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OPERATOR ALGEBRA QUANTUM HOMOGENEOUS SPACES OF UNIVERSAL GAUGE GROUPS SNIGDHAYAN MAHANTA AND VARGHESE MATHAI Abstract. In this paper, we quantize universal gauge groups such as SU(∞), as well as 1 1 their homogeneous spaces, in the σ-C∗-algebra setting. More precisely, we propose concise 0 definitions of σ-C∗-quantum groups and σ-C∗-quantum homogeneous spaces and explain 2 theseconceptshere. Atthesametime,weputthesedefinitionsinthemathematicalcontext n a of countably compactly generated spaces as well as C∗-compact quantum groups and ho- J mogeneous spaces. We also study the representable K-theory of these spaces and compute 6 it for the quantum homogeneous spaces associated to the universal gauge group SU(∞). 2 ] A If H is a compact and Hausdorff topological group, then the C∗-algebra of all continu- Q ous functions C(H) admits a comultiplication map ∆ : C(H) → C(H)⊗ˆC(H) arising from . h the multiplication in H. This observation motivated Woronowicz (see, for instance, [23]), t a m amongst others such as Soibelman [19], to introduce the notion of a C∗-compact quantum [ group in the setting of operator algebras as a unital C∗-algebra with a coassociative comulti- 2 plication, satisfying a few other conditions. If the group H is only locally compact then the v situation becomes significantly more difficult. One of the reasons is that the multiplication 3 9 map m : H × H → H is no longer a proper map and one needs to introduce multiplier 8 5 algebras of C∗-algebras to obtain a comultiplication, see for instance, Kustermans-Vaes [7]. . 2 For an excellent and thorough introduction to this theory the readers are referred to, for 1 0 instance, [8]. In the sequel we show that if H = l−i→mnHn is a countably compactly generated 1 group, i.e., if H ⊂ H are compact and Hausdorff topological groups for all n ∈ N and if : n n+1 v H is the direct limit, then a story similar to the compact group case goes through using the i X general framework of σ-C∗-algebras as systematically developed by Phillips [12, 13], moti- r a vated by some earlier work by Arveson, Mallios, Voiculescu, amongst others. There is a clean formulation of, what we call, σ-C∗-quantum groups, which are noncommutative generaliza- tions of C(H). Examples of countably compactly generated groups are U(∞) = lim U(n), −→n SU(∞) = lim SU(n), where U(n) (resp. SU(n)) are the unitary (resp. special unitary) −→n 2000 Mathematics Subject Classification. 46L80,58B32, 58B34,46L65. Key words and phrases. σ-C∗-quantum groups, σ-C∗-quantum homogeneous spaces, universal gauge groups, operator algebras, K-theory. Acknowledgements. BothauthorsgratefullyacknowledgesupportundertheAustralianResearchCouncil’s Discovery Projects funding scheme. 1 groups. They are also known in the physics literature as universal gauge groups, see Harvey- Moore [5] and Carey-Mickelsson [4]. Such spaces are not locally compact and hence the ex- isting literature on quantum groups cannot handle them. Moreover, locally compact groups that are not compact, are also not countably compactly generated. We also discuss in detail the interesting example of the quantum version of the universal special unitary group, C(SU (∞)). q A pro C∗-algebra is an inverse limit of C∗-algebras and ∗-homomorphisms, where the inverse limit is constructed inside the category of all topological ∗-algebras and continuous ∗-homomorphisms. For the general theory of topological ∗-algebras one may refer to, for instance, [9]. The underlying topological ∗-algebra of a pro C∗-algebra is necessarily complete and Hausdorff. It is not a C∗-algebra in general; it would be so if, for instance, the directed set is finite. If the directed set is countable, then the inverse limit is called a σ-C∗-algebra. One can choose a linearly directed cofinal subset inside any countable directed set and the passage to a cofinal subsystem does not change the inverse limit. Therefore, we shall always identify a σ-C∗-algebra A ∼= lim A , where n ∈ N. The inverse limit could have also been ←−n n constructed inside the category of C∗-algebras; however, the two results will not agree. For instance, if H = lim H as above, then the inverse limit lim C(H ) inside the category of −→n n ←−n n topological ∗-algebras is C(H), whereas that inside the category of C∗-algebras is C (H), b ∼ i.e., the norm bounded functions on H. It is known that C (H) = C(βH), where βH is the b Stone–Cˇech compactification of H. Therefore, if one wants to model a space via its algebra of all continuous functions then the former inverse limit is the appropriate one. Henceforth, the inverse limits are always constructed inside the category of topological ∗-algebras. It is known that any ∗-homomorphism between two pro C∗-algebras is automatically continuous, provided the domain is a σ-C∗-algebra (see Theorem 5.2. of [12]). Furthermore, the category of commutative and unital σ-C∗-algebras with unital ∗-homomorphisms (automatically continuous) is contravariantly equivalent to the category of countably compactly generated and Hausdorff spaces with continuous maps via the functor X 7→ C(X) (see Proposition 5.7. of [12]). If A ∼= lim A , B ∼= lim B are two σ-C∗-algebras, then the minimal tensor ←−n n ←−n n product is defined to be A⊗ˆ B = lim A ⊗ˆ B . Henceforth, A⊗ˆB will always denote min ←−n n min n the minimal or spatial tensor product between σ-C∗-algebras. We next outline the contents of the paper. §1 initiates the concept of a σ-C∗-quantum group, where the interesting example of the quantum version of the universal special unitary group, C(SU (∞)), is discussed in detail. §2 initiates the concept of a σ-C∗-quantum q homogeneous space, where some interesting examples of quantum version of homogeneous spaces associated to the universal special unitary group, SU(∞), are discussed in detail. §3 contains the computation of the representable K-theory of C(SU (∞)) as well as some of q the quantum homogeneous spaces associated to it. 2 1. σ-C∗-quantum groups In this section, we define the concept of a σ-C∗-quantum group and explain it here. We also discuss in detail the interesting example of the quantum version of the universal special unitary group, C(SU (∞)). q If H is a countably compactly generated and Hausdorff topological group, although the multiplication map m : H × H → H is not proper, we get an induced comultiplication map m∗ : C(H) → C(H × H) ∼= C(H)⊗ˆC(H), which will be coassociative owing to the associativity of m. Motivated by the definition of Woronowicz (see also Definition 1 of [7]), we propose: Definition 1. A unital σ-C∗-algebra A is called a σ-C∗-quantum group if there is a unital ∗-homomorphism ∆ : A → A⊗Â which satisfies coassociativity, i.e., (∆⊗îd)∆ = (id⊗ˆ∆)∆ and such that the linear spaces ∆(A)(A⊗ˆ1) and ∆(A)(1⊗Â) are dense in A⊗Â. Lemma 1. Let {An,θn : An → An−1}n∈N be a countable inverse system of C∗-algebras and let B ⊂ A be dense subsets for all n such that θ (B ) ⊂ B . Then lim B is a dense n n n n n−1 ←−n n subset of the σ-C∗-algebra lim A . ←−n n Proof. The assertion follows from the Corollary to Proposition 9 in §4-4 of [2]. (cid:3) Example 1. Obviously, any C∗-compact quantum group is a σ-C∗-quantum group. Let {An,θn : An → An−1}n∈N be a countable inverse system of C∗-compact quantum groups with θ surjective and unital for all n. Furthermore, let us assume that the comultiplication n homomorphisms ∆ form a morphism of inverse systems of C∗-algebras {∆ } : {A } → n n n {A ⊗Â }. Then (A,∆) = (lim A ,lim ∆ ) is a σ-C∗-quantum group. Indeed, the density n n ←−n n ←−n n of the linear spaces ∆(A)(A⊗ˆ1) and ∆(A)(1⊗Â) inside A⊗Â follow from the above Lemma. Our next goal is to outline the construction of the quantum universal special unitary group, C(SU (∞)). Recall that for q ∈ (0,1), the C∗-algebra C(SU (n)) is the universal q q C∗-algebra generated by n2 +2 elements G := {un : i,j = 1,...,n}∪{0,1}, which satisfy n ij the following relations (1) 0∗ = 02 = 0, 1∗ = 12 = 1, 01 = 0 = 10, 1un = un1 = un, 0un = un0 = 0 for all i,j ij ij ij ij ij n n (2) un(un )∗ = δ 1, (un)∗un = δ 1 ik jk ij ki kj ij k=1 k=1 X X n n n (3) ··· E un ···un = E 1 i1i2···in j1i1 jnin j1j2···jn iX1=1iX2=1 iXn=1 3 where 0 whenever i ,i ,··· ,i are not distinct; 1 2 n E := i1i2···in (−q)ℓ(i1,i2,···,in). ( Here δ 1 = 0 if i 6= j, where 0 denotes the zero element in the generating set of C(SU (n)) ij q and, for any permutation σ, ℓ(σ) = Card{(i,j)|i < j,σ(i) > σ(j)}. The C∗-algebra C(SU (n)) has a C∗-compact quantum group structure with the comultiplication ∆ given q by ∆(0) := 0⊗0, ∆(1) := 1⊗1 and ∆(un) := un ⊗un . ij ik kj k X It is known that C(SU (n)) is a type-I C∗-algebra [3], whence it is nuclear. Therefore, there q is a unique choice for the C∗-tensor product in the definition of the comultiplication. There is a surjective ∗-homomorphism θ : C(SU (n)) → C(SU (n−1)) defined on the generators n q q by θ (x) := x if x = 0,1 n un−1 if 1 ≤ i,j ≤ n−1, θ (un) := ij n ij δ 1 otherwise, ( ij such that the following diagram commutes for all n > 2 C(SU (n)) ∆n // C(SU (n))⊗ˆC(SU (n)) q q q θn θn⊗ˆθn (cid:15)(cid:15) (cid:15)(cid:15) ∆n−1 C(SU (n−1)) // C(SU (n−1))⊗ˆC(SU (n−1)). q q q One can verify this assertion by a routine calculation on the generators. Consequently, for n > 2 the families {C(SU (n)),θ } and {C(SU (n))⊗ˆC(SU (n)),θ ⊗ˆθ } form countable q n q q n n inversesystemsofC∗-algebrasand{∆ } : {C(SU (n))}→{C(SU (n))⊗ˆC(SU (n))}becomes n q q q a morphism of inverse systems of C∗-algebras. We construct the underlying σ-C∗-algebra of the universal quantum gauge group as the inverse limit C(SU (∞)) = limC(SU (n)). q ←− q n In fact, C(SU (∞)) is a σ-C∗-quantum group, since it is the inverse limit of C∗-compact q quantum groups, where the comultiplication ∆ on C(SU (∞)) is defined to be ∆ = lim ∆ q ←−n n (see the Example above). If G is a set of generators and R a set of relations, such that the pair (G,R) is admissible (see Definition 1.1. of [1]), then one can always construct a universal C∗-algebra C∗(G,R). For instance, the universal C∗-algebra generated by the set {1,x}, subject to the relations 4 {1∗ = 12 = 1, 1x = x1 = x, x∗x = 1 = xx∗}, is isomorphic to C(S1). The generators and relations of C(SU (n)) described above are also admissible. q Remark 1. All matrix C∗-compact quantum groups considered, for instance, in [22, 23], such that the relations put a bound on the norm of each generator, are of the form C∗(G,R), where (G,R) is an admissible pair of generators and relations. Let {(Gi,Ri)}i∈N be a countable family of admissible pairs of generators and relations, so that C∗(G ,R ) exist for all i. Let F(G) denote the associative nonunital complex ∗- i i algebra (freely) generated by the concatenation of the elements of G G∗ and finite C- linear combinations thereof, where denotes disjoint union and G∗ = {g∗|g ∈ G} (formal ` adjoints). We call a relation in R algebraic if it is of the form f = 0 (or can be brought to ` that form), where f ∈ F(G). For instance, if G = {1,x}, then x∗x = 1 is algebraic, whereas kxk 6 1 is not. If (G,R) is a pair of generators and relations, then a representation ρ of (G,R) in a (pro) C∗-algebra B is a set map ρ : G → B, such that ρ(G) satisfies the relations R inside B. If (G,R) is a weakly admissible pair of generators and relations (see Definition 1.3.4. of [13]), then onecanconstruct the universal pro C∗-algebra C∗(G,R) (see Proposition 1.3.6. of ibid.). It is known that any combination (even the empty set) of algebraic relations is weakly admissible (see Example 1.3.5.(1) of ibid.). We further make the following hypotheses: (a) There are surjective maps θ : G → G , so that one may form the inverse limit in i i i−1 the category of sets G = lim G , with canonical projection maps p : G → G . We also ←−i i i i require the surjections θ to admit sections s : G → G satisfying θ ◦s = id , i i−1 i−1 i i i−1 Gi−1 so that we get canonical splittings γ : G → G satisfying p ◦ γ = id . The map γ i i i i Gi i sends g → {h }, where i j g if j = i, i (4) h = θ ◦···◦θ (g ) if j = i−n, n > 0, j i−n+1 i i   s ◦···◦s (g ) if j = i+m, m > 0. i+m−1 i i  (b) We require that for all i the iterated applications of θ ’s and s ’s on G satisfy R for all  j k i i j 6 i and k > i. Thesurjective mapsθ inducesurjective ∗-homomorphismsθ : C∗(G ,R ) → C∗(G ,R ); i i i i i−1 i−1 consequently, {C∗(Gi,Ri),θi}i∈N forms a countable inverse system of C∗-algebras. We may form the inverse limit lim C∗(G ,R ), which is by construction a σ-C∗-algebra. Let (G,R) ←−i i i be a pair of generators and relations, where G = lim G and R denotes the set of relations ←−i i {γ (G ) satisfies R for all i}. A representation ρ of (G,R) in a (pro) C∗-algebra B is a set i i i map ρ : G → B, such that ρ◦γ (G ) satisfies R inside B for all i. We assume that (G,R) is i i i a weakly admissible pair, so that one can construct the universal pro C∗-algebra C∗(G,R). 5 Theorem 1. There is an isomorphism of pro C∗-algebras C∗(G,R) ∼= lim C∗(G ,R ). ←−i i i Proof. It suffices to show that lim C∗(G ,R ) is a universal representation of (G,R), i.e., ←−i i i there is a map ι : G → lim C∗(G ,R ) such that ι◦γ (G ) satisfies R inside lim C∗(G ,R ) ←−i i i i i i ←−i i i for all i and given any representation ρ of the pair (G,R) in a pro C∗-algebra B, there is a unique continuous ∗-homomorphism κ : lim C∗(G ,R ) → B making the following diagram ←−i i i commute: G = ←lim−iGi TTTTTTTριTTTTTT// Tl←iTm−TTiTCT ∗((cid:15)(cid:15)Gκi,Ri) ** B. Themapι : G → lim C∗(G ,R )isdefinedasg 7→ {p (g)},whichisarepresentationof(G,R) ←−i i i i due to the Hypothesis (b) above. The construction of the universal pro C∗-algebra C∗(G,R) (resp. C∗-algebra C∗(G ,R )) is defined via a certain Hausdorff completion of F(G) (resp. i i F(G )) with respect to representations in pro C∗-algebras (resp. C∗-algebras) satisfying R i (resp. R ). The surjective maps θ induce ∗-homomorphisms θ : F(G ) → F(G ), whence i i i i i−1 we may construct the ∗-algebra lim F(G ) (purely algebraic inverse limit). By the above ←−i i Lemma it suffices to define κ on coherent sequences of the form {w } ∈ lim F(G ), which i ←−i i then extends uniquely to a ∗-homomorphism on the entire lim C∗(G ,R ). Thanks to the ←−i i i maps ρ ◦ γ : G → B, ρ extends uniquely to a ∗-homomorphism lim F(G ) → B. Now i i ←−i i there is a unique choice for κ({w }) forced by the compatibility requirement, i.e., κ({w }) = i i ρ({w }). By construction κ is a ∗-homomorphism and it is automatically continuous, since i lim C∗(G ,R ) is a σ-C∗-algebra. (cid:3) ←−i i i In the example of C(SU (∞)), one could try to define the section maps s : G → G q n−1 n−1 n as 0 7→ 0, 1 7→ 1, un−1 7→ un. ij ij But the Hypothesis (b) will not be satisfied and hence the above Theorem is unfortu- nately not applicable. However, the Theorem could be of independent interest as it can be applied to inverse systems, where the structure ∗-homomorphisms admit sections (also ∗-homomorphisms). Let G := {wn : i,j = 1,...,n}∪{0,1} be a set of generators satisfying the relations R n ij n 0∗ = 02 = 0, 1∗ = 12 = 1, 01 = 0 = 10, 1wn = wn1 = wn, 0wn = wn0 = 0, kwnk 6 1 ij ij ij ij ij ij for all i,j. The pair (G ,R ) is an admissible pair for all n, so that there is a universal n n C∗-algebra C∗(G ,R ). There are surjective maps θ : G → G given by n n n n n−1 6 θ (x) := x if x = 0,1 n wn−1 if 1 ≤ i,j ≤ n−1, θ (wn) := ij n ij δ 1 otherwise. ( ij making {C∗(G ,R ),θ } an inverse system of C∗-algebras and surjective ∗-homomorphisms. n n n There are obvious sections s : G → G sending 0 7→ 0, 1 7→ 1 and wn−1 7→ wn n−1 n−1 n ij ij giving rise to maps γ : G → G = lim G as described above (see Equation (4)). There n n ←−n n are surjective ∗-homomorphisms π : C∗(G ,R ) → C(SU (n)) for all n > 2 given on the n n n q generators by π (x) = x for x = 0,1 and π (wn) = un, which produce a morphism of inverse n n ij ij systems {π } : {C∗(G ,R )} → {C(SU (n))}. Indeed, it follows from the Relations (1), (2) n n n q and (3) that the norms of the generators of C(SU (n)) do not exceed 1 in any representation. q Consequently, there is a surjective ∗-homomorphism of σ-C∗-algebras (see [15] 1.6. Lemma) ∗ limπ : limC (G ,R ) → C(SU (∞)). ←− n ←− n n q n n However, the authors cannot provide a good description of the kernel at the moment. Let us set G = lim G and let R denote the set of relations {γ (G ) satisfies R for all n}. Note ←−n n n n n that kxk 6 1 viewed as a relation for a representation in a pro C∗-algebra B means that p(x) 6 1 for all C∗-seminorms p on B. The family of pairs (G ,R ) satisfy Hypotheses n n (b) and the pair (G,R) is weakly admissible (see Example 1.3.5.(2) of [13]), so that the above Theorem applies, i.e., lim C∗(G ,R ) ∼= C∗(G,R). As a corollary, we deduce that ←−n n n the elements of (lim π )(G) provide explicit generators of C(SU (∞)). ←−n n q 2. σ-C∗-quantum homogeneous spaces In this section we define the concept of a σ-C∗-quantum homogeneous space and explain it here. We also discuss in detail the interesting examples of the quantum versions of the homogeneous spaces associated to the universal special unitary group, SU(∞). Let {An,θn : An → An−1}n∈N and {Bn,ψn : Bn → Bn−1}n∈N be countable inverse systems of C∗-compact quantum groups with θ and ψ surjective and unital for all n. Furthermore, n n let us assume that the comultiplication homomorphisms ∆A,∆B form morphisms of inverse n n systems of C∗-algebras {∆A} : {A } → {A ⊗Â } and {∆B} : {B } → {B ⊗ˆB }. Then n n n n n n n n (A,∆A) = (lim A ,lim ∆A) and (B,∆B) = (lim B ,lim ∆B) are σ-C∗-quantum groups ←−n n ←−n n ←−n n ←−n n by the discussion in the above section (see Example 1). 7 Suppose now that there are compatible ∗-homomorphisms, θ′ : A → B , that is, such n n n that the following diagrams commute θ′n (5) A // B n n θn ψn (cid:15)(cid:15) θ′n−1 (cid:15)(cid:15) An−1 // Bn−1. and ∆An (6) A // A ⊗Â n n n θn′ θn′⊗ˆθn′ (cid:15)(cid:15) ∆Bn (cid:15)(cid:15) B // B ⊗ˆB . n n n Then, after Nagy [10] one calls the C∗-subalgebras H = f ∈ A (θ′⊗îd)∆A(f) = 1⊗ˆf ⊂ A n n n n n as C∗-compact quantum homo(cid:8)geneous(cid:12)spaces for all n ∈ N. I(cid:9)t is pointed out by Nagy that (cid:12) a parallel theory can be developed for H˜ = f ∈ A (id⊗ˆθ′)∆A(f) = f⊗ˆ1 ⊂ A . By n n n n n assumption, one has the following commutative diagram for all n, (cid:8) (cid:12) (cid:9) (cid:12) ∆n (7) A // A ⊗Â n n n θn θn⊗ˆθn (cid:15)(cid:15) ∆n−1 (cid:15)(cid:15) An−1 // An−1⊗Ân−1. A similar commutative diagram holds for {Bn,ψn : Bn → Bn−1}n∈N. Then we have Lemma 2. In the notation above, the ∗-homomorphism θ : A → A restricts to a n n n−1 ∗-homomorphism of C∗-compact quantum homogeneous spaces H → H for all n ∈ N. n n−1 Proof. Let f ∈ H . Then (θ′⊗îd)∆A(f) = 1⊗ˆf and we compute, n n n (θ′ ⊗îd)∆A (θ (f)) = (θ′ ⊗îd)(θ ⊗ˆθ )∆A(f) n−1 n−1 n n−1 n n n = (θ′ ⊗îd)(id⊗ˆθ )(θ ⊗îd)∆A(f) n−1 n n n = (θ′ ⊗îd)(id⊗ˆθ )1⊗ˆf n−1 n = 1⊗ˆθ (f), n showing that θ (f) ∈ H . (cid:3) n n−1 Now we define (8) H := limH ←− n n 8 to be a σ-C∗-quantum homogeneous space, where the inverse limit is once again taken inside the category of topological ∗-algebras. Remark 2. We remark here that this definition can be generalized as follows: Let (A,∆A) and (B,∆B) be σ-C∗-quantum groups, and θ′ : A −→ B be a ∗-homomorphism. Then one can also call the σ-C∗-subalgebra H = f ∈ A (θ′⊗îd)∆A(f) = 1⊗ˆf ⊂ A a σ-C∗-quantum homogeneous(cid:8)space.(cid:12)However we will not b(cid:9)e discussing these here. (cid:12) Our next goal is to outline the construction of the quantum homogeneous space associated to the universal gauge group SU(∞). Recall that there is a surjective ∗-homomorphism θ : C(SU (n)) → C(SU (n−1)) defined on the generators by n q q θ (x) := x if x = 0,1 n un−1 if 1 ≤ i,j ≤ n−1, θ (un) := ij n ij δ 1 otherwise. ( ij Then the quantum spheres (cf. [10]) are by fiat C(S2n−1) = f ∈ C(SU (n)) (θ ⊗îd)∆ (f) = 1⊗ˆf , q q n n and come with induced ∗-homom(cid:8)orphisms (cid:12) (cid:9) (cid:12) θ : C(S2n−1) → C(S2n−3). n q q Then the quantum homogeneous space associated to the universal gauge group SU(∞) is defined to be C(S∞) := limC(S2n−1). q ←− q n Itisshownin[17]thatC(S2n−1)isisomorphictoagroupoidC∗-algebra,whichisindependent q of q. Example 2. Another example is that of the C∗-quantum projective space (see, for instance, [18]), C(CPn) = C∗({v∗v |v = un+1 , v = un+1 , 1 6 i,j 6 n+1}) ⊂ C(S2n+1). q i j i (n+1)i j (n+1)j q Moreover, there is a short exact sequence of C∗-algebras relating the C∗-quantum projective spaces (see Corollary 2 of [18]), viz., (9) 0 → K → C(CPn) → C(CPn−1) → 0, q q for n > 1 and C(CP0) ≃ C. We define the C∗-quantum infinite projective space as q C(CP∞) = limC(CPn). q ←− q n 9 3. Representable K-theory of σ-C∗-quantum homogeneous spaces The appropriate K-theory for σ-C∗-algebras is representable K-theory, denoted by RK. In this section, we compute the representable K-theory of C(SU (∞)) as well as some of the q quantum homogeneous spaces associated to it. The RK-theory agrees with the usual K-theory of C∗-algebras if the input is a C∗-algebra and many of the nice properties that K-theory satisfies generalise to RK-theory. Let us briefly recall some of the basic facts about σ-C∗-algebras and RK-theory after Phillips [14] and Weidner [21]. (1) The RK-theory is homotopy invariant and satisfies Bott 2-periodicity. (2) If A is a C∗-algebra, then there is a natural isomorphism RK (A) ∼= K (A). i i (3) There is a natural isomorphism RK (A⊗ˆK) ∼= RK (A), where K denotes the algebra i i of compact operators on a separable Hilbert space. (4) If {An}n∈N is a countable inverse system of σ-C∗-algebras with surjective homomorphisms(whichcanalwaysbearranged), thentheinverselimitexistsasaσ-C∗-algebra and there is a Milnor lim1-sequence ←− 0 → lim1RK (A ) → RK (lim A ) → lim RK (A ) → 0. ←−n 1−i n i ←−n n ←−n i n Here we recall that Sheu [16] and Soibelman–Vaksman [20] have computed the K-theory of the C∗-quantum spheres, viz., K (C(S2n−1)) ≃ Z and K (C(S2n−1)) ≃ Z. 0 q 1 q Theorem 2. RK (C(S∞)) ≃ Z and RK (C(S∞)) ≃ {0}. 0 q 1 q Proof. There is a short exact sequence for all n (see Corollary 8 of [16]), 0 → C(T)⊗ˆK −→ C(S2n−1) −θ→n C(S2n−3) → 0. q q This gives rise to a 6-term exact sequence involving the topological K-theory groups (10) Z d1 // Z d2 // Z OO d6 d3 (cid:15)(cid:15) Z oo Z oo Z d5 d4 Sheu argues that d = 0 (see Section 7 of ibid.) from which it follows that d = d = 0 and 1 3 5 d = d = d = id or −id. The differential d (resp. d ) is the homomorphism induced by θ 2 4 6 2 5 n between the K -groups (resp. K -groups). By properties (2) and (4) above, one obtains the 0 1 following exact sequence of abelian groups 10