aa r X i v : . [ m a t h . K T ] A p r UNSUSPENDED CONNECTIVE E -THEORY OTGONBAYAR UUYE
Abstract.
We prove connective versions of results by Shulman [Shu10]and D˘ad˘arlat-Loring [DL94]. As a corollary, we see that two separable C ∗ -algebras of the form C ( X ) ⊗ A , where X is a based, connected,finite CW-complex and A is a unital properly infinite algebra, are bu -equivalent if and only if they are asymptotic matrix homotopy equiva-lent. Introduction
Let S denote the Connes-Higson asymptotic homotopy category of separa-ble C ∗ -algebras (c.f. [CH90, GHT00]). Let Σ denote the suspension functorΣ B := C ( R ) ⊗ B and let K denote the algebra of compact operators on aseparable Hilbert space. E -theory is the bivariant K -theory defined by E ( A, B ) := S (Σ A, Σ B ⊗ K ) . (1)In this paper, we prove connective extensions of the following two closelyrelated results. Theorem 0.1 (Shulman [Shu10]) . Let A be a separable C ∗ -algebra. Then qA ⊗ K is S -equivalent to Σ A ⊗ K . Theorem 0.2 (D˘ad˘arlat-Loring [DL94, Theorem 4.3]) . Let A and B beseparable C ∗ -algebras. If the abelian monoid S ( A, A ⊗ K ) is a group, thenthe suspension functor induces an isomorphism S ( A, B ⊗ K ) ∼ = E ( A, B ⊗ K ) . (2)See Theorem 3.8 and Theorem 3.11 for the precise statements. Consider-ing stable algebras, we obtain Theorem 0.1 and Theorem 0.2, respectively.We note that this gives new and more conceptual, if not simpler, proofs ofthe theorems.We refer to [Tho03] and references therein for details of connective E -theory and its applications. Acknowledgement.
This research is supported by the Danish National Re-search Foundation (DNRF) through the Centre for Symmetry and Deforma-tion at the University of Copenhagen. The author wishes to thank Tatiana
Date : September 26, 2018. Our proof of Theorem 0.2 is closely related to the remark at end of Section 4 in [DL94,Theorem 4.3].
Shulman and Hannes Thiel for stimulating conversations on the topics pre-sented here. The author also wishes to thank Takeshi Katsura for insightfulcomments. 1.
Asymptotic Matrix Homotopy Category
We start by fixing some notation.
Notation 1.1. (i) Let A and B be C ∗ -algebras. We write A ⋆ B , A × B and A ⊗ B for the free product, direct product/sum and maximal tensorproduct of A and B , respectively.(ii) For n ≥
1, let M n denote the C ∗ -algebra of n × n complex matrices.For n , m ≥
1, we write ⊕ for the operation ⊕ : M n × M m → M n + m , ( a, b ) (cid:18) a b (cid:19) (3)and, i m,n , for m ≥ n , for the inclusion i m,n : M n ֒ → M m , a a ⊕ . (4)We identify C with M and M n ⊗ M m with M nm for n , m ≥
1, and K with the colimit of M n along i m,n .(iii) For k ≥
0, let Σ k denote the C ∗ -algebra C ( R k ) of continuous functionson R k vanishing at infinity. We identify Σ with C and Σ k ⊗ Σ l withΣ k + l for k , l ≥ Definition 1.2.
Let A and B be separable C ∗ -algebras. We define m ( A, B )as the colimit m ( A, B ) := colim n →∞ S ( A, B ⊗ M n ) (5)along (id B ⊗ i m,n ) ∗ .We summarize some properties of m that are well-known and/or easy tocheck. Statements (i)-(iii) say, essentially, that m is a homotopy invariant,matrix stable category enriched over the abelian monoids. Proposition 1.3.
Let A , B , C and D stand for separable C ∗ -algebras andlet m , n ≥ .(i) Homotopic ∗ -homomorphisms A → B define the same class in m ( A, B ) .(ii) The composition S ( B, C ⊗ M m ) × S ( A, B ⊗ M n ) → S ( A, C ⊗ M mn ) (6)( g, f ) ( g ⊗ id M n ) ◦ f (7) gives m a category structure, with the identity morphism on A repre-sented by id A ⊗ i n, : A → A ⊗ M n .(iii) The addition S ( A, B ⊗ M n ) × S ( A, B ⊗ M m ) → S ( A, B ⊗ M n + m ) (8)( f , g ) f ⊕ g (9) NSUSPENDED CONNECTIVE E -THEORY 3 gives m ( A, B ) the structure of an abelian monoid, bilinear with respectto composition.(iv) The tensor product S ( A, B ⊗ M n ) × S ( C, D ⊗ M m ) → S ( A ⊗ C, B ⊗ D ⊗ M nm ) (10)( f , g ) f ⊗ g (11) defines a natural bilinear functor ⊗ : m ( A, B ) × m ( C, D ) → m ( A ⊗ C, B ⊗ D ) . (12) (v) For any A and C , the functor F ( B ) := m ( A, B ⊗ C ) is split exact.Proof. We prove only the last statement (v). This follows from [DL94,Proposition 3.2] and [Wei94, Theorem 2.6.15]. (cid:3)
Definition 1.4 (c.f. [Tho03, Definition 4.4.14]) . We call m the asymptoticmatrix homotopy category of separable C ∗ -algebras. Lemma 1.5 (Cuntz [Cun87, Proposition 3.1(a)]) . For any separable C ∗ -algebras B and C , the natural map B ⋆ C → B × C (13) is an m -equivalence. (cid:3) Corollary 1.6.
For any separable C ∗ -algebras B, C and D , the natural map ( B ⊗ D ) ⋆ ( C ⊗ D ) → ( B ⋆ C ) ⊗ D (14) is an m -equivalence.Proof. The following diagram is commutative( B ⊗ D ) ⋆ ( C ⊗ D ) / / (cid:15) (cid:15) ( B ⋆ C ) ⊗ D (cid:15) (cid:15) ( B ⊗ D ) × ( C ⊗ D ) / / ( B × C ) ⊗ D . (15)The vertical maps are m -equivalences by Lemma 1.5 and the bottom hori-zontal map is an isomorphism. (cid:3) Notation 1.7.
Let B be a separable C ∗ -algebra. Following Cuntz, we write qB for the kernel of the folding map B ⋆ B id ⋆ id / / B .We note that the short exact sequence0 / / qB / / B ⋆ B / / B / / split-exact . Proposition 1.8.
For any separable C ∗ -algebras B and D , the natural map σ B,D : q ( B ⊗ D ) → qB ⊗ D (17) is an m -equivalence. OTGONBAYAR UUYE
Proof.
Fix A and let F denote the functor F ( B ) := m ( A, B ).We apply F to the following commutative diagram of split-exact se-quences:0 / / q ( B ⊗ D ) / / σ B,D (cid:15) (cid:15) B ⊗ D ⋆ B ⊗ D / / (cid:15) (cid:15) B ⊗ D / / / / qB ⊗ D / / ( B ⋆ B ) ⊗ D / / B ⊗ D / / . (18)By Corollary 1.6, F induces isomorphism on the middle map. Since F issplit exact, it follows that F ( σ B,D ) is an isomorphism. Now the proof followsfrom Yoneda’s Lemma. (cid:3)
Remark . Let Ho denote the homotopy category of C ∗ -algebras and let n denote the matrix homotopy category with morphisms n ( A, B ) := colim n Ho ( A, B ⊗ M n ) . (19)Then, in Lemma 1.5 and Corollary 1.6, we actually have n -equivalences.However, the map σ B,D from Proposition 1.8 is not an n -equivalence ingeneral. For instance, let T denote the reduced Toepliz algebra. Then T is KK -contractible, hence q ( T ) ⊗ K is contractible i.e. homotopy equivalentto the zero algebra 0 (c.f. [Cun84]). However, q C ⊗ T ⊗ K has a non-trivialprojection, hence not contractible. It follows that σ C ,T : q ( T ) → q C ⊗ T is not an n -equivalence.Indeed, for any A and B , we have a natural isomorphism Ho ( A, B ⊗ K ) ∼ = n ( A, B ⊗ K ) . (20)Hence if f : A → B is an n -equivalence, then f ⊗ id K : A ⊗ K → B ⊗ K is ahomotopy equivalence. Remark . Let A and B be separable C ∗ -algebras.(i) We have a natural isomorphism S ( A, B ⊗ K ) ∼ = m ( A, B ⊗ K ) . (21)It follows that if f ∈ m ( A, B ) is an m -equivalence, then f ⊗ id K is an S -equivalence from A ⊗ K to B ⊗ K .(ii) Tensoring with K gives an isomorphism S ( A, B ⊗ K ) ∼ = S ( A ⊗ K , B ⊗ K ) . (22)2. Matrix Homotopy Symmetry
The following definition is inspired by [DL94].
Definition 2.1.
A separable C ∗ -algebra A is matrix homotopy symmetric ifid A ∈ m ( A, A ) has an additive inverse: there is n ≥ η : A → A ⊗ M m such that i n, ⊕ η : A → A ⊗ M n + m is null-homotopic. NSUSPENDED CONNECTIVE E -THEORY 5 Remark . (i) If the monoid m ( A, A ) is a group, then A is matrix ho-motopy symmetric. Conversely, if A is matrix homotopy symmetric,then m ( A, B ) and m ( B, A ) are abelian groups for any B .(ii) If A is matrix homotopy symmetric, then so is A ⊗ D for any D .(iii) If A is m -equivalent to B and A is matrix homotopy symmetric, thenso is B . Example 2.3. (i) The algebra Σ is matrix homotopy symmetric. Infact, the algebra C ( X ), of continuous functions vanishing at the basepoint, is matrix homotopy symmetric for any based, connected , finiteCW-complex X (c.f. [DN90, Proposition 3.1.3] and the discussion pre-ceding it).(ii) The algebra qB is matrix homotopy symmetric for any B , by taking n = m = 1 and η = τ the flip-map on qA (c.f. [Cun87, Proposition1.4]). Notation 2.4.
Let B be a separable C ∗ -algebra. Let π B : qB → B denotethe composition π B : qB (cid:31) (cid:127) / / B ⋆ B id ⋆ / / B . (23)We remark that q is functorial (for ∗ -homomorphisms) and for any ∗ -homomorphism f : A → B , we have a commutative diagram qA q ( f ) / / π A (cid:15) (cid:15) qB π B (cid:15) (cid:15) A f / / B . (24)From our point of view, the following is the key ingredient that underliesboth Theorem 0.1 and Theorem 0.2.
Proposition 2.5.
Let A be a separable C ∗ -algebra. Then the followingstatements are equivalent:(a) The algebra A is matrix homotopy symmetric.(b) For any B and D , we have ( π B ⊗ id D ) ∗ : m ( A, qB ⊗ D ) ∼ = m ( A, B ⊗ D ) . (c) The map π A : qA → A is an m -equivalence.(d) The map π C ⊗ id A : q C ⊗ A → A is an m -equivalence.Proof. The statements (c) and (d) are equivalent by Proposition 1.8.Since qA is matrix homotopy symmetric (c.f. Example ii), it follows fromRemark 2.2 that (c) implies (a).Suppose that A is matrix homotopy symmetric. Then the functor F ( B ) := m ( A, B ⊗ D ) is a homotopy invariant, split exact, matrix stable functor withvalues in abelian groups. Hence ( π B ) ∗ : F ( qB ) → F ( B ) is an isomorphismfor all B , by [Cun87, Proposition 3.1], i.e. (a) implies (b).The remaining implication, (b) ⇒ (c), follows from Yoneda’s Lemma. (cid:3) OTGONBAYAR UUYE
As a corollary, we now prove Theorem 0.1. In view of Proposition 1.8, itis enough to prove the following. See also Theorem 3.8.
Theorem 2.6 (Bott Periodicity) . Let u : q C → Σ ⊗ M ∈ m ( q C , Σ ) denotethe Bott element. Then u ⊗ id K : q C ⊗ K → Σ ⊗ M ⊗ K ∼ = Σ ⊗ K (25) is an m -equivalence (equivalently, an S -equivalence).Proof. We have a commutative diagram q ( q C ) q ( u ) / / π q C (cid:15) (cid:15) q (Σ ⊗ M ) π Σ2 ⊗ M (cid:15) (cid:15) q C u / / Σ ⊗ M . (26)The vertical maps are m -equivalences by Proposition 2.5 and the map q ( u ) ⊗ id K is a homotopy equivalence (in particular, an m -equivalence) by KK -theoretic Bott Periodicity. It follows that u ⊗ id K is an m -equivalence. (cid:3) Bott Invertibility
Definition 3.1.
Let u : q C → Σ ⊗ M ∈ m ( q C , Σ ) denote the Bott ele-ment. We say that a separable C ∗ -algebra D is Bott inverting if the element u ⊗ id D ∈ m ( q C ⊗ D, Σ ⊗ D ) (27)is an m -equivalence. Remark . (i) If D is Bott inverting, then so is D ⊗ B for any B .(ii) If D is m -equivalent to B and D is Bott inverting, then so is B .First we show that there are plenty of algebras that are Bott inverting.See Example 3.10 for an example that is not Bott inverting.
Lemma 3.3.
Let D be a separable C ∗ -algebra. Suppose that for some n ≥ ,the inclusion id D ⊗ i n, : D ֒ → D ⊗ M n (28) factors in S through a Bott inverting algebra . Then D is Bott inverting.Proof. Let D f / / B g / / D ⊗ M n (29) The map π q C is in fact an n -equivalence by [Cun87, Theorem 1.6]. NSUSPENDED CONNECTIVE E -THEORY 7 be a factorization of the inclusion id D ⊗ i n, : D ֒ → D ⊗ M n , with B Bottinverting. Then the following diagram is commutative in S : q C ⊗ D id q C ⊗ f / / u ⊗ id D (cid:15) (cid:15) q C ⊗ B id q C ⊗ g / / u ⊗ id B (cid:15) (cid:15) q C ⊗ D ⊗ M nu ⊗ id D ⊗ Mn (cid:15) (cid:15) Σ ⊗ M ⊗ D id Σ2 ⊗ M ⊗ f / / Σ ⊗ M ⊗ B id Σ2 ⊗ M ⊗ g / / Σ ⊗ M ⊗ D ⊗ M n . (30)Since i n, is invertible in m , and u ⊗ id B is invertible by assumption, itfollows that u ⊗ id D is invertible. (cid:3) Definition 3.4.
We say that a C ∗ -algebra D is stable if D ∼ = D ⊗ K .By Bott Periodicity (Theorem 2.6) and Remark 3.2, stable algebras areBott inverting. Lemma 3.5 (Kirchberg) . Let D be a separable C ∗ -algebra. If D contains astable full C ∗ -subalgebra, then map id D ⊗ i , : D ֒ → D ⊗ M (31) factors through a stable algebra.Proof. See the proof of [Tho03, Lemma 4.4.7]. (cid:3)
Combining Lemma 3.3 and Lemma 3.5, we get the following.
Corollary 3.6.
All separable C ∗ -algebras that contain a stable full C ∗ -subalgebra are Bott inverting. In particular, all separable unital properlyinfinite C ∗ -algebras are Bott inverting. (cid:3) Remark . Same methods show that comparison map from algebraic totopological K -theory K alg ∗ ( D ) → K top ∗ ( D ) (32)is an isomorphism if D has a stable full C ∗ -subalgebra (c.f. [SW90]).Now we are ready to state and prove the connective versions of Theo-rem 0.1 and Theorem 0.2, which we recover by considering stable algebras. Theorem 3.8.
Let A be a separable C ∗ -algebra. If D is Bott inverting, thenwe have equivalences qA ⊗ D ∼ = m q C ⊗ A ⊗ D ∼ = m Σ ⊗ A ⊗ D. (33) Proof.
Follows from Proposition 1.8 and Bott invertibility. (cid:3)
Definition 3.9 (A. Thom [Tho03, Theorem 4.2.1]) . Let A and B be sepa-rable C ∗ -algebras. For n ∈ Z , we define bu n ( A, B ) as the colimit bu n ( A, B ) := colim k →∞ m (Σ k ⊗ A, Σ k + n ⊗ B ) (34)along the suspension maps. The connective E -category bu is the categorywith morphisms bu ( A, B ). OTGONBAYAR UUYE
Let X and Y be based, connected, finite CW-complexes. Then from theproof of Theorem [Tho03, Theorem 4.2.1], we see that bu n ( C ( X ) , C ( Y )) ∼ = kk n ( Y, X ) (35)in the notation of [DN90, DM00].
Example 3.10.
Let X be a based, connected, finite CW-complex and let D = C ( X ). Then, for any k ≤
0, we have bu k ( D, C ) ∼ = 0 by [DN90,Corollary 3.4.3].We claim that D is Bott inverting if and only if D is m -contractible.Indeed, first note that, by Proposition 2.5, the mapid Σ ⊗ π C : Σ ⊗ q C → Σ (36)is an m -equivalence, thus π C : q C → C is a bu -equivalence. Now supposethat D is Bott inverting. Then bu k ( D, C ) ∼ = bu k ( q C ⊗ D, C ) ∼ = bu k − ( D, C ) . (37)for any k ∈ Z . Hence the map 0 : D → m -equivalence by[DM00, Theorem 2.4]. The converse is clear.In particular, for any k ≥
0, the algebra Σ k is not Bott inverting.
Theorem 3.11.
Let A and B be a Bott inverting separable C ∗ -algebras. If A is matrix homotopy symmetric, then we have a natural isomorphism m ( A, B ) ∼ = bu ( A, B ) . (38) Proof.
Suppose that A is matrix homotopy symmetric. By Proposition 2.5,we have isomorphisms m ( A, q C ⊗ B ) ∼ = (cid:15) (cid:15) ∼ = / / m ( q C ⊗ A, q C ⊗ B ) ∼ = (cid:15) (cid:15) m ( A, B ) ∼ = / / m ( q C ⊗ A, B ) . (39)and by Bott invertibility, we have m ( q C ⊗ A, q C ⊗ B ) ∼ = m (Σ ⊗ A, Σ ⊗ B ) . (40)Now it is easy to check that the composition m ( A, B ) → m ( q C ⊗ A, q C ⊗ B ) → m (Σ ⊗ A, Σ ⊗ B ) (41)is the double suspension Σ . (cid:3) Corollary 3.12.
Let A and B be a matrix homotopy symmetric, Bott in-verting separable C ∗ -algebras. Then A and B are bu -equivalent if and onlyif they are m -equivalent. (cid:3) NSUSPENDED CONNECTIVE E -THEORY 9 References [CH90] Alain Connes and Nigel Higson,
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Department of Mathematical Sciences, University of Copenhagen, Univer-sitetsparken 5, DK-2100 Copenhagen E, Denmark
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