aa r X i v : . [ m a t h . OA ] O c t K-THEORY OF EQUIVARIANT QUANTIZATION
XIANG TANG AND YI-JUN YAO
Abstract.
Using an equivariant version of Connes’ Thom Isomorphism,we prove that equi-variant K -theory is invariant under strict deformation quantization for a compact Lie groupaction. Introduction
Let α be a strongly continuous action of R n on a C ∗ -algebra A , and J be a skew-symmetricmatrix on R n . Rieffel [10] constructed a strict deformation quantization A J of A via oscillatoryintegrals(1) a × J b := Z R n × R n α Ju ( a ) α v ( b ) e πiu · v dudv, for u, v ∈ R n , and a, b ∈ A ∞ (the smooth subalgebra of A for α ). Such a construction gives riseto many interesting examples of noncommutative manifolds, e.g. quantum tori, θ -deformationof S , etc. In [11], Rieffel proved that the K -theory of A J is equal to the K -theory of the originalalgebra A , by using Connes’ Thom isomorphism of K -theory.In this paper, we are interested in examples that the algebra A is also equipped with a stronglycontinuous action β by a compact group G . When the two actions commute, the results in [11]naturally generalize to the equivariant setting. An easy observation is that, as the G -action β commutes with the R n -action α , naturally α can be lifted to a strongly continuous action ˜ α onthe crossed product algebra A ⋊ β G . Rieffel’s construction (1) applies to the R n -action ˜ α on A ⋊ β G , and defines a quantization algebra ( A ⋊ β G ) J . By the commutativity between α and β ,we easily check that β lifts to a strongly continuous action ˜ β on A J , and A J ⋊ ˜ β G is isomorphicto ( A ⋊ β G ) J . Now by the results on the K -theory of strict deformation quantization [11], weconclude that K • ( A ⋊ β G ) = K • (( A ⋊ β G ) J ) = K • ( A J ⋊ ˜ β G ) . In this paper, we generalize the above discussion of equivariant quantization to a situationwhere the actions α and β do not commute. Define GL ( J ) to be the group of invertible matrices g such that g t J g = J , and SL n ( R , J ) := SL n ( R ) T GL ( J ). We remark that when J is the standardskew-symmetric matrix on R n , GL ( J ) is the linear symplectic group. Let ρ : G → SL n ( R , J )be a group homomorphism such that(2) β g α x = α ρ g ( x ) β g , for any g ∈ G, x ∈ R n . When ρ is a trivial group homomorphism, the actions α and β commute.A natural example of such a system appears as follows. Example 1.1.
Let G = Z = Z / Z act on R n by reflection with respect to the origin. Let Z n be the integer lattice in R n . The n -torus T n = R n / Z n inherits an action of Z from the Z action on R n . The group R n acts on R n by translation and descends to act on T n . Let A be the C ∗ -algebra of continuous functions on T n , and J be the standard symplectic matrixon R n . The action α (and β ) of R n (and Z ) on A is the dual action of the corresponding actions on T n . We easily check that Eq. (2) holds in this case with ρ being the natural inclusion Z ֒ → SL n ( R , J ) . Different from the case where the actions α and β commute, for a nontrivial ρ : G → SL n ( R , J ), the R n -action α on A does not lift naturally to an action on A ⋊ β G . Therefore, wecannot apply Rieffel’s deformation construction to the algebra A ⋊ β G . Nevertheless, a simplecalculation shows that β g ( a × J b ) = β g ( a ) × J β g ( b ) , β g ( a ∗ ) = β g ( a ) ∗ , which shows that the G -action β is still well-defined on A J . Accordingly, we can consider thecrossed product algebra A J ⋊ β G . Applying this construction to Ex. 1.1, we obtain A J ⋊ β Z ,which is well studied in literature, e.g. [5], [6], [8], and [14].In this paper, we prove the following theorem about the K -theory groups of A J ⋊ β G . Theorem 1.2.
If the actions α , β and the group homomorphism ρ satisfy (2), then K • ( A J ⋊ β G ) ∼ = K • ( A ⋊ β G ) , • = 0 , . The proof of this theorem will be presented in the next section. As applications of our theo-rem, we recover some results of [5] on the computation of the K -groups of Z i -quantum tori for i = 2 , , ,
6, and we also apply these results to the θ -deformation [4] of S . Acknowledgments:
We would like to thank Professors S. Echterhoff and N. Higson for explain-ing the relationship between the equivariant Thom isomorphism theorem (Theorem 2.1) and theConnes-Kasparov conjecture. We also want to thank Professor H. Li for interesting discussionsand comments which greatly helped us to improve the readability of the paper. We thank Pro-fessor H. Oyono-Oyono for helping us to remove a separability assumption in a previous version.And we are grateful to an anonymous referee for pointing out one mistake and various placesto improve accuracy in a previous version of this paper. Tang’s research is partially supportedby NSF grant 0900985. Tang would like to thank the School of Mathematical Sciences of FudanUniversity and the Max-Planck Institute for their warm hospitality of his visits. Yao’s researchis partially supported by NSF grant 0903985 and NSFC grant 10901039 and 11231002.2.
Proof of the main theorem
Our proof of Theorem 1.2 is an equivariant generalization of Rieffel’s proof in [11]. We firstprove the theorem under the assumption that A is separable. Following [11], we will decomposeour proof into 3 steps. Step I.
Following the notations in [11], we let B A be the space of smooth A -valued functionson R n whose derivatives together with themselves are bounded on R n . Let S A be the space of A -valued Schwartz functions on R n . The integral h f, g i A := Z f ( x ) ∗ g ( x ) dx defines an A -valued inner product on S A . Rieffel generalized the definition to B A by usingoscillatory integrals. Namely, given J , we define a product on B A by( F × J G )( x ) := Z F ( x + J u ) G ( x + v ) e πiu · v dudv, F, G ∈ B A . Furthermore, B A acts on S A by( L JF f )( x ) := Z F ( x + J u ) f ( x + v ) e πiu · v dudv, F ∈ B A , f ∈ S A . -THEORY OF EQUIVARIANT QUANTIZATION 3 The above two integrals are both oscillatory ones. Via the A -valued inner product on S A , wecan equip B A with the operator norm k k J , and obtain a pre- C ∗ -algebra ( B AJ , × J , k k J ). Denotethe corresponding C ∗ -algebra by B AJ . Meanwhile, S A viewed as a ∗ -ideal of B AJ (cf. Rieffel, [10]),denoted by S AJ , can be completed into S AJ .With an action α of R n on A , Rieffel [11, Prop. 1.1] introduced a strongly continuous R n -action ν on B AJ and also on S AJ by( ν t ( F ))( x ) := α t ( F ( x − t )) . The fixed point subalgebra of this action ν is identified [11, Prop. 2.14] with the C ∗ -subalgebraof B AJ generated by elements ˜ a ( x ) := α x ( a ) , a ∈ A ∞ , which is exactly A J .In [11, Thm. 3.2], it is proved that A J is strongly Morita equivalent to S AJ ⋊ ν R n . Wewill generalize this theorem to the equivariant setting with the G -action β . We introduce the G -action β on B AJ by β g ( F )( x ) := β g ( F ( ρ g − ( x ))) . The exactly same arguments as in [11, Prop. 1.1] prove that the G -action β is strongly continuouson S A , therefore so is it on S AJ . Proposition 2.1.
The crossed product algebras A J ⋊ β G and ( S AJ ⋊ ν R n ) ⋊ ¯ β G are stronglyMorita equivalent.Proof. We will apply Combes’ theorem [1, Sec. 6] on equivariant Morita equivalence after provingthat the G -actions β and ¯ β are Morita equivalent, which will imply the Morita equivalence weseek.According to [1], two G -actions β , β on A and B are Morita equivalent if there is a strongMorita equivalence bimodule X between A and B such that there is a G -action β on X satisfying β g ( aξ ) = β g ( a ) β g ( ξ ) , β g ( ξb ) = β g ( ξ ) β g ( b ) , A h β g ( ξ ) , β g ( ξ ) i = β g ( A h ξ , ξ i ) , h β g ( ξ ) , β g ( ξ ) i B = β g ( h ξ , ξ i B ) . for ξ, ξ , ξ ∈ X .Rieffel [11] constructed a Morita equivalence bimodule between A J and S AJ ⋊ ν R n . We recallit now. Let C ∞ ( R n , A ) be the C ∗ -algebra of A -valued functions on R n that vanish at infinity.Let τ be the R n -action on B AJ by translation, ( τ t F )( x ) = F ( t + x ), and µ be the action of R n on C ∞ ( R n , A ) by µ s ( f )( x ) = e πis · x f ( x ) . Define an action α of R n on S AJ by α t ( F )( x ) = α t ( F ( x )) . Both µ and τ act on C ∞ ( R n , A ) and their combination gives an action of the Heisenberg group H of dimension 2 n + 1 on C ∞ ( R n , A ). This Heisenberg group action commutes with α and definesan H × R n -action σ on C ∞ ( R n , A ). Define X to be the subspace of C ∞ ( R n , A ) of σ -smoothvectors. Rieffel [11, Prop. 2.2] proved that X is a ∗ -subalgebra of S AJ for any J , and a suitablecompletion X of X serves as a strong Morita equivalence bimodule, which we refer to [11] fordetails. XIANG TANG AND YI-JUN YAO
Define a right A J -module structure on X by identifying A J with the subspace of ν -invariantvectors in B AJ , i.e. f · a := f × J ˜ a, for a ∈ A ∞ where ˜ a ∈ C ∞ ( R n , A ) is defined by ˜ a ( x ) = α x ( a ). The algebra S AJ ⋊ ν R n acts on X by ψ ( f ) := Z ψ ( t ) × J ν t ( f ) dt, ψ ∈ S AJ ⋊ ν R n , f ∈ X . We define an S AJ ⋊ ν R n -valued inner product on X by S AJ ⋊ ν R n h f, g i ( x ) := f × J ν x ( g ∗ ) , x ∈ R n , f, g ∈ X , and an A J -valued inner product on X by h f, g i A J := (cid:18)Z α t ( f ∗ × J g ( − t )) dt (cid:19) , f, g ∈ X . We also know from [11] that (cid:16) X , S AJ ⋊ ν R n h , i , h , i A J (cid:17) is a strong Morita equivalence bimodulebetween S AJ ⋊ ν R n and A J .We easily check the following identities between the actions β g α t = α ρ g ( t ) β g , β g τ t = τ ρ g ( t ) β g , β g µ t = µ ( ρ Tg ) − ( t ) β g , g ∈ G, t ∈ R n . where ρ Tg is the transpose of ρ g . These identities show that the G -action β on C ∞ ( R n , A )preserves the subspace X of σ -smooth vectors. Using the property that β and β act stronglycontinuously on S AJ , we can easily check that β and β are Morita equivalent G -actions in thesense of Combes [1]. Therefore, A J ⋊ β G is strongly Morita equivalent to ( S AJ ⋊ ν R n ) ⋊ β G . (cid:3) As A is separable, A has a countable approximate identity. This implies that [11, Cor. 3.3] A J (and S AJ ) has a countable approximate identity. Accordingly, A J ⋊ β G (and ( S AJ ⋊ ν R n ) ⋊ β G )also has a countable approximate identity, and therefore has strictly positive elements. Thistogether with the above Morita equivalence result shows that A J ⋊ β G and ( S AJ ⋊ ν R n ) ⋊ β G arestably isomorphic. As stably isomorphic C ∗ -algebras have isomorphic K -groups, we concludethat K • ( A J ⋊ β G ) ∼ = K • (( S AJ ⋊ ν R n ) ⋊ β G ) . Step II.
As we know, one powerful tool in dealing with the K -theory of C ∗ -algebras is Connes’Thom isomorphism, which remains to this day one of the few ways to prove isomorphism resultsof K -groups for crossed products. Let C n be the complex Clifford algebra associated with R n .We first observe that the semidirect product group R n ⋊ ρ G is amenable, hence by Kasparov [7, §
6, Thm. 2.] we know that for a separable G - C ∗ -algebra B , there exists an isomorphismfrom KK i ( C , B ⋊ ( R n ⋊ G )) to KK i ( C , (( B ⊗ C n ) ⋊ G ). In other words, we need to use thefollowing equivariant Thom isomorphism Theorem, which is a generalization of Connes’ Thomisomorphism Theorem [3]. This is a key ingredient of the whole approach. Theorem 2.1.
Let R n and G act strongly continuously on a separable C ∗ -algebra B with theactions denoted by α and β . Let ρ : G → GL ( n, R ) . If the actions α and β satisfy Equation (2),then K • (( B ⋊ α R n ) ⋊ β G ) ∼ = K G • (cid:0) B ⋊ α R n (cid:1) ∼ = K G • ( B ⊗ C n ) ∼ = K • (( B ⊗ C n ) ⋊ β G ) , where C n is the complex Clifford algebra associated with R n . In [7, §
6, Thm. 2.], the connectivity of the group G is assumed. But this assumption can be easily droppedusing the same idea of the proof. -THEORY OF EQUIVARIANT QUANTIZATION 5 Taking B = S AJ in the above theorem which is separable (as A is separable), we conclude that K • (( S AJ ⊗ C n ) ⋊ ¯ β G ) is isomorphic to K • (( S AJ ⋊ ν R n ) ⋊ β G ). Step III.
Rieffel proved [10, Prop. 5.2] that there is an isomorphism(3) S AJ ∼ = A ⊗ K ⊗ C ∞ ( V ) , where K is the algebra of compact operators on an infinite dimensional separable Hilbert space H , and V is the kernel of J in R n . Let U be the orthogonal complement of V in R n . It iseasy to check that U is a J -invariant subspace, and both U and V are G -invariant subspaces.As G is compact, there is a G -invariant complex structure on U compatible with J | U (vieweda symplectic form on U ). Without loss of generality, we will just assume that G preserves thestandard Euclidean structure on U . The key observation in the proof of [10, Prop. 5.2] is thatwhen A is the trivial C ∗ -algebra C and J invertible, S C J is naturally identified as the space ofcompact operators, still denoted by K , on the subspace H of L ( U ) generated by elements g (¯ z ) e − k z k , where g is an anti-holomorphic function. As H is a G -invariant subspace, we can conclude thatRieffel’s isomorphism (3) is G -equivariant (note that G acts on K by conjugation). By Combes’result on G-equivariant Morita equivalence, ( A ⊗ K ⊗ C ∞ ( V ) ⊗ C n ) ⋊ ¯ β G is strongly Moritaequivalent to ( A ⊗ C ∞ ( V ) ⊗ C n ) ⋊ ¯ β G .Now we look at the decomposition of R n as V ⊕ U . The Clifford algebra C n associated with R n is G -equivariantly isomorphic to C V ⊗ C U , where C V and C U are the complex Clifford algebrasassociated with V and U , respectively. Notice that J restricts to define a symplectic form on U , and that the action of G preserves both the restricted J and the metric on U . Thereforethe G -action on U is spin c . Hence, the algebra ( A ⊗ C ∞ ( V ) ⊗ C n ) ⋊ ¯ β G is KK -equivalent to( A ⊗ C ∞ ( V ) ⊗ C V ) ⋊ ¯ β G . Again by the G -equivariant Thom isomorphism Thm. 2.1 for thetrivial V action on A , we conclude that K • (( S AJ ⊗ C n ) ⋊ ¯ β G ) = K • (( A ⊗ K ⊗ C ∞ ( V ) ⊗ C n ) ⋊ ¯ β G )= K • ( (cid:0) A ⊗ C ∞ ( V ) ⊗ C V (cid:1) ⋊ ¯ β G ) = K • ( A ⋊ β G ) . Summarizing Step I-III, we have the following equality, K • ( A J ⋊ β G ) Step I === K • (( S AJ ⋊ ν R n ) ⋊ ¯ β G ) Step II === K • (( S AJ ⊗ C n ) ⋊ ¯ β G ) Step III === K • ( A ⋊ β G ) . This completes the proof of Theorem 1.2 under the assumption that A is separable. For a general C ∗ -algebra A , we can write A as an inductive limit of a net A I of separable R n ⋊ ρ G -algebras.Then A J is an inductive limit of the net A IJ of separable G -algebras. As K -groups commuteswith inductive limit, we conclude that K • ( A J ⋊ β G ) = lim I K • ( A IJ ⋊ β G ) = lim I K • ( A I ⋊ β G ) = K • ( A ⋊ β G ) . This completes the proof of Theorem 1.2 for general C ∗ -algebras.3. Examples
In this section, we discuss some applications of Theorem 1.2.
XIANG TANG AND YI-JUN YAO
Noncommutative toroidal orbifolds.
We identify a 2-torus T by R / Z . R acts onitself by translation and induces an action α on T . For θ ∈ R , we consider the symplectic form J = θdx ∧ dx on R . The group SL ( Z ) acts on R preserving the lattice Z and thereforealso acts on T , which is denoted by β . Inside SL ( Z ), there are cyclic subgroups generated by σ = (cid:18) − − (cid:19) , σ = (cid:18) − −
11 0 (cid:19) σ = (cid:18) −
11 0 (cid:19) , σ = (cid:18) −
11 1 (cid:19) . The element σ i generates a cyclic subgroup Z i of SL ( Z ) of order i = 2 , , ,
6. In this example,the group SL ( R , J ) is identical to the group SL ( R ). Define ρ : Z i → SL ( R ) to be theinclusion. And it is straightforward to check the actions β of Z i on T , ρ of Z i on R , and α of R on T satisfy Eq. (2). As is explained in Sec. 1, the group Z i naturally acts on Rieffel’sdeformation A J , which is the quantum torus A θ . Theorem 1.2 states that K • ( A J ⋊ Z i ) = K • ( A ⋊ Z i ) . We recover with a completely different proof the result of [5, Cor. 2.2]. We have brought thequestion of computation of K -groups of these noncommutative orbifolds to a purely topologicalsetting, and we refer to [5] and references therein for the explicit computation of the K -groupsof the undeformed algebras A ⋊ Z i , i = 2 , , ,
6. For example, when i = 2, the K -groups of A ⋊ Z are K • ( A ⋊ Z ) ∼ = (cid:26) Z , • = 0 , , • = 1 . Theta deformation.
Consider a 4-sphere S centered at (0 , , , ,
0) in R with radius 1.In coordinates, it is the set (cid:8) ( x , · · · , x ) | x + x + x + x + x = 1 (cid:9) . Defines T -action on S by, for 0 ≤ t , t < π , (cid:0) ( t , t ) , ( x , · · · , x ) (cid:1) −→ ( x , · · · , x ) cos( t ) sin( t ) 0 0 0 − sin( t ) cos( t ) 0 0 00 0 cos( t ) sin( t ) 00 0 − sin( t ) cos( t ) 00 0 0 0 1 . The same formula as above also defines an R -action α on S . The action β of Z on S is byreflection ( σ , ( x , · · · , x )) −→ ( x , − x , x , − x , x ) . The group Z also acts on R by reflection ρ : σ −→ (cid:18) − − (cid:19) . On R , for θ ∈ R , consider the same symplectic form J = θdx ∧ dx . It is easy to check thatthe actions α, β, ρ satisfy Eq. (2). Consider the algebra C ( S ) of continuous functions on S .Rieffel’s construction defines a deformation C ( S θ ) of C ( S ) by J and the action α , which is the θ -deformation [4] introduced by Connes and Landi. As is explained in Sec. 1, Z acts stronglycontinuously on C ( S θ ). Theorem 1.2 states that K • ( C ( S ) ⋊ Z ) = K • ( C ( S θ ) ⋊ Z ) . -THEORY OF EQUIVARIANT QUANTIZATION 7 The K -theory of C ( S ) ⋊ Z can be computed [9] topologically as the Grothendieck group ofthe monoid of all isomorphism classes of Z -equivariant vector bundles on S .Notice that the quotient S / Z is an orbifold homeomorphic to S . As an orbifold, S / Z [12]has a good covering { U i } such that each U i and any none empty finite intersection U i ∩· · ·∩ U i k isa quotient of a finite group action on R . Such a good covering allows to compute the topological Z -equivariant K -theory of S by the ˇCech cohomology on S / Z of the sheaf K • Z introduced bySegal [13]. The restriction of K • Z to an open chart U of S / Z is defined to be the Z -equivariant K -theory of π − ( U ) with π the canonical projection S → S / Z . Locally, when U is sufficientlysmall, we can compute K • Z ( U ) to be K • ( π − ( U ) σ ) ⊕ K • ( U ), where π − ( U ) σ is the σ -fixedpoint submanifold. When • = 0, it is equal to Z | π − ( U ) σ ⊕ Z U , and when • = 1, it is zero.Gluing this local computation by the Mayer-Vietoris sequence, we conclude that K ( C ( S θ ) ⋊ Z ) = Z , K ( C ( S θ ) ⋊ Z ) = 0 . Remark 3.1.
We observe that in the above example, the group Z is not essential. Our com-putations generalize to K • ( C ∞ ( S θ ) ⋊ Z i ) , for i = 3 , , . References [1] Combes, F., Crossed products and Morita equivalence,
Proc. London Math. Soc. (3) 49 (1984), no. 2, 289–306.[2] Connes, A., C ∗ -alg´ebres et g´eom´etrie diff´erentielle. (French) C. R. Acad. Sci. Paris S´er.
A-B 290 (1980), no.13, A599–A604.[3] Connes, A., An analogue of the Thom isomorphism for crossed products of a C ∗ -algebra by an action of R ., Adv. in Math , 39 (1981), no. 1, 31–55.[4] Connes, A., Landi, G., Noncommutative manifolds, the instanton algebra and isospectral deformations,
Comm. Math. Phys.
221 (2001), no. 1, 141–159.[5] Echterhoff, S., L¨uck, W., Phillips, N. C., and Walters, S., The structure of crossed products of irrationalrotation algebras by finite subgroups of SL ( Z ), J. Reine Angew. Math.
639 (2010), 173–221.[6] Farsi, C., Watling, N., Symmetrized non-commutative tori,
Math. Ann.
296 (1993), 739-741.719.[7] Kasparov, G., K-theory, group C ∗ -algebras, and higher signatures (conspectus). Novikov conjectures, indextheorems and rigidity , Vol. 1 (Oberwolfach, 1993), 101-146, London Math. Soc. Lecture Note Ser., 226,Cambridge Univ. Press, Cambridge, (1995).[8] Kumjian, A., On the K-theory of the symmetrized noncommutative torus,
C.R. Math. Rep. Acad. Sci. Canada
12, 87-89 (1990).Berlin, 1987.[9] Phillips, N. C.,
Equivariant K -theory for proper actions , Pitman Research Notes in Mathematics Series, 178,Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc.,New York, (1989).[10] Rieffel, M., Deformation quantization for actions of R d , Mem. Amer. Math. Soc.
106 (1993), no. 506.[11] Rieffel, M., K -groups of C ∗ -algebras deformed by actions of R d , J. Funct. Anal.
116 (1993), no. 1, 199–214.[12] Moerdijk, I., Pronk, D. A. Simplicial cohomology of orbifolds,
Indag. Math.
10 (1999), no. 2, 269–293.[13] Segal, G., Equivariant K -theory, Inst. Hautes ´Etudes Sci. Publ. Math. no. 34 (1968) 129–151.[14] Walters, S., Projective modules over the non-commutative sphere,
J. London Math. Soc. (2) 51 (1995), no.3, 589-602.(2) 51 (1995), no.3, 589-602.