AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES K-theory for C∗-algebras: The hexagonal exact sequence ∗ Eduard Vilalta Vila Resum (CAT) Universitat Autònoma de Aquest treball té com a objectiu introduir el lector a la teoria K per C ∗ -àlgebres Barcelona demostrant-ne dos dels seus resultats centrals coneguts: La periodicitat de Bott evilalta@mat.uab.cat i la successió exacta cı́clica de sis termes. Aquests dos resultats constitueixen una eina essencial de cara al càlcul explı́cit dels K -grups d’una C ∗ -àlgebra, i han ∗Corresponding author estat utilitzats amb èxit en l’estudi de diverses famı́lies. De cara a enunciar-los, ens desviem lleugerament de la literatura estàndard i introduı̈m la notació K 0 , que permet simplificar els resultats i definicions necessàries per entendre les seves demostracions. Abstract (ENG) The aim of this work is to introduce the reader to C ∗ -algebraic K -theory whilst proving two of its main known results: Bott periodicity and the hexagonal exact sequence. These constitute a determinant tool for the explicit computation of the K -groups of a C ∗ -algebra, and have been used succesfully to study a variety of families. In order to state them, we deviate slightly from the standard literature and introduce the notation K 0 , which allows us to simplify the results and definitions needed to understand their proofs. Keywords: Operator Algebras, Non- commutative Topology, K -Theory. MSC (2010): 46L05, 46L35, 46L80, Acknowledgement 46L85. The author was partially supported by grant Received: October 13, 2019. 2017/COLAB/00487 of the Spanish Minis- Accepted: November 15, 2019. terio de Educación, Cultura y Deporte. http://reportsascm.iec.cat Reports@SCM 5 (2020), 1–10; DOI:10.2436/20.2002.02.18. 1 K -theory for C ∗ -algebras 1. Introduction The development of C ∗ -algebraic K -theory was initiated in the early 1970s, when G.A. Elliott classified the so-called approximately finite dimensional C ∗ -algebras by using their ordered K0 groups, see [2]. Since then, C ∗ -algebraic K -theory has become an important tool in the treatment of operator algebras, and has been used succesfully to classify a considerably large family of separable and simple C ∗ -algebras. In analogy to the topological K -theory developed by Aityah–Hirzebruch, in C ∗ -algebraic K -theory one defines a family of functors Kn from the category of C ∗ -algebras to that of abelian groups, thus assigning to every C ∗ -algebra A a family of groups Kn (A). The computation of these groups, usually known as the K -groups of the algebra, provides useful information on the structure of the sets of projections and unitaries of A. Towards this computation, and in contrast to algebraic K -theory, there exist a number of tools that make the treatment of the K -groups of a C ∗ -algebra manageable. Amongst them, there are two that are of particular importance: The first one, known as Bott periodicity, is the C ∗ -algebraic equivalent to the periodicity obtained in topological K -theory, and states that all K -groups of even and odd subscripts are isomorphic to K0 and K1 , respectively; see [1, Ch. 9]. The second result, which is a consequence of the first one, allows us to construct a hexagonal exact sequence from any exact sequence of C ∗ -algebras. In particular, the existence of such a sequence implies that one can compute the K -groups of a C ∗ -algebra by studying the K -groups of one of its ideals and its corresponding quotient; see [1, § 9.3]. Therefore, the aim of this work is to introduce the reader to C ∗ -algebraic K -theory whilst proving these two results. More explicitly, for any exact sequence of C ∗ -algebras ϕ φ 0 /I /A /B / 0, (1) we wish to obtain the associated hexagonal exact sequence K1 (ϕ) K1 (φ) K1 (I ) / K1 (A) / K1 (B) O (2) δ0 K0 (φ) K0 (ϕ)  δ1 K0 (B) o K0 (A) o K0 (I ). To this end, we will assume without loss of generality (see, for example, [4, § 1.1.5]) that (1) is of the form 0 /I i /A π / A/I / 0, (3) where I is an ideal of A and i, π are the usual inclusion and quotient mappings. The remainder of this paper has been divided into three parts, where some familiarity with C ∗ -algebras is assumed. All the C ∗ -algebraic background needed for these sections can be found in many textbooks; see, for example, [3]. In Section 2 we recall the definitions of the K -functors as well as some of their properties. We also introduce the notation Kn0 , which will allow us to shorten the definitions of this section and the proofs of Section 3. As the aim of Section 2 is for the reader to get acquainted with the basics of K -theory, we omit all the proofs. 2 http://reportsascm.iec.cat Eduard Vilalta Vila The index map, denoted by δ1 , is then defined in Section 3, where we first prove that the K 0 -functors are exact and invariant under homotopy. As these functors are equivalent to the K -functors, the construction of the index map together with this fact gives us a six-term exact sequence which is not yet cyclic. Finally, in Section 4 we prove Bott periodicity and construct the hexagonal exact sequence by closing the sequence obtained in Section 3. Since the proofs of some of the preliminary lemmas in this section are rather long and arduous, we only provide a reference for them. 2. An overview of C∗-algebraic K-theory In this first section we briefly review the concepts and results of C ∗ -algebraic K -theory that will be used in the subsequent sections. All the proofs can be found in [4, 5]. Throughout this paper, A and B will denote C ∗ -algebras and, given any two square matrices a, b over A, we will refer to the matrix diag(a, b) by a ⊕ b. 2.1 The projection group K0 and the unitary group K1 We begin our overview defining the first two K -groups. As we will later prove, these are the only ones up to isomorphism. Proposition 2.1. Let Pn (A) be the sets of projections in Mn (A) and denote by P∞ (A) their disjoint union. Then, by writting p ∼0 q if and only if p = vv ∗ and q = v ∗ v for some rectangular matrix v , one gets that (P∞ (A)/ ∼0 , ⊕) is a commutative monoid with the class of 0 as its unit. Definition 2.2. For any unital C ∗ -algebra A, the group K0 (A) is defined to be the Grothendieck group of the monoid above, where we denote the class of an element p ∈ P∞ (A) by [p]0 . If A does not have a unit, we define the group K0 (A) as the kernel of the map K0 (π) : K0 (Ã) → K0 (C), where K0 (π)([p]0 ) = [π(p)]0 and π is the usual projection map from à to C applied entry-wise (here, we use à to denote the unitification of A). Remark 2.3. It can be shown that every element in K0 (A) is of the form [p]0 −[1n ⊕0n ]0 for some projection p ∈ P∞ (Ã) whose scalar part s(p) is 1n ⊕ 0n . Moreover, if A is unital, it follows from its construction that every element in K0 (A) is of the form [p]0 − [q]0 for some p, q ∈ P∞ (A), where we can assume that both projections are of the same size. As A is equipped with a norm, we can study its induced topology. In particular, we say that two elements a and b are homotopic in a subset S ⊂ A, in symbols a ∼h b, if there exists a continuous path in S going from a to b. For example, given two projections p, q homotopic in Pn (A), one can see that [p]0 = [q]0 . Conversely, if [p]0 = [q]0 , then p ⊕ 0s ∼h q ⊕ 0t for some positive integers s and t. We will also say that two ∗-homomorphisms ϕ0 and ϕ1 from A to B are homotopic if there is a continuous map t 7→ ϕt from [0, 1] to the ∗-homomorphisms from A to B such that t 7→ ϕt (a) is a homotopy for each a ∈ A. Moreover, two C ∗ -algebras A and B are said to be homotopic if there exist two ∗-homomorphisms φ and ϕ such that φ ◦ ϕ ∼h idB and ϕ ◦ φ ∼h idA . Proposition 2.4. Let A be a unital C ∗ -algebra and consider the set U∞ (A) = ∪n Un (A), where Un (A) denotes the set of all unitary n × n matrices over A. Then, the equivalence relation “u ∼1 v if and only if u ⊕ 1n ∼h v ⊕ 1m in UN (A) for some suitable integers n, m, and N” makes (U∞ (A)/ ∼1 , ⊕) into a commutative group with the class of 1 as its unit. Reports@SCM 5 (2020), 1–10; DOI:10.2436/20.2002.02.18. 3 K -theory for C ∗ -algebras Definition 2.5. Given a unital C ∗ -algebra A, the group K1 (A) is the commutative group defined above, where we refer to the class of a unitary u ∈ U∞ (A) as [u]1 . If A does not have a unit, we define K1 (A) := K1 (Ã). + (Ã), where Remark 2.6. It can be proven that every element in K1 (A) is of the form [u]1 with u ∈ U∞ + (Ã) is the set of unitaries whose scalar part is of norm 1. U∞ Example 2.7. It is easy to see that two elements p, q ∈ P∞ (C) are equivalent under ∼0 if and only if dim(Im(p)) = dim(Im(q)). Thus, it follows that K0 (C) ∼ = Z. Moreover, recall that a unitary u in a unital C ∗ -algebra is homotopic to 1 in U(A) if and only if its spectrum is not T; see [4, Lem. 2.1.3(ii)]. Therefore, as all unitaries in U∞ (C) have finite spectrum, they must be equivalent to 1 under ∼1 . This implies that K1 (C) = 0. One can also adapt these arguments to see that K0 (B(H)) = K1 (B(H)) = 0 for any separable infinite dimensional Hilbert space H. 2.2 Suspension functor and higher index K-groups Once the K0 and K1 groups have been defined, one can make use of the suspension functor S to define two families of groups: the higher index K -groups and the K 0 -groups. Even though these two families turn out to be the same, the introduction of the K 0 -groups allows us to simplify both the definitions and proofs regarding the properties of the higher index K -groups. Recall that the suspension functor S is an exact covariant functor mapping a C ∗ -algebra A to SA := {f ∈ C (T, A) | f (1) = 0}, and a ∗-homomorphism φ : A → B to the ∗-homomorphism Sφ from SA to SB defined as Sφ(f ) = φ ◦ f . Definition 2.8. By using the notation S 0 = id and S n = S n−1 ◦ S, we define the higher index K -groups 0 Kn (A) = K1 (S n−1 A) and the K 0 -groups Kn (A) = K0 (S n (A)). 0 0 0 Now let φ : A → B be a ∗-homomorphism. We denote by Kn (φ) : Kn (A) → Kn (B) the group homo- 0 0 morphism Kn (φ)([p]0 − [s(p)]0 ) = [S n φ̃(p)]0 − [S n φ̃(s(p))]0 . One can check that this definition makes Kn into functors. A proof of the theorem below can be found in [5, Thm. 7.2.5]. 0 Theorem 2.9. Given any C ∗ -algebra A, consider the map θA,n : Kn (A) → Kn (A) defined as θA,n ([u]1 ) = [w (1m ⊕ 0m )w ∗ ]0 − [1m ⊕ 0m ]0 , + ^ u ∈ Um (S n−1 A), where w is a homotopy between 12m and u ⊕ u ∗ in U2m (S n−1 A). Then, θA,n is an isomorphism for every integer n ≥ 1. Definition 2.10. Given a ∗-homomorphism φ : A → B, we define Kn (φ) : Kn (A) → Kn (B) as Kn (φ) = −1 0 θB,n ◦ Kn (φ) ◦ θA,n . Together with this definition, the K -groups also become functors. 3. Homotopy invariance and the index map The goal of this section is to define the map δ1 from (2) and prove that the two rows together with the 0 right column of (2) form an exact sequence. However, we will first show that the Kn functors are invariant 4 http://reportsascm.iec.cat Eduard Vilalta Vila under homotopy, as this is one of the main tools used in the explicit computation of the K groups of a C ∗ -algebra. Note that, by their definition and Theorem 2.9, this will imply that the functors Kn are also invariant under homotopy. 0 Theorem 3.1. Given two homotopic ∗-homomorphisms ϕ0 and ϕ1 from A to B, we have that Kn (ϕ0 ) = 0 Kn (ϕ1 ), for every n ≥ 0. Proof. Fix n ∈ N and let q be an element in Pk (Sgn A) for some k. Then, write q as the sum q = p+α1 n A,k Sg with p ∈ Mk (S A) and α ∈ Mk (C). For every t ∈ [0, 1], define the elements pt = S n ϕt (p) and n qt = S]n ϕ (q), where t 7→ ϕ is the homotopy from ϕ to ϕ . t t 0 1 As S]n ϕ is a ∗-homomorphism, it follows that q is a projection for every t. Moreover, one gets that t t qt = pt + α1Sg n B,k and, consequently, that t → 7 qt is continuous if and only if t 7→ pt is continuous. Now let δp : [0, 1] × Tn → A be the map defined as δ(p)(t, (z1 , ... , zn )) = pt (z1 )(z2 ) · · · (zn ), and note that, for any two pairs (t1 , ξ1 ), (t2 , ξ2 ) ∈ [0, 1] × Tn , one gets kδp (t1 , ξ1 ) − δp (t2 , ξ2 )k ≤ kϕt1 (p(ξ1 )) − ϕt2 (p(ξ1 ))k + kp(ξ1 ) − p(ξ2 )k. Thus, since p is continuous and t 7→ ϕt is a homotopy, we have that δp is also continuous. Furthermore, as δp has compact support, the map is uniformly continuous. It then follows that t 7→ pt is continuous and that t 7→ qt is a homotopy of projections. Therefore, one gets [q0 ]0 = [q1 ]0 for any q ∈ P∞ (Sg n A), which implies the equality K (S ] n ϕ ) = K (S]n ϕ ), from which 0 0 0 1 the desired result follows. Example 3.2. Let X be a compact, Hausdorff, and contractible topological space. Then, the K0 and K1 groups of the C ∗ -algebra C (X , C) are isomorphic to Z and 0, respectively, as C (X , C) is homotopic to C. Recall that X is contractible if there exists a point x0 and a continuous map c : X × [0, 1] → X such that c(x, 0) = x and c(x, 1) = x0 for every x ∈ X . Then, a pair of functions giving the homotopy between C (X , C) and C are z 7→ z1C (X ,C) and f 7→ f (x0 ). For more details, see [4, Ex. 3.3.6]. Proposition 3.3. For any exact sequence of the form (3), the induced sequence Kn (i) Kn (π) Kn (I ) / Kn (A) / Kn (A/I ) (4) is exact for every n. Proof. As we have previously noted, it follows from their definition and Theorem 2.9 that proving the result 0 for Kn is equivalent to proving it for Kn . Moreover, by using the functoriality and exactness of S, one can see that the diagram 0 Kn (I ) / K 0 (A) / K 0 (A/I ) n n    K0 (Im(Sn i)) / K0 (S n A) / K0 (S n A/Im(S n i)) is commutative and has isomorphisms as columns, for every n ≥ 0. Thus, we only need to prove that ker(K0 (π)) ⊂ Im(K0 (i)), as we can restrict ourselves to n = 0 and the other inclusion is clear. Reports@SCM 5 (2020), 1–10; DOI:10.2436/20.2002.02.18. 5 K -theory for C ∗ -algebras Now, given an element [p]0 − [s(p)]0 in ker(K0 (π)), find a unitary u ∈ UN (A/Ig ) such that, for suitable integers n, k, N, u(π̃ ⊕ 1n ⊕ 0k ) = s(p) ⊕ 1n ⊕ 0k . Then, by taking a unitary w homotopic to 12N in U2N (A) such that π̃(w ) = u ⊕ u ∗ , we can define the projection r = w (p ⊕ 1n ⊕ 0k+N )w ∗ . As π̃(r ) ∈ M∞ (C1A ) by construction, it follows that r ∈ M∞ (Ĩ ). In particular, we have that [p]0 − [s(p)]0 = [r ]0 − [s(r )]0 ∈ Im(K0 (i)), as required. Theorem 3.4. For any exact sequence of the form (3), there exists a group homomorphism δ1 such that the following sequence is exact: K1 (i) K1 (π) K1 (I ) / K1 (A) / K1 (A/I ) (5) K0 (π) K0 (i)  δ1 K0 (A/I ) o K0 (A) o K0 (I ). Proof. Given an element [u]1 ∈ K1 (A/I ) with u ∈ Uk+ (A/I g ), we define its image through the index map δ1 as δ1 ([u]1 ) = [w (1n ⊕ 0k )w ∗ ]0 + [1n ⊕ 0k ]0 , where v ∈ Uk+ (A/I g ) is such that u ⊕ v ∼h 1n+k in U + (A/I k g ), and w is a unitary lift of u ⊕ v . It can be proven that δ1 is indeed a well defined group homomorphism; see, for example, [5, Prop. 8.1.3]. Then, it follows from Proposition 3.3 that we only need to prove the equalities Im(δ1 ) = ker(K0 (i)) and ker(δ1 ) = Im(K1 (π)). Moreover, note that the inclusions Im(δ1 ) ⊆ ker(K0 (i)) and Im(K1 (π)) ⊆ ker(δ1 ) are clear. g ) for some m, and let w be a unitary lift of u ⊕ u ∗ . As Thus, let [u]1 ∈ ker(δ1 ) with u ∈ U + (A/I m δ1 ([u]1 ) = 0, we can find an integer k and a matrix v ∈ M2(k+2m) (Ĩ ) such that vv ∗ = 12n − q ⊕ 1k ⊕ 0n and v ∗ v = 12n − (1m ⊕ 0m ) ⊕ (1k ⊕ 0n ), where q = w (1m ⊕ 0m )w ∗ and n = k + 2m. By using the previous two equalities together with vv ∗ v = v , it is easy to check that π̃(v ) = 0m ⊕ X for some X ∈ M2n−m (C1à ). Therefore, there exists a complex (2n + m) × (2n + m) matrix U such that π̃(qw ⊕ v ) = u ⊕ U. As U ∼h 12n+m , it follows that [u]1 = [u]1 + [U]1 = K1 (π)([qw ⊕ v ]1 ) ∈ Im(K1 (π)). Now let x = [p]0 − [1n ⊕ 0n ]0 ∈ ker(K0 (i)) with p ∈ P2n (Ĩ ). Then, there exists an integer k ∈ N for which p ⊕ 1k ⊕ 0m ∼0 (1n ⊕ 0n ) ⊕ 1k ⊕ 0m =: sk , with m = 3(2n + k). Moreover, we can find a complex matrix P such that Psk P ∗ = 1n+k ⊕ 0n+m and, consequently, we have q := P(p ⊕ 1k ⊕ 0m )P ∗ ∼0 1n+k ⊕ 0n+m =: d. Take w ∈ U∞ + (Ã) such that wqw ∗ = d and note that π̃(w ) commutes with d. It follows that π̃(w ) = a ⊕ b + for some a ∈ Un+k (Ã). Finally, as w is a unitary lift of a ⊕ b, we get x = [q]0 − [d]0 = [wdw ∗ ]0 − [d]0 = δ1 (π̃(a)) ∈ Im(δ1 ), as desired. 6 http://reportsascm.iec.cat Eduard Vilalta Vila 4. Bott periodicity and the hexagonal exact sequence We begin this section by noting that Mn (SA) f can be identified with the set of continuous functions f ∈ C (T, Mn (Ã)) such that f (1) ∈ Mn (C1à ). In particular, we can write f = {f ∈ C (T, Un (Ã)) | f (1) ∈ Mn (C1 )}. Un (SA) à By using this identification, we can now define the Bott map. What follows is a combination of [4, Ch. 11] and [5, Ch. 9]: Definition 4.1. Let A be a unital C ∗ -algebra and take p ∈ Pn (A). We define fp ∈ Un (SA) f as the map from T to A such that fp (z) = zp + (1n − p). The Bott map β A : K0 (A) → K1 (SA) is then defined as βA ([p]0 − [q]0 ) = [fp fq∗ ]1 for any element [p]0 − [q]0 in K0 (A). Remark 4.2. It can be proven that βA is a well defined homomorphism; see, e.g., [4, § 11.1]. If A is not unital, we define the Bott map of A to be the only homomorphism for which the following diagram is commutative: / K0 (A) / K0 (Ã) o / /0 0 K0 (C) βA βà βC    / K1 (SA) / K1 (S Ã) o / / 0. 0 K1 (SC) In particular, it follows that we only need to prove that the Bott map is an isomorphism in the unital case. Thus, we will assume from now on that A has a unit. We will first prove that βA is surjective. Let GL0 (Mn (A)) be the set of invertible n × n matrices that are homotopic to the identity. Then, define the following sets: Invn0 := C (T, GL0 (Mn (A))), Xm Polnm := {f ∈ Invn0 | f(z) = ai zi , ai ∈ Mn (A)}, i=0 Xm Trignm := {f ∈ Invn0 | f(z) = ai zi , ai ∈ Mn (A)}. i=−m Remark 4.3. One can check that Un (SA) f is a subset of Invn for every n, and that, if two unitaries are 0 homotopic in Invn0 , then they are also homotopic in Un (SA); f see [4, § 11.2]. As we have already mentioned in the introduction, we omit the proof of the next lemma. Lemma 4.4 ([5, Lem. 9.2.3–9.2.7]). For every unital C ∗ -algebra A and every integer n ∈ N, we have: (i) for every f ∈ Invn0 , there exists an integer m and an element h ∈ Trignm such that f ∼h h in Invn0 ; (ii) for every integer m there exists a continuous map µnm from Polnm to Polmn+n 1 such that µnm (f ) is homotopic to f ⊕ 1mn in Invn0 , for every f ; Reports@SCM 5 (2020), 1–10; DOI:10.2436/20.2002.02.18. 7 K -theory for C ∗ -algebras (iii) for any degree one polynomial f ∈ Poln1 there exists an element γ(f ) of the form fp such that f ∼h γ(f ) in Poln1 ; moreover, the map f 7→ γ(f ) is continuous. Proposition 4.5. The Bott map is surjective. f By Lemma 4.4(1), we can find an element h ∈ Trign such Proof. Let [f ]1 ∈ K1 (SA) with f ∈ Un (SA). m that f ∼h h in Invn0 . As z −N ⊕ 1M−1 ∼h f1∗N ⊕0M−N in UM (SA) f for every pair N, M such that N ≤ M, we get (hz N )z −N ⊕ 1M ∼h (hz N ⊕ 1M )fp∗N for every N ≤ M, and where pN = 1N ⊕ 0M−N . In particular, if N is large enough, hz N is polynomial. Thus, for any such N, Lemma 4.4(2) ensures that we can find a degree one polynomial r such that hz N ⊕ 1t ∼h r for some t. Moreover, it follows from Lemma 4.4(3) that there exists some element fp homotopic to r . By now adding some extra 1’s in the diagonal, we have that f ⊕ 1M+nt ∼h fp⊕0M fp∗N and, consequently, that βA ([p]0 − [pN ]0 ) = [f ]1 . We now prove that βA is injective. Once again, we will omit the proof of the following lemma. Lemma 4.6 ([5, § 9.1.2 & Lem. 9.2.10]). For every unital C ∗ -algebra A and every integer n ∈ N, we have: (i) the map π : {fp | p ∈ Pn (A)} → Pn (A) sending an element fp to p is continuous; (ii) for any homotopy f 7→ ft in Invn0 , there exists a positive integer N such that f 7→ ft can be uniformly approximated by a homotopy c 7→ ct in TrignN that is piecewise linear. In particular, if f0 , f1 ∈ TrignN , one can set c0 = f0 and c1 = f1 . Proposition 4.7. The Bott map is injective. Proof. Let [p]0 − [q]0 ∈ K0 (A) be such that βA ([p]0 − [q]0 ) = 0 or, equivalently, such that [fp fq∗ ]1 = 0. Then, by possibly adding some zeros diagonally, we have that fp ∼h fq . By Lemma 4.6(2), we can find a polynomial homotopy between z N fp and z N fq for some integer N. Moreover, once again adding some zeros and ones diagonally, and using Lemma 4.4(2,3), we obtain a homotopy t 7→ fpt such that p0 = p and p1 = q. Finally, Lemma 4.6(1) gives a homotopy between p and q, from which we get that [p]0 = [q]0 and, therefore [p]0 − [q]0 ∈ ker(βA ). Combining Propositions 4.5 and 4.7 above, one gets the desired result: Theorem 4.8. For any C ∗ -algebra A, the Bott map βA is an isomorphism between the groups K0 (A) and K2 (A). Consequently, all the K -groups Kn (A) of even subindexes are isomorphic to K0 (A), and those with odd subindexes are isomorphic to K1 (A). Other proofs of this result are indeed possible, such as the recent one in [6]. In it, Voiculescu’s almost commuting matrices are used to define a homomorphism αA : K1 (SA) → K0 (A), which is then shown, with the help of Atiyah’s rotation trick, to be the mutual inverse of βA (the author thanks the anonymous referee for this reference). With Bott periodicity at hand, we can now construct the hexagonal exact sequence. 8 http://reportsascm.iec.cat Eduard Vilalta Vila Theorem 4.9. For any exact sequence of the form (3), there exists a group homomorphism δ0 such that the following hexagonal sequence is exact K1 (i) K1 (π) K1 (I ) / K1 (A) / K1 (A/I ) O δ0 δ1 K0 (π) K0 (i)  K0 (A/I ) o K0 (A) o K0 (I ). Proof. Given an exact sequence 0 → I → A → A/I → 0, consider the suspended sequence 0 → SI → SA → S(A/I ) → 0 0 0 and its corresponding index δ1 from Theorem 3.4. Then, define δ0 as the composition θI−1 ,1 ◦ δ1 ◦ βA/I , −1 where θI ,1 is the isomorphism from Theorem 2.9. K2 (A/I ) ? 0 θI−1 ,1 ◦δ1  K1 (i) K1 (π) βA/I K1 (I ) / K1 (A) / K1 (A/I ) δ1 K0 (π) K0 (i)  K0 (A/I ) o K0 (A) o K0 (I ) By Theorem 3.4, we only need to prove that the sequence is exact at K1 (I ) and K0 (A/I ). To do this, simply note that the following diagram is commutative K0 (π) δ0 K1 (i) K0 (A) / K0 (A/I ) / K1 (I ) / K1 (A) O O O O βA βA/I θI θA  K1 (Sπ)  δ1  K0 (Si)  K1 (SA) / K1 (S(A/I )) / K0 (SI ) / K0 (SA), and that all of its columns are isomorphisms. As the second row is exact by Theorem 3.4, so is the first one. Example 4.10. Let H be an infinite dimensional separable Hilbert space and consider the Calkin algebra Q(H) = B(H)/K (H), where K (H) is the algebra of compact operators on H. Then, the exact sequence 0 / K (H) i / B(H) π / Q(H) /0 induces, by Theorem 4.9, the hexagonal exact sequence K1 (i) K1 (π) K1 (K (H)) / K1 (B(H)) / K1 (Q(H)) O δ0 K0 (π) K0 (i)  δ1 K0 (Q(H)) o K0 (B(H)) o K0 (K (H)). Moreover, recall from Example 2.7 that K0 (B(H)) = K1 (B(H)) = 0. Thus, δ0 and δ1 are isomorphisms. By using that the K -groups are stable (see [4, Prop. 6.4.1 & Prop. 8.2.8]), one can also see that K0 (K (H)) ∼ = K0 (C) ∼ = Z and that K1 (K (H)) ∼= K1 (C) = 0. Therefore, the K0 and K1 groups of the Calkin algebra are isomorphic to 0 and Z respectively. Reports@SCM 5 (2020), 1–10; DOI:10.2436/20.2002.02.18. 9 K -theory for C ∗ -algebras References [1] B. Blackadar, “K -Theory for operator alge- [4] M. Rørdam, F. Larsen, and N. Laustsen, “An in- bras”, Springer-Verlag, New York, 1986. troduction to K -theory for C ∗ -algebras”, Cam- bridge University Press, 2000. [2] G.A. Elliott, “On the classification of induc- tive limits of sequences of semisimple finite- [5] N.E. Wegge-Olsen, “K -Theory and C ∗ -alge- dimensional algebras”, Journal of Algebra 38 bras: a friendly approach”, Oxford University (1976), 29–44. Press, 1993. [3] G.J. Murphy, “C ∗ -Algebras and operator the- [6] R. Willett, “Bott periodicity and almost com- ory”, Academic Press, 1990. muting matrices”, preprint, 2019. 10 http://reportsascm.iec.cat