ) = s ( e ). A loop is an edge e with r ( e ) = s ( e ) (a cycle with one edge). v v n e e e e e nn e e e e e e Figure 2.
Graph Σ n of the graph C*-algebra C ( S n +1 q ) Definition 4.1.
The graph C*-algebras C ∗ ( E ) of a row-finite graph E is the universalC*-algebra generated by mutually orthogonal projections (cid:8) P v : v ∈ E (cid:9) and partialisometries (cid:8) S e : e ∈ E (cid:9) with relations (Cuntz-Krieger relations): S ∗ e S e = P r ( e ) for all e ∈ E (cid:88) e ∈ E : s ( e )= v S e S ∗ e = P v for all v ∈ E that are not sinks. C ∗ ( E ) admits C ( S ) co-action ρ : C ∗ ( E ) → C ∗ ( E ) ⊗ C ( S ) defined by ρ ( S e ) := S e ⊗ u , ρ ( P v ) := P v ⊗ Theorem 4.2.
Let E be a row-finite graph, A a C*-algebra with a continuous action of U (1) and ω : C ∗ ( E ) → A a U (1) -equivariant ∗ -homomorphism. If ω ( P v ) (cid:54) = 0 for all v ∈ E , then ω is injective. Let Σ and Γ be the graphs in Figure 1. It is well known that [15]: (i) there is anisomorphism C ∗ (Σ ) → C ( S ) defined on generators by S e (cid:55)→ u , P v (cid:55)→ , (4.1)with u the unitary generator of C ( S ); (ii) there is an isomorphism C ∗ (Γ ) → T definedon generators by S e (cid:55)→ t t ∗ , S e (cid:55)→ t (1 − tt ∗ ) , P v (cid:55)→ tt ∗ , P v (cid:55)→ − t ∗ . (4.2)where t is the right unilateral shift. These isomorphisms intertwine the U (1) gauge actionon the graph C*-algebras with the natural action on C ( S ) risp. T .The C*-algebra C ( S n +1 q ) [17] can be presented as graph C*-algebra of the graph inFigure 2. Such a graph has n + 1 vertices v , . . . , v n and an edge e ij : v i → v j for all i ≤ j . Note that all vertices here are targets of edges. Therefore the corresponding graphC*-algebra C ( S n +1 q ) is fully generated by partial isometries corresponding to the edges.By removing the edge e nn from the graph in Figure 2 we obtain the graph Γ n of the2 n -dimensional noncommutative closed ball C ( B nq ). Abusing notation we denote by the HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 13 same symbols edges and vertices of the two graphs, and the generators of both graphC*-algebras. By removing v n and all the edges with target v n from Γ n we obtain thegraph Σ n − of S n − q .We can view B nq as half equator of S n +1 q , and S n − q as boundary of B nq . Two C*-algebra morphisms r n : C ( S n +1 q ) → C ( B nq ) and ∂ n : C ( B nq ) → C ( S n − q ) , the restriction to half equator and boundary map, can be defined in terms of the projec-tions and partial isometries in Definition 4.1 as follows: r n sends the generators S e , P v of C ( S n +1 q ) to the homonymous generators S e , P v of C ( B nq ) for all v ∈ Σ n , e ∈ Σ n (cid:114) { e nn } ,and sends S e nn ∈ C ( S n +1 q ) to P v n ∈ C ( B nq ); ∂ n sends S e , P v ∈ C ( B nq ) to S e , P v ∈ C ( S n − q ) for all v ∈ Γ n (cid:114) { v n } , e ∈ Γ n (cid:114) r − ( v n ), sends P v n to 0 and S e to 0 if r ( e ) = v n .Observe that ∂ n is U (1)-equivariant while r n is not. The composition ∂ n ◦ r n is U (1)-equivariant as well.In parallel to (3.1), it was proved in [2] that we have a U (1)-equivariant pullbackdiagram C ( S n +1 q ) • C ( S n − q ) • C ( B nq ) ⊗ C ( S ) • C ( S n − q ) ⊗ C ( S ) • ∂ n ◦ r n ( r n ⊗ id) ◦ δδ ∂ n ⊗ id (4.3)4.2. A pullback structure of the Hong-Szyma´nski quantum even balls.
In thissection we are going to prove that, for all n ≥
1, there is a pullback diagram: C ( B nq ) ∂ n (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) ρ n (cid:47) (cid:47) T ⊗ nσ n − (cid:15) (cid:15) (cid:15) (cid:15) C ( S n − q ) (cid:31) (cid:127) ω n − (cid:47) (cid:47) C ( S n − H ) (4.4)Here ∂ n is the boundary map in § σ n − the map in Lemma 2.5. We now constructthe remaining two maps. The vertical maps are surjective, while the horizontal maps aregoing to be injective. Proposition 4.3.
Two U (1) -equivariant injective ∗ -homomorphism ρ n : C ( B nq ) → T ⊗ n and ω n − : C ( S n − q ) → C ( S n − H ) are defined by: ρ n ( S e ij ) := t i t j t ∗ j j − (cid:89) k =0 (1 − t k t ∗ k ) ∀ ≤ i ≤ j < n, (4.5a) ρ n ( S e in ) := t i n − (cid:89) k =0 (1 − t k t ∗ k ) ∀ ≤ i < n, (4.5b) ω n − ( S e ij ) := s i s j s ∗ j j − (cid:89) k =0 (1 − s k s ∗ k ) ∀ ≤ i ≤ j < n. (4.5c) With ρ n and ω n − defined as above, (4.4) is a commutative diagram.Proof. One checks with an explicit computation that Cuntz-Krieger relations are satisfied,so that ρ n and ω n − define ∗ -homomorphisms. For all 0 ≤ i ≤ j < n one has ρ n ( S e ij ) ∗ ρ n ( S e ij ) = t j t ∗ j j − (cid:89) k =0 (1 − t k t ∗ k ) =: ρ n ( P v j ) , and for all 0 ≤ i < n : ρ n ( S e in ) ∗ ρ n ( S e in ) = n − (cid:89) k =0 (1 − t k t ∗ k ) =: ρ n ( P v n ) . Since the t i ’s are commuting isometries, and (1 − t k t ∗ k ) t k = 0, it follows that the projections ρ n ( P v j ) are mutually orthogonal. Since ρ n ( S e ij ) ρ n ( S e ij ) ∗ = t i ρ n ( P v j ) t ∗ i one also has: n (cid:88) j = i ρ n ( S e ij ) ρ n ( S e ij ) ∗ = t i (cid:18) n (cid:88) j = i ρ n ( P v j ) (cid:19) t ∗ i , for all i (cid:54) = n . By induction on i from n to lower values one proves that n (cid:88) j = i ρ n ( P v j ) = i − (cid:89) k =0 (1 − t k t ∗ k ) , (4.6)that means n (cid:88) j = i ρ n ( S e ij ) ρ n ( S e ij ) ∗ = t i i − (cid:89) k =0 (1 − t k t ∗ k ) t ∗ i = t i t ∗ i i − (cid:89) k =0 (1 − t k t ∗ k ) = ρ n ( P v i )for all i (cid:54) = n . Cuntz-Krieger relations are then satisfied by the elements ρ n ( S e ij ) , ρ n ( P v j )),proving that the ∗ -homomorphism ρ n is well-defined. Similarly one proves that ω n − iswell-defined.Equivariance of ρ n and ω n − is obvious, while injectivity follows from Theorem 4.2.Finally, one explicitly checks on generators that the diagram is commutative (in particularone may notice that σ n − ρ n ( S e in ) = 0 since (cid:81) n − k =0 (1 − s k s ∗ k ) = 0 in C ( S n − H )). (cid:3) Lemma 4.4.
Im( ρ n ) ⊃ K ( (cid:96) ( N n )) for all n ≥ . By convention, an empty sum is 0 and an empty product is 1.
HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 15
Proof.
Im( ρ n ) is a C*-subalgebra of T ⊗ n . It contains a non-zero compact operator, ρ ( P v n ) = (cid:81) n − k =0 (1 − t k t ∗ k ) ∈ K ( (cid:96) ( N n )). We now prove that it is irreducible, so thatfrom [5, Corollary I.10.4] it will follow that Im( ρ n ) ⊃ K ( (cid:96) ( N n )).Let a ∈ B ( (cid:96) ( N )) ⊗ n . We must show that a is in the commutant of Im( ρ n ) iff it isproportional to the identity. It follows from (4.6) that x i := n (cid:88) j = i ρ n ( S e ij ) = t i n (cid:88) j = i ρ n ( P v j ) = t i i − (cid:89) k =0 (1 − t k t ∗ k )for all 0 ≤ i < n . Since a commutes with x = t , it follows that its first leg is proportionalto the identity. By the same argument, since it commutes with x = (1 − t t ∗ ) t , its secondleg must be proportional to the identity as well. By repeating the argument n times onereaches the conclusion. (cid:3) Proposition 4.5. (4.4) is a pullback diagram.Proof.
It is enough to prove that [14, Prop. 3.1]:(i) ker ∂ n ∩ ker ρ n = { } , (ii) Im( ρ n ) = σ − n − (cid:0) ω n − (cid:0) C ( S n − q ) (cid:1)(cid:1) , (iii) ∂ n (ker ρ n ) = ker ω n − . Points (i) and (iii) are trivial, since ρ n and ω n − are both injective. In (ii), the inclusionIm( ρ n ) ⊂ σ − n − (cid:0) ω n − (cid:0) C ( S n − q ) (cid:1)(cid:1) follows from the commutativity of the diagram (4.4).We have to prove the opposite inclusion.Take any x ∈ σ − n − (cid:0) ω n − (cid:0) C ( S n − q ) (cid:1)(cid:1) . Then σ n − ( x ) = ω n − ( y ) for some y ∈ C ( S n − q ).Since ∂ n is surjective, y = ∂ n z for some z ∈ C ( B n q ). From σ n − ( ρ n ( z )) = ω n − ( ∂ n ( z ))) = ω n − ( y ) = σ n − ( x )we deduce that x − ρ n ( z ) ∈ ker σ n − . From Lemma 2.5 and Lemma 4.4, ker σ n − = K ( (cid:96) ( N n )) ⊂ Im( ρ n ). Thus x − ρ n ( z ) = ρ n ( t ) for some t ∈ C ( B nq ). But then x = ρ n ( z + t )is in the image of ρ n , thus proving the thesis. (cid:3) As a byproduct of previous proposition, we get a pullback realization of Vaksman-Soibelman quantum spheres and projective spaces in terms of non-spherical balls.
Proposition 4.6.
There exists pullback diagrams C ( S n +1 q ) C ( S n − q ) T ⊗ n ⊗ C ( S ) C ( S n − H ) ⊗ C ( S ) C ( C P nq ) C ( C P n − q ) T ⊗ n C ( S n − H ) Proof.
Recall that in (2.3) the outer rectangle is a pullback diagram if both inner squaresare pullbacks. The U (1)-invariant part of (4.3) gives the pullback diagram: C ( CP nq ) C ( CP n − q ) C ( B nq ) C ( S n − q ) ∂ n If we attach it to the pullback diagram (4.4) we get the second diagram in the Proposition.To get the first one we attach (4.3) to the diagram obtained by tensoring (4.4) everywherewith C ( S ) and tensoring all maps with the identity on C ( S ) (this is a pullback diagramby [14, Thm. 3.9]). (cid:3) A comparison theorem.Proposition 4.7.
The map ω n : C ( S n +1 q ) → C ( S n +1 H ) in Prop. 4.3 induces an isomor-phism in K-theory. Its restriction to U (1) -fixed point algebras induces an isomorphism K ∗ ( C ( CP nq )) → K ∗ ( C ( CP n T )) as well.Proof. Let us begin by constructing a commutative diagram: C ( S n +1 q ) • C ( S n − q ) • C ( B nq ) ⊗ C ( S ) • C ( S n − q ) ⊗ C ( S ) • C ( S n +1 H ) • C ( S n − H ) • ⊗ T • T ⊗ n ⊗ C ( S ) • C ( S n − H ) ⊗ C ( S ) • ω n ω n − ⊗ T ρ n ⊗ id ω n − ⊗ id (4.7)where the left diamond is the pullback diagram (4.3) and the right diamond is the pullbackdiagram (3.2) for k = 0. In order to check commutativity, let us rewrite the four faces: HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 17 C ( S n +1 q ) C ( S n +1 H ) C ( B nq ) ⊗ C ( S ) T ⊗ n ⊗ C ( S ) ω n ( r n ⊗ id) δ φ ◦ p ρ n ⊗ id C ( S n +1 q ) C ( S n +1 H ) C ( S n − q ) C ( S n − H ) ⊗ T ω n ∂ n ◦ r n p ω n − ⊗ T C ( B nq ) ⊗ C ( S ) T ⊗ n ⊗ C ( S ) C ( S n − q ) ⊗ C ( S ) C ( S n − H ) ⊗ C ( S ) ρ n ⊗ id ∂ n ⊗ id σ n − ⊗ id ω n − ⊗ id C ( S n − q ) C ( S n − H ) ⊗ T C ( S n − q ) ⊗ C ( S ) C ( S n − H ) ⊗ C ( S ) ω n − ⊗ T δ φ ◦ (id ⊗ σ ) ω n − ⊗ id where p and p are the maps in Lemma 3.1 and φ is the gauge map a ⊗ f (cid:55)→ a (0) ⊗ a (1) f (cf. [10, Section 2.3]). The third diagram is simply (4.4) tensored everywhere by C ( S ).Commutativity of the other three diagrams can be explicitly checked on generators. Letus start with the first diagram. Firstly φ ◦ p ( s i ) = (cid:26) t i ⊗ u ∀ i = 0 , . . . , n − ⊗ u if i = n Then one checks using (4.5) that φ ◦ p ◦ ω n ( S e ij ) = (cid:26) ρ n ( S e ij ) ⊗ u if i (cid:54) = nρ n ( P v n ) ⊗ u if i = n Since ( r n ⊗ id) ◦ δ ( S e ij ) = (cid:26) S e ij ⊗ u if i (cid:54) = nP v n ⊗ u if i = n clearly φ ◦ p ◦ ω n = ( ρ n ⊗ id) ◦ ( r n ⊗ id) ◦ δ . We now pass to the second diagram. Here p ◦ ω n ( S e ij ) = (cid:26) ρ n ( S e ij ) ⊗ ∀ ≤ i ≤ j < n j = n (for j = n one has (cid:81) n − k =0 (1 − s k s ∗ k ) = 0 in C ( S n − H )). On the other hand ∂ n ◦ r n ( S e ij ) = (cid:26) S e ij ∀ ≤ i ≤ j < n j = n hence p ◦ ω n = ω n − ◦ ∂ n ◦ r n ⊗ T . Finally we consider the fourth diagram. Firstly wenotice that φ ◦ (id ⊗ σ )( s i ⊗
1) = s i ⊗ u for all i = 0 , . . . , n −
1, and then φ ◦ (id ⊗ σ ) ◦ ( ω n − ⊗ T ) = φ ◦ ω n − ⊗ C ( S ) the latter is equal to ( ω n − ⊗ id) δ by equivariance of ω n − . This can also be checkedexplicitly on generators: φ ◦ ω n − ( S e ij ) ⊗ C ( S ) = ω n − ( S e ij ) ⊗ u = ( ω n − ⊗ id) δ ( S e ij ) for all 0 ≤ i ≤ j < n .Now that we proved that (4.7) is commutative, we can use Theorem 2.3 and inductionon n to prove that ω n induces an isomorphism in K-theory. For n = 0, ω : C (Σ ) → C ( S )is the isomorphism sending S e to u . Assume by inductive hypothesis that, for some n ≥ ω n − induces an isomorphism in K-theory. The map ρ n induces an isomorphismfor trivial reasons: it is a unital *-homomorphism and both domain and codomain have K = Z [1] and K = 0.Recall that if f : A → B and g : C → D are ∗ -homomorphisms, then K ∗ ( A ⊗ B ) ∼ = K ∗ ( A ) ⊗ K ∗ ( B ) by Kunneth formula (the tensor product on the right hand side is thegraded tensor product of graded abelian groups) and under this isomorphism ( f ⊗ g ) ∗ = f ∗ ⊗ g ∗ . It follows that ω n − ⊗ id and ρ n ⊗ id induce isomorphisms in K-theory.Finally, note that both ω n − and the map 1 T : C → T induce isomorphisms in K-theory (the latter because K ( T ) = Z [1] and K ( T ) = 0), hence ω n − ⊗ T induces anisomorphism in K-theory as well.It follows from Theorem 2.3 that ω n induces an isomorphism in K-theory, thus com-pleting the inductive step.Concerning the fixed point algebras, from (4.7) we get the commutative diagram: C ( CP nq ) C ( CP n − q ) C ( B nq ) C ( S n − q ) C ( CP n T ) (cid:0) C ( S n − H ) ⊗ T (cid:1) U (1) T ⊗ n C ( S n − H ) ω n ω n − ⊗ T ρ n ω n − (4.8)where now ω n and ω n − ⊗ T are restricted and corestricted to the fixed point algebras. Wecan use again Theorem 2.3 to prove that ω n : C ( CP nq ) → C ( CP n T ) induces an isomorphismin K-theory. For n = 0, ω : C (Γ ) → C is the isomorphism sending P v to 1. Assume theclaim is true for ω n − , n ≥ ω n − : C ( S n − q ) → C ( S n − H ) and ρ n : C ( B nq ) →T ⊗ n in (4.8) induce an isomorphism in K-theory. The map ω n − ⊗ T : C ( CP n − q ) → (cid:0) C ( S n − H ) ⊗ T (cid:1) U (1) is the composition of the map ω n − : C ( CP n − q ) → C ( CP n − T ),which induces an isomorphism in K-theory by inductive hypothesis, and the restriction-corestriction to U (1)-fixed point algebras of the map ⊗ T : C ( S n − H ) → C ( S n − H ) ⊗ T ,that gives an isomorphism in K-theory by Lemma 3.3. From Theorem 2.3, we concludethat ω n : C ( CP nq ) → C ( CP n T ) induces an isomorphism in K-theory. (cid:3) HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 19
Corollary 4.8.
The isomorphism induced on K-theory by the U (1) -equivariant map ω n : C ( S n +1 q ) → C ( S n +1 H ) identifies the set of distinguished generators coming from noncom-mutative line bundles associated with the corresponding quantum U (1) -principal bundles S n +1 q and S n − H over quantum complex projective spaces CP nq and CP n T , respectively.Proof. By virtue of Proposition 4.3 and Proposition 4.7 it follows immediately from thetheorem “
Pushforward commutes with association ” from [9]. (cid:3)
We know [2, Theorem 2.3] that K ( C ( S n +1 q )) ∼ = Z is generated by the class of theunitary S e nn + (1 − S e nn S ∗ e nn ). Its image through ω n , as a corollary of previous proposition,gives a generator of K ( C ( S n +1 H )). Using s n s ∗ n (cid:81) n − k =0 (1 − s k s ∗ k ) = (cid:81) n − k =0 (1 − s k s ∗ k ) in C ( S n +1 H ) such a unitary can be written in the following form: Corollary 4.9. K ( C ( S n +1 H )) ∼ = Z is generated by the unitary U := s n (cid:81) n − k =0 (1 − s k s ∗ k ) . The Atiyah-Todd picture
The classical case revisited.
The classical theorem of Atiyah-Todd says that the K -group K ( CP n ) = K ( C ( CP n )) equipped with the ring structure defined via the ten-sor product of vector bundles over CP n (or, equivalently, the tensor product of finitelygenerated projective left C ( CP n )-modules, which are automatically C ( CP n )-bimodules)fits into the following commutative square of rings: Z [ t, t − ] (cid:15) (cid:15) ∼ = (cid:47) (cid:47) R ( U (1)) (cid:15) (cid:15) Z [ x ] / ( x n +1 ) ∼ = (cid:47) (cid:47) K ( CP n ) . (5.1)Here the left vertical arrow is given by t (cid:55)→ x , the right vertical arrow is induced by theassociated vector bundle construction, the top isomorphism maps t into the fundamentalrepresentation of U (1) in the representation ring R ( U (1)), and the bottom isomorphismmaps x to the K-theory element [L ] − [1], where L denotes the Hopf line bundle on CP n associated with the fundamental representation of U (1). Below, for any k ∈ Z , we denoteby L k the k -th tensor power of L when k is non-negative, and the | k | -th tensor power ofL − when k is negative and where L − is the Hopf line bundle on CP n associated withthe dual of the fundamental representation of U (1). Equivalently, L k is the Hopf linebundle on CP n associated with the k -th tensor power of the fundamental representationof U (1), where negative tensor powers refer to tensor powers of the dual of the fundamentalrepresentation of U (1).Since the elements (1 + x ) k , k = 0 , . . . , n , form a basis of the free Z -module Z [ x ] / ( x n +1 )and the assignment (1 + x ) (cid:55)→ [L ] gives an isomorphism of rings, the classes[L ] , . . . , [L n ] (5.2)form a basis of the free Z -module K ( CP n ). We call this basis the Atiyah-Todd basis . Our next step is to unravel how the classes [L k ], for k = − k = n + 1, can beexpressed in the Atiyah-Todd basis. Note first that the equality0 = x n +1 = ((1 + x ) − n +1 = n +1 (cid:88) k =0 ( − n +1 − k ( n +1 k ) (1 + x ) k (5.3)in Z [ x ] / ( x n +1 ) translates to the equality n +1 (cid:88) k =0 ( − n +1 − k ( n +1 k ) [L k ] = 0 (5.4)in K ( CP n ). Thus we obtain[L n +1 ] = n (cid:88) k =0 ( − n − k ( n +1 k ) [L k ] , (5.5)which we will refer to as the first Atiyah-Todd identity .Furthermore, since (1 + x ) is invertible in Z [ x ] / ( x n +1 ) and the initial equality (5.3) canbe rewritten as (1 + x ) n +1 (cid:88) k =1 ( − − k ( n +1 k ) (1 + x ) k − = 1 , we obtain (1 + x ) − = n +1 (cid:88) k =1 ( − − k ( n +1 k ) (1 + x ) k − = n (cid:88) k =0 ( − k (cid:0) n +1 k +1 (cid:1) (1 + x ) k (5.6)in Z [ x ] / ( x n +1 ). This equality translates to K ( CP n ) as[ L − ] = n (cid:88) k =0 ( − k (cid:0) n +1 k +1 (cid:1) [ L k ] . (5.7)We will refer to (5.7) as the second Atiyah-Todd identity .In [1, Prop. 3.3 and 3.4], the additive version of the bottom isomorphism in the diagram(5.1) was established for the Vaksman-Soibelman quantum complex projective spaces CP nq .It yields a noncommutative version of the Atiyah-Todd basis (5.2). All this seems inter-esting because Atiyah-Todd’s method to prove the existence of the commutative diagram(5.1) uses the ring structure of K-theory, which is missing in the noncommutative setting.In the forthcoming subsection devoted to the multipullback noncommutative deformationof the complex projective spaces, not only we obtain an analogue of the Atiyah-Toddbasis (5.2), but also we establish analogues of the Atiyah-Todd identities (5.5) and (5.7),which are lacking in [1, Prop. 3.3 and 3.4]. HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 21
The multipullback noncommutative deformation.
Although the K -group ofa noncommutative C*-algebra does not have an intrinsic ring structure, it turns out that,much as in the diagram (5.1), the abelian group K ( C ( CP n T )) is a free module of rankone over the representation ring R ( U (1)) divided by the ideal generated by the ( n + 1)-stpower of the formal difference between the fundamental representation and the trivial one-dimensional representation. The basis of this free module is the K -class of C ( CP n T ). Themodule structure comes from tensoring finitely generated projective C ( CP n T )-modules bythe bimodules associated with the quantum Hopf U (1)-principal bundle S n +1 H → CP n T .Moreover, we will show that, despite the aforementioned lack of an intrinsic ring structure,we still enjoy analogs of the Atiyah-Todd identities (5.4) and (5.7).Recall that, we denote by S n +1 H the multipullback (2 n +1)-dimensional quantum sphere[10] and by CP n T the corresponding multipullback quantum complex projective space [8],whose C*-algebra we identify with a U (1)-fixed-point subalgebra of C ( S n +1 H ) (see [10]).Next, let ∂ n +1 : T ⊗ ( n +1) −→ T ⊗ ( n +1) / K ⊗ ( n +1) ∼ = C (cid:0) S n +1 H (cid:1) (see [10, Lemma 5.1]) (5.8)be the canonical quotient map, and let P k := k (cid:88) i =1 e ii ∈ K ⊂ T , P ⊥ k := I − P k ∈ K + ⊂ T , k ∈ N . (5.9)Here e ij with i, j ∈ N represents a matrix unit in K which we identify with K ( (cid:96) ( N )),and K + stands for the minimal unitization of K . Note that, according to the standardsummation-over-the-empty-set convention, P := 0, so P ⊥ = I . For finite square matrices P, Q ∈ M ∞ ( A ) with entries in a unital C*-algebra A , we use the notion P ∼ A Q to denotethat they are unitarily equivalent over A , and use P (cid:1) Q to denote their diagonal directsum.Furthermore, for 0 ≤ j ≤ n and k ≥
0, we define the projections E jk := ∂ n +1 (cid:0) ( ⊗ j P ) ⊗ P ⊥ k ⊗ ( ⊗ n − j I ) (cid:1) ∈ C ( CP n T ) . (5.10)Note that E nk = ∂ n +1 (( ⊗ n P ) ⊗ P ⊥ k ) = ∂ n +1 (( ⊗ n P ) ⊗ I ) since ∂ n +1 (( ⊗ n P ) ⊗ P k ) = 0.In this spirit, for the sake of forthcoming recursive formulas, we adopt the notation E n +1 k := ∂ n +1 ( ⊗ n +1 P ) = 0. Now, recall from [16, Theorem 4] and the remark belowthis theorem that, for j = 0 , . . . , n , the classes [ E j ] form a basis of the free Z -module K ( C ( CP n T )) ∼ = Z n +1 : K ( C ( CP n T )) = n (cid:77) j =0 Z [ E j ] . (5.11)Next, remembering that E jk ∈ C ( CP n T ) (they are all U (1)-invariant), we will fol-low an argument used in [16] to establish (cid:8) [ ∂ n (( ⊗ j I ) ⊗ ( ⊗ n − j P ))] (cid:9) Let S be the generating isometry of the Toeplitz algebra T identified with theunilateral shift on the Hilbert space (cid:96) ( N ) . For any k ≥ and n ≥ , u k := (cid:32) P k ⊗ I S k ⊗ ( S k ) ∗ ( S k ) ∗ ⊗ S k I ⊗ P k (cid:33) ∈ M ( T ⊗ ) is a self-adjoint unitary conjugating ( e kk ⊗ I ) (cid:1) to (cid:1) ( P ⊗ P ⊥ k ) .Proof. First, we verify that the self-adjoint element u k ∈ M ( T ⊗ ) is unitary: (cid:32) P k ⊗ I S k ⊗ ( S k ) ∗ ( S k ) ∗ ⊗ S k I ⊗ P k (cid:33) (cid:32) P k ⊗ I S k ⊗ ( S k ) ∗ ( S k ) ∗ ⊗ S k I ⊗ P k (cid:33) = (cid:32) P k ⊗ I + S k ( S k ) ∗ ⊗ ( S k ) ∗ S k P k S k ⊗ ( S k ) ∗ + S k ⊗ ( S k ) ∗ P k ( S k ) ∗ P k ⊗ S k + ( S k ) ∗ ⊗ P k S k ( S k ) ∗ S k ⊗ S k ( S k ) ∗ + I ⊗ P k (cid:33) = (cid:32) P k ⊗ I + P ⊥ k ⊗ I ⊗ ( S k ) ∗ + S k ⊗ ⊗ S k + ( S k ) ∗ ⊗ I ⊗ P ⊥ k + I ⊗ P k (cid:33) = (cid:32) I ⊗ I I ⊗ I (cid:33) . (5.13)Next, u k conjugates ( e kk ⊗ I ) (cid:1) (cid:1) (cid:0) P ⊗ P ⊥ k (cid:1) because (cid:32) P k ⊗ I S k ⊗ ( S k ) ∗ ( S k ) ∗ ⊗ S k I ⊗ P k (cid:33) (cid:32) e kk ⊗ I 00 0 (cid:33) (cid:32) P k ⊗ I S k ⊗ ( S k ) ∗ ( S k ) ∗ ⊗ S k I ⊗ P k (cid:33) = (cid:32) e k ⊗ S k (cid:33) (cid:32) P k ⊗ I S k ⊗ ( S k ) ∗ ( S k ) ∗ ⊗ S k I ⊗ P k (cid:33) = (cid:32) e ⊗ P ⊥ k (cid:33) . (5.14) (cid:3) Lemma 5.2. For any ≤ j ≤ n and any k ≥ , [ E jk +1 ] = [ E jk ] − [ E j +1 k ] . Proof. First, note that the statements are true for j = n because E nk = ∂ n +1 (( ⊗ n P ) ⊗ I ) = E nk +1 (5.15)is independent of k , and E n +1 k := 0. Hence, we can assume 0 ≤ j < n .Furthermore, since P ⊥ k = P ⊥ k +1 + e kk and the summands are orthogonal projections, weobtain E jk = ∂ n +1 (( ⊗ j P ) ⊗ P ⊥ k ⊗ ( ⊗ n − j I )) ∼ C ( CP n T ) ∂ n +1 (( ⊗ j P ) ⊗ P ⊥ k +1 ⊗ ( ⊗ n − j I )) (cid:1) ∂ n +1 (( ⊗ j P ) ⊗ e kk ⊗ ( ⊗ n − j I ))= E jk +1 (cid:1) ∂ n +1 (( ⊗ j P ) ⊗ e kk ⊗ ( ⊗ n − j I )) . (5.16) HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 23 Therefore, to finish the proof, it suffices to show that[ ∂ n +1 (( ⊗ j P ) ⊗ e kk ⊗ ( ⊗ n − j I ))] = [ E j +1 k ] . (5.17)To this end, we take advantage of Lemma 5.1 to conclude that ( ⊗ j P ) ⊗ u k ⊗ ( ⊗ n − j − I )conjugates (( ⊗ j P ) ⊗ e kk ⊗ ( ⊗ n − j I )) (cid:1) (cid:1) (( ⊗ j P ) ⊗ P ⊗ P ⊥ k ⊗ ( ⊗ n − j − I )) = 0 (cid:1) E j +1 k . (5.18)Here the tensor product ( ⊗ j P ) ⊗ u k ⊗ ( ⊗ n − j − I ) is understood entrywise with respect tothe matrix u k .Finally, since ∂ n +1 ( a ij ) is U (1)-invariant for each entry a ij of ( ⊗ j P ) ⊗ u k ⊗ ( ⊗ n − j − I ),we have ∂ n +1 ( a ij ) ∈ C ( CP n T ), so ∂ n +1 ((( ⊗ j P ) ⊗ e kk ⊗ ( ⊗ n − j I )) (cid:1) ∼ C ( CP n T ) (cid:1) E j +1 k . (5.19)Passing to the K -classes, we obtain (5.17), as needed. (cid:3) Having shown the recursive relation (5.12), we are ready to prove: Lemma 5.3. For any k ≥ , [ L k ] = k (cid:88) j =0 ( − j (cid:0) kj (cid:1) [ E j ] . Proof. It is known that, for k ≥ 0, the modules L k are represented, respectively, by theprojections ∂ n +1 (cid:0) P ⊥ k ⊗ ( ⊗ n I ) (cid:1) =: E k (see [16, Theorem 6]). Starting from l = 0, weprove inductively, for 0 ≤ l ≤ k with k ≥ L k ] = l (cid:88) j =0 ( − j (cid:0) lj (cid:1) [ E jk − l ] . (5.20)Equation (5.20) is clearly true for l = 0. Now, for 0 < l ≤ k , taking advantage of theinduction hypothesis and the recursive relation (5.12) in Lemma 5.2, we compute:[ L k ] = l − (cid:88) j =0 ( − j (cid:0) l − j (cid:1) [ E jk − l +1 ]= l − (cid:88) j =0 ( − j (cid:0) l − j (cid:1) (cid:0) [ E jk − l ] − [ E j +1 k − l ] (cid:1) = l − (cid:88) j =0 (cid:0) ( − j (cid:0) l − j (cid:1) [ E jk − l ] + ( − j +1 (cid:0) l − j (cid:1) [ E j +1 k − l ] (cid:1) = [ E k − l ] + l − (cid:88) j =1 ( − j (cid:0)(cid:0) l − j (cid:1) [ E jk − l ] + (cid:0) l − j − (cid:1) [ E jk − l ] (cid:1) + ( − l (cid:0) l − l − (cid:1) [ E lk − l ]= [ E k − l ] + l − (cid:88) j =1 ( − j (cid:0) lj (cid:1) [ E jk − l ] + ( − l [ E lk − l ] (5.21) = l (cid:88) j =0 ( − j (cid:0) lj (cid:1) [ E jk − l ] . (5.22)This proves (5.20), which, for l = k , becomes the desired equality. (cid:3) Now we are ready to prove the following main result of the present subsection. Theorem 5.4. For any n ∈ N , we have noncommutative analogs of the Atiyah-Todd basisand identities: K ( C ( CP n T )) = n (cid:77) k =0 Z [ L k ] , (5.23)[ L n +1 ] = n (cid:88) k =0 ( − n − k ( n +1 k ) [ L k ] , (5.24)[ L − ] = n (cid:88) k =0 ( − k (cid:0) n +1 k +1 (cid:1) [ L k ] . (5.25) Proof. To begin with, note that (5.23) follows immediately from Lemma 5.3 and (5.11)because the expansion coefficients ( − j (cid:0) kj (cid:1) in Lemma 5.3 form a matrix in GL n +1 ( Z ).(The matrix is lower-triangular of determinant ± n +1 (cid:88) k =0 ( − n +1 − k ( n +1 k ) [ L k ]= n +1 (cid:88) k =0 ( − n +1 − k ( n +1 k ) (cid:32) k (cid:88) j =0 ( − j (cid:0) kj (cid:1) [ E j ] (cid:33) = n +1 (cid:88) j =0 n +1 (cid:88) k = j ( − n +1+ j − k ( n +1 k ) (cid:0) kj (cid:1) [ E j ]= n +1 (cid:88) j =0 (cid:32) n +1 (cid:88) k = j ( − n +1+ j − k ( n + 1)! k !( n + 1 − k )! k ! j !( k − j )! (cid:33) [ E j ]= n +1 (cid:88) j =0 ( n + 1)! j !( n + 1 − j )! (cid:32) n +1 (cid:88) k = j ( − n +1+ j − k ( n + 1 − j )!( n + 1 − k )!( k − j )! (cid:33) [ E j ]= n +1 (cid:88) j =0 ( n + 1)! j !( n + 1 − j )! ( − j (cid:32) n +1 − j (cid:88) k =0 ( − n +1 − j − k ( n + 1 − j )!( n + 1 − j − k )! k ! (cid:33) [ E j ]= n +1 (cid:88) j =0 ( n + 1)! j !( n + 1 − j )! ( − j (1 + ( − n +1 − j [ E j ] = 0 . (5.26) HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 25 Finally, to prove (5.25), we recall from [16] that the class [ L − ] can be represented bythe projection (cid:1) nj =0 E j . Thus (5.25) becomes n (cid:88) k =0 ( − k (cid:0) n +1 k +1 (cid:1) [ L k ] = n (cid:88) j =0 [ E j ] . (5.27)The left-hand-side can be computed as follows: n (cid:88) k =0 ( − k (cid:0) n +1 k +1 (cid:1) [ L k ] = n (cid:88) k =0 ( − k (cid:0) n +1 k +1 (cid:1) ( k (cid:88) j =0 ( − j (cid:0) kj (cid:1) [ E j ])= n (cid:88) j =0 n (cid:88) k = j ( − k + j ( n + 1)!( k + 1)!( n − k )! k ! j !( k − j )! [ E j ]= n (cid:88) j =0 ( n + 1)! j !( n − j )! (cid:32) n (cid:88) k = j ( − k + j ( n − j )!( n − k )!( k − j )! 1 k + 1 (cid:33) [ E j ]= n (cid:88) j =0 ( n + 1)! j !( n − j )! (cid:32) n − j (cid:88) k =0 ( − k ( n − j )!( n − j − k )! k ! 1 k + j + 1 (cid:33) [ E j ]= n (cid:88) j =0 ( n + 1)! j !( n − j )! (cid:32) n − j (cid:88) k =0 ( − k (cid:0) n − jk (cid:1) k + j + 1 (cid:33) [ E j ] . (5.28)Hence it remains to show that, for all 0 ≤ j ≤ n ,( n + 1)! j !( n − j )! (cid:32) n − j (cid:88) k =0 ( − k (cid:0) n − jk (cid:1) k + j + 1 (cid:33) = 1 . To this end, we introduce auxiliary polynomials over Q : f j ( x ) := n − j (cid:88) k =0 ( − k (cid:0) n − jk (cid:1) k + j + 1 x k + j +1 , (5.29)which can be evaluated and formally differentiated and integrated. Now our goal can berephrased as follows: j !( n − j )!( n + 1)! = f j (1) . To compute this, note first that f (cid:48) j ( x ) = n − j (cid:88) k =0 ( − k (cid:0) n − jk (cid:1) x k + j = ( − n − j x j ( x − n − j . (5.30)Therefore, as f j (0) = 0 because k, j ≥ 0, we obtain: f j (1) = (cid:90) ( − n − j x j ( x − n − j dx = ( − n − j j + 1 x j +1 ( x − n − j (cid:12)(cid:12)(cid:12)(cid:12) − (cid:90) ( − n − j ( n − j ) j + 1 x j +1 ( x − n − j − dx = ( − n +1 − j ( n − j ) j + 1 (cid:90) x j +1 ( x − n − j − dx. (5.31) Iterating this kind of integration by parts, we infer that f j (1) = ( − n +( n − j ) − j ( n − j )!( j + 1)( j + 2) · · · ( j + ( n − j )) (cid:90) x j +( n − j ) ( x − dx = ( n − j )!( j + 1)( j + 2) · · · n (cid:90) x n dx = ( n − j )!( j + 1)( j + 2) · · · n ( n + 1)= j !( n − j )!( n + 1)! , (5.32)as desired. (cid:3) Acknowledgements This paper is part of the international project “Quantum Dynamics” supported by EUgrant H2020-MSCA-RISE-2015-691246. The international project was also co-financed bythe Polish Ministry of Science and Higher Education from the funds allocated for science inthe years 2016-2019 through the grants W2/H2020/2016/317281 and 328941/PnH/2016(Piotr M. Hajac), W43/H2020/2016/319577 and 329915/PnH/2016 (Tomasz Maszczyk),W30/H2020/2016/319460 and 329390/PnH/2016 (Bartosz Zieli´nski). Also, Piotr M. Ha-jac, Tomasz Maszczyk and Bartosz Zieli´nski thank the University of Kansas, Lawrence,for its hospitality, and Piotr M. Hajac thanks the University of Naples Federico II for thehospitality and sponsorship. References [1] Arici, F., Brain, S., and Landi, G. The Gysin sequence for quantum lens spaces. J. Noncommut.Geom. 9 , (2015), 1077–1111.[2] Arici, F., D’Andrea, F., Hajac, P. M., and Tobolski, M. An equivariant pullback structureof trimmable graph C*-algebras. To appear in the Journal of Noncommutative Geometry .[3] Arici, F., Kaad, J., and Landi, G. Pimsner algebras and Gysin sequences from principal circleactions. Journal of Noncommutative Geometry 10 , 1 (2016), 29–64.[4] Calow, D., and Matthes, R. Covering and gluing of algebras and differential algebras. Journalof Geometry and Physics 32 , 4 (2000), 364—-396.[5] Davidson, K. R. C*-algebras by example , vol. 6 of Fields Institute Monographs . American Mathe-matical Society, Providence, RI, 1996.[6] D (cid:44) abrowski, L., Hajac, P. M., and Neshveyev, S. Noncommutative Borsuk-Ulam-type conjec-tures revisited. To appear in Journal of Noncommutative Geometry .[7] Farsi, C., Hajac, P. M., Maszczyk, T., and Zielinski, B. Rank-two milnor idempotents forthe multipullback quantum complex projective plane. arXiv preprint arXiv:1708.04426 (2017).[8] Hajac, P. M., Kaygun, A., and Zieli´nski, B. Quantum complex projective spaces from Toeplitzcubes. Journal of Noncommutative Geometry 6 , 3 (2012), 603–621.[9] Hajac, P. M., and Maszczyk, T. Pullbacks and nontriviality of associated noncommutativevector bundles. arXiv preprint arXiv:1601.00021 (2015). HE K-THEORY OF MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE SPACES 27 [10] Hajac, P. M., Nest, R., Pask, D., Sims, A., and Zieli´nski, B. The K-theory of twistedmultipullback quantum odd spheres and complex projective spaces. Journal of NoncommutativeGeometry 12 , 3 (2018), 823–863.[11] Hong, J. H., and Szyma´nski, W. Noncommutative balls and mirror quantum spheres. Journalof the London Mathematical Society 77 , 3 (2008), 607–626.[12] Hong, J. H., and Szyma´nski, W. Quantum spheres and projective spaces as graph algebras. Communications in Mathematical Physics 232 , 1 (Dec 2002), 157–188.[13] Meyer, U. Projective quantum spaces. Letters in Mathematical Physics 35 , 2 (1995), 91–97.[14] Pedersen, G. K. Pullback and pushout constructions in C*-algebra theory. Journal of FunctionalAnalysis 167 (1999), 243–344.[15] Raeburn, I. Graph algebras . No. 103 in CBMS Regional Conference Series in Mathematics. Amer-ican Mathematical Society, Providence, RI, 2005.[16] Sheu, A. J.-L. Vector bundles over multipullback quantum complex projective spaces. arXivpreprint arXiv:1705.04611 (2017).[17] Vaksman, L. L., and Soibel’man, Y. S. Algebra of functions on the quantum group SU ( n + 1)and odd-dimensional quantum spheres. Algebra i analiz 2 , 5 (1990), 101–120.[18] Woronowicz, S. L. Twisted SU (2) group. an example of a non-commutative differential calculus. Publications of the Research Institute for Mathematical Sciences 23 , 1 (1987), 117–181.(F. D’Andrea) Universit`a di Napoli “Federico II” and I.N.F.N. Sezione di Napoli, Comp-lesso MSA, Via Cintia, 80126 Napoli, Italy. Email address : [email protected] (P.M. Hajac) Instytut Matematyczny, Polska Akademia Nauk, ul. ´Sniadeckich 8, Warszawa,00-656 Poland Email address : [email protected] (T. Maszczyk) Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097Warszawa, Poland Email address : [email protected] (A. Sheu) Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence,KS 66045, U.S.A. Email address : [email protected] (B. Zieli´nski) Department of Computer Science, University of (cid:32)L´od´z, Pomorska 149/15390-236 (cid:32)L´od´z, Poland Email address ::