aa r X i v : . [ m a t h . OA ] M a y Approximate ideal structures and K-theory
Rufus WillettMay 12, 2020
Abstract
We introduce a notion of approximate ideal structure for a C ˚ -algebra, and use it as a tool to study K -theory groups. The notionis motivated by the classical Mayer-Vietoris sequence, by the theoryof nuclear dimension as introduced by Winter and Zacharias, and bythe theory of dynamical complexity introduced by Guentner, Yu, andthe author. A major inspiration for our methods comes from recentwork of Oyono-Oyono and Yu in the setting of controlled K -theoryof filtered C*-algebras; we do not, however, use that language in thispaper.We give two main applications. The first is a vanishing resultfor K -theory that is relevant to the Baum-Connes conjecture. Thesecond is a permanence result for the K¨unneth formula in C ˚ -algebra K -theory: roughly, this says that if A can be decomposed into a pair ofsubalgebras p C, D q such that C , D , and C X D all satisfy the K¨unnethformula, then A itself satisfies the K¨unneth formula. Contents The product map 347 The inverse Bott map 398 Surjectivity of the product map 459 Injectivity of the product map 49A Nuclear dimension 54B Finite dynamical complexity 58
Approximate ideals structures and long exact sequences
Let C and D be C ˚ -subalgebras of a C ˚ -algebra A . There is a naturalsequence of maps K p C X D q ι Ñ K p C q ‘ K p D q σ Ñ K p A q B K p C X D q ι Ñ K p C q ‘ K p D q (1)of K -theory groups where the solid arrows labeled ι and σ are defined re-spectively by ι p κ q : “ p κ, ´ κ q and σ p κ, λ q : “ κ ` λ . The dashed arrow labeled B does not exist in general, but in the very special case that C and D areideals in A such that A “ C ` D , one can canonically fill it in. Indeed, thedashed arrow is then a boundary map in a six-term exact sequence K p C X D q ι / / K p C q ‘ K p D q σ / / K p A q B (cid:15) (cid:15) K p A q B O O K p C q ‘ K p D q σ o o K p C X D q ι o o . This is the C ˚ -algebraic analogue of the classical Mayer-Vietoris sequenceassociated to a cover of a topological space by two open sets.The main technical tools developed in this paper are partial exactnessresults for the sequence in line (1) that hold under less rigid assumptionsthan C and D being ideals. These tools have interesting consequences evenfor many simple C ˚ -algebras, where there are no non-trivial ideals. Looking2t the diagram in line (1) in more detail, K p C X D q ι Ñ K p C q ‘ K p D q loooooooomoooooooon p III q σ Ñ K p A q loomoon p II q B K p C X D q looooomooooon p I q ι Ñ K p C q ‘ K p D q (2)we establish partial exactness results at each of the three places marked (I),(II), and (III), under progressively more stringent assumptions. Exactnessat point (I) is the easiest to prove, and is automatic: if ι p κ q “ κ P K p C X D q , one can always canonically construct a class in K p A q thatis the ‘reason’ for its being zero in some sense.For exactness in the positions marked (II) and (III) in line (2), we needmore assumptions. Here are the technical definitions. Definition 1.1.
Let A be a C ˚ -algebra, and let C be a set of pairs p C, D q of C ˚ -subalgebras of A . Then A admits an approximate ideal structure over C iffor any δ ą F of A there exists a positive contraction h in the multiplier algebra of A and a pair p C, D q P C such that:(i) }r h, a s} ă δ for all a P F ;(ii) d p ha, C q ă δ and d pp ´ h q a, D q ă δ for all a P F ;(iii) d pp ´ h q ha, C X D q ă δ and d pp ´ h q h a, C X D q ă δ for all a P F .The pair t h, ´ h u should be thought of as a ‘partition of unity’ on A ,splitting it into two ‘parts’ C and D that are simpler than the original. Wediscuss examples below, but keep the discussion on an abstract level for now.These conditions allow us to prove a version of exactness at position (II)in line (2): roughly this says that if A admits an approximate ideal structureover C , then for any class r u s in K p A q one can find a pair p C, D q P C andbuild a class Bp u q P K p C X D q such that if Bp u q “
0, then r u s is in the imageof σ .The first of our main results is as follows. Theorem 1.2.
Say that A admits an approximate ideal structure over a set C such that for all p C, D q P C , the C ˚ -algebras C , D , and C X D have trivial K -theory. Then A has trivial K -theory. This result is already quite powerful: for example, it allows one to reprovethe main theorem on the Baum-Connes conjecture of Guentner, Yu, and the3uthor from [16] without the need for the controlled K -theory methods usedthere.In order to get our results on the K¨unneth formula, we need an exactnessproperty at position (III) in line (2); unfortunately, this needs the strongerassumption on A defined below. Definition 1.3.
Let A be a C ˚ -algebra and C a set of pairs p C, D q of C ˚ -subalgebras of A . Then A admits a uniform approximate ideal structure over C if it admits an approximate ideal structure over C , and if in addition thefollowing property holds. For all ǫ ą δ ą C ˚ -algebra B , if c P C b B and d P D b B satisfy } c ´ d } ă δ , then thereexists x P p C X D q b B with } x ´ c } ă ǫ and } x ´ d } ă ǫ .The above definition is satisfied, for example, if all the pairs p C, D q P C are pairs of ideals. However, this is too much to ask if one wants applicationsthat go beyond well-understood cases. There are non-trivial examples, butwe will discuss these until later.Here is our second main theorem. Theorem 1.4.
Let A be a C ˚ -algebra. Assume that A admits a uniformapproximate ideal structure over C , and that for each p C, D q P C , C , D , and C X D satisfy the K¨unneth formula. Then A satisfies the K¨unneth formula. Before moving on to examples, let us digress slightly to give backgroundon the K¨unneth formula for readers unfamiliar with this.
The K¨unneth formula
One of the main results in this paper is about the K¨unneth formula, whichconcerns the external product map ˆ : K ˚ p A b B q Ñ K ˚ p A q b K ˚ p B q in C ˚ -algebra K -theory. This product is as a special case of the very generalKasparov product, but can also be defined in an elementary way: see forexample [19, Section 4.7]. A C ˚ -algebra A is said to satisfy the K¨unnethformula if for any C ˚ -algebra B with free abelian K -groups, the productmap above is an isomorphism.Study of the K¨unneth formula seems to have been initiated by Atiyah[1] in the commutative case, and in general by Schochet [30]. In particular,4hese authors showed (in the relevant contexts) that A satisfies the K¨unnethformula in the above sense if and only if for any B there is a canonical shortexact sequence0 Ñ K ˚ p A q b K ˚ p B q ˆ Ñ K ˚ p A b B q Ñ Tor p K ˚ p A q , K ˚ p B qq Ñ . This short exact sequence is a useful computational tool, so it is desirableto know for which C ˚ -algebras the K¨unneth formula holds. One can see theK¨unneth formula as a sort of ‘dual form’ of the universal coefficient theorem(UCT). Thus another motivation for studying the K¨unneth formula is as itforms a simpler proxy for the UCT.The class of C ˚ -algebras known to satisfy the K¨unneth formula is large.Atiyah [1] essentially showed that commutative C ˚ -algebras satisfy the K¨unnethformula. It follows that any C ˚ -algebra that is KK -equivalent to a com-mutative C ˚ -algebra satisfies the K¨unneth formula. The class of such C ˚ -algebras is exactly the class satisfying the UCT . Hence the UCT is strongerthe K¨unneth formula.The UCT is in fact strictly stronger that the K¨unneth formula: this fol-lows from combining work of Chabert, Echterhoff, and Oyono-Oyono [7],of Lafforgue [21], and of Skandalis [31]. Indeed, it follows from the ‘go-ing down functor’ machinery of [7] that if G is any group that satisfies theBaum-Connes conjecture with coefficients, then C ˚ r p G q satisfies the K¨unnethformula. Thanks to [21], this applies in particular when G is a hyperbolicgroup. On the other hand, results of [31] imply that if G is an infinite,hyperbolic, property (T) group, then C ˚ r p G q does not satisfy the UCT.Other results extending the range of validity of the K¨unneth formulainclude work of B¨onicke and Dell’Aiera [4], which extends the results of [7]from groups to groupoids; and work of Oyono-Oyono and Yu [25] which usesthe methods of controlled K -theory developed by those authors [24], andbased on older ideas of Yu [36]. The work of Oyono-Oyono and Yu was themain technical inspiration for this paper, and we say more on this below.Despite all these positive results, there are known to be C ˚ -algebras thatdo not satisfy the K¨unneth formula. The only way we know to produce For this and the next paragraph, all C ˚ -algebras are separable. This is implicit in the original work of Rosenberg and Schochet [29], and was madeexplicit by Skandalis in [31, Proposition 5.3]. The result as stated here is not exactly in Skandalis’s paper [31], but it follows fromSkandalis’s ideas, plus more recent advances in geometric group theory: see [18, Theorem6.2.1] for a discussion of the version stated. K -exact C ˚ -algebras: see thediscussion in [7, Remark 4.3 (1)]. We do not know of an exact C ˚ -algebrathat does not satisfy the K¨unneth formula. Examples
Our definitions were motivated partly by the theory of nuclear dimension.Indeed, we can weaken Definition 1.1 as follows.
Definition 1.5. A C ˚ -algebra A admits a weak approximate ideal structure over C if the conditions from Definition 1.1 are satisfied, with condition (iii)on intersections omitted.In Appendix A, we show that if A is a (separable) C ˚ -algebra of nucleardimension one, then A admits a weak approximate ideal structure over a classof pairs of subhomogeneous C ˚ -subalgebras with very simple structure. Thisresult is not enough to deduce K -theoretic consequences with our currenttechniques; nonetheless, it provides evidence that our conditions are naturalfrom the point of view of general C ˚ -algebra structure theory.In Appendix B, we discuss examples coming from groupoids. In jointwork with Guentner and Yu [16, Appendix A], we introduced a notion ofa decomposition of an ´etale groupoid. In Appendix B, we show that suchdecompositions naturally give rise to approximate ideal structures of theassociated reduced groupoid C ˚ -algebras, and moreover that we get uniformapproximate ideal structures in this way if the groupoids involved are ample.We use this to show that a large class of reduced groupoid C ˚ -algebras satisfythe K¨unneth formula . Inspiration and motivation
This paper was inspired by the work of Oyono-Oyono and Yu in [25] on theK¨unneth formula in controlled K -theory. It owes a great deal to their work,both conceptually and in some technical details: in particular, the key ideato use a sort of approximate Mayer-Vietoris sequence comes directly from[25], and the difficult proof of Proposition 5.7 is based closely on their work. This result was pointed out to us by Wilhelm Winter. Similar results have been proved recently (and earlier than the current work) byOyono-Oyono using the methods of controlled K -theory. K -theory, only usual K -theory groups. We do not use filtrations on our C ˚ -algebras, and we do not need (nor do we get results on) a ‘controlled’ versionof the K¨unneth formula. It is not clear to us what the difference is betweenthe range of validity of our results and those of [25]; we suspect that there isa large overlap.We were motivated largely by the theory of nuclear dimension [35]: wewanted to narrow the gap between the sort of structural results that one canuse to deduce K -theoretic consequences, and the sort of structural resultsthat are known for C ˚ -algebras of finite nuclear dimension. Outline of the paper
Section 2 introduces a general notion of ‘boundary classes’, and shows thatsuch classes have good properties with respect to the sequence of maps inline (2): roughly, we prove a weak form of exactness at position (II) in line(2). The discussion in Section 2 does not give a construction of boundaryclasses: this is done in Section 3 using approximate ideal structures. We thenprove Theorem 1.2, our first main goal of the paper.In Section 4, we prove exactness at position (I) in line (2); this is simplerthan exactness at position (II), but is postponed until later as it is not neededfor the proof of Theorem 1.2. We also collect together some other technicalresults on the boundary map that are needed later. Exactness at position(III) in line (2) is handled in Section 5: this is the most difficult of ourexactness properties, both to prove and to use.Section 6 recalls some facts about the product in K -theory, and provesthat the products maps interact well with our boundary classes. Section7 recalls material about the inverse Bott map that we need for the techni-cal proofs. We prove Theorem 1.4 in Sections 8 and 9, which handle thesurjectivity and injectivity halves respectively.Finally, there are two appendices that discuss examples. The first of these,Appendix A shows that C ˚ -algebras of nuclear dimension one have weak ap-proximate ideal structures. Appendix B gives examples of (uniformly) ap-proximate ideal structures coming from groupoid theory, and briefly discussesconsequences for the Baum-Connes conjecture and K¨unneth formula.7 otation and conventions Throughout, if A is a C ˚ -algebra (or more generally, Banach algebra), then r A denotes A itself if A is unital, and the unitization of A if it is not unital.If X is a subspace of a C ˚ -algebra A , then r X is the subspace of r A spannedby X and the unit. There is an ambiguity here about what happens when C is a C ˚ -subalgebra of A , and C has its own unit which is not the unit of A : we adopt the convention that in this case, r C means the C ˚ -subalgebraof A generated by C and the unit of r A . This convention will always, andonly, apply to C ˚ -subalgebras called C , D and C X D (plus suspensions andmatrix algebras of these), so we hope it causes no confusion.We use 1 n and 0 n to denote the unit and zero element of M n p r A q when itseems helpful to avoid ambiguity, but drop the subscripts whenever thingsseem more readable without. We use the usual ‘top-left corner’ identificationof M n p A q with M m p A q for n ď m , usually without comment. We also usethe usual ‘block sum’ convention that if a P M n p A q , and b P M m p A q , then a ‘ b : “ ˆ a b ˙ P M n ` m p A q . The symbol b as applied to C ˚ -algebras always denotes the spatial tensorproduct. If X is a closed subspace of a C ˚ -algebra A , and B is a C ˚ -algebra,then X b B denotes the closure of the algebraic tensor product X d B inside A b B . For a C ˚ -algebra A , SA : “ C p R qb A is its suspension, S A : “ S p SA q its double suspension, and for a closed subspace X of A , SX : “ C p R q b X .We always denote the compact operators on ℓ p N q by K , so in particular A b K is the stabilisation of K .It is typical in C ˚ -algebra K -theory to the K and K groups as gener-ated by certain equivalence classes of projections and unitaries respectively.However, we will need to work more generally with equivalence classes ofidempotents and projections. This is because one typically has more con-crete formulas available in the latter context. Readers unfamiliar with thisapproach can find the necessary background in [2, Chapters II, III and IV],for example.We have attempted to keep the paper self-contained and elementary, notassuming much any background beyond basic C ˚ -algebra K -theory . Al-though using only elementary language is often desirable in its own right, we Modulo the comments above about invertibles and idempotents.
Acknowledgments
This work was started during a sabbatical visit to the University of M¨unster.I would like to thank the members of the mathematics department there fortheir warm hospitality.I would like to particularly thank Cl´ement Dell’Aiera, Dominik Enders,Sabrina Gemsa, Herv´e Oyono-Oyono, Ian Putnam, Aaron Tikuisis, StuartWhite, Wilhelm Winter, and Guoliang Yu for numerous enlightening conver-sations relevant to the topics of this paper.The support of the US NSF through grants DMS 1564281 and DMS1901522 is gratefully acknowledged.
In this section, we work in the context of general Banach algebras. This isnot needed for our applications, but we hope it clarifies what goes into theresults; it also makes no difference to the proofs.
Definition 2.1.
Let A be a Banach algebra, and let C and D be Banachsubalgebras. We define maps on K -theory by ι : K ˚ p C X D q Ñ K ˚ p C q ‘ K ˚ p D q , κ ÞÑ p κ, ´ κ q . and σ : K ˚ p C q ‘ K ˚ p D q Ñ K ˚ p A q , p κ, λ q ÞÑ κ ` λ. With notation as above, assume for a moment that C and D are (closed,two-sided) ideals in A such that A “ C ` D . Then there is a Mayer-Vietorisboundary map B : K p A q Ñ K p C X D q that fits into a long exact sequence ¨ ¨ ¨ ι Ñ K p C q ‘ K p D q σ Ñ K p A q B Ñ K p C X D q ι Ñ K p C q ‘ K p D q σ Ñ ¨ ¨ ¨ . Our aim in this section is to get analogous results for more general Banachsubalgebras C and D : for at least some classes r u s P K p A q , we want to9non-canonically) construct a ‘boundary class’ Bp u q P K p C X D q that hassimilar exactness properties with respect to ι and σ .The next two lemmas concern ‘almost idempotents’. We would guessresults like these are well-known to experts, but could not find what weneeded in the literature. Lemma 2.2.
For any ǫ, c ą there exists δ P p , { q with the followingproperty. Let A be a Banach algebra and e P A satisfy } e ´ e } ă δ and } e } ď c . Let χ be the characteristic function of t z P C | Re p z q ą { u .Then χ p e q (defined via the holomorphic functional calculus) is a well-definedidempotent, and satisfies } χ p e q ´ e } ă ǫ .Proof. First note that if δ P p , { q and if z P C satisfies | z ´ z | ă δ ,then | z || z ´ | ă δ , and so either | z | ă ? δ , or | z ´ | ă ? δ . Hence by thepolynomial spectral mapping theorem, if } e ´ e } ă δ , then the spectrum of e is contained in the union of the balls of radius ? δ and centered at 0 and 1respectively. As ? δ ă {
2, it follows that χ is holomorphic on the spectrumof e . Hence χ p e q makes sense under the assumptions, and is an idempotentby the functional calculus.Let now r “ ? δ ă {
2, and let γ and γ be positively oriented circlescentered on 0 and 1 respectively, and of radius r . Then by the above remarks,if } e ´ e } ă δ we have that γ Y γ is a positively oriented contour on which χ is holomorphic, and that has winding number one around each point of thespectrum of e . Hence by definition of the holomorphic functional calculus χ p e q ´ e “ πi ż γ Y γ p χ p z q ´ z qp z ´ e q ´ dz. Estimating the norm of this using that | χ p z q ´ z | “ r for z P γ Y γ gives } χ p e q ´ e } ď π ż γ Y γ r }p z ´ e q ´ }| dz | . (3)Let us estimate the term }p z ´ e q ´ } for z P γ Y γ . Set w “ ´ z . Thenwe have that w ´ e is also invertible, and }p z ´ e q ´ } “ }p w ´ e qp w ´ e q ´ p z ´ e q ´ }ď p c ` | w |q}|pp z ´ z q ´ p e ´ e qq ´ }ď p c ` q}|pp z ´ z q ´ p e ´ e qq ´ } . (4)10ow, we have that for z P γ Y γ , | z ´ z | “ | z || z ´ | ě r “ ? δ ą δ ą } e ´ e } . Hence using the Neumann series inverse formula pp z ´ z q ´ p e ´ e qq ´ “ z ´ z ´ ´ e ´ ez ´ z ¯ ´ “ z ´ z ÿ n “ ´ e ´ ez ´ z ¯ n we get the estimate }pp z ´ z q ´ p e ´ e qq ´ } ď | z ´ z | ´ } e ´ e } ď r ´ δ “ ? δ ´ δ . Combining this with line (4), we see that for z P γ Y γ , }p z ´ e q ´ } ď c ` ? δ ´ δ . To complete the proof, substituting the above estiumate into line (3) givesthat } χ p e q ´ e } ď π ż γ Y γ r p c ` q? δ ´ δ | dz | “ π ` Length p γ q ` Length p γ q ˘ r p c ` q? δ ´ δ . Substituting in Length p γ q “ Length p γ q “ πr and r “ ? δ we get } χ p e q ´ e } ď ? δ p c ` q ´ ? δ , which is enough to complete the proof. Definition 2.3.
Let A be a Banach algebra, let X be a subset of A , let a P A , and let ǫ ą
0. The element a is ǫ -in X , denoted a P ǫ X , if there exists x P X with } a ´ x } ď ǫ . Lemma 2.4.
Let A be a Banach algebra and B a Banach subalgebra. Thenfor all c ą and all ǫ P p , c ` q there exists δ ą with the following property.(i) Say n ě and say e P M n p A q is an idempotent which is δ -in M n p B q and such that } e } ď c . Then there is an idempotent f P M n p B q with } e ´ f } ă ǫ . Moreover, the class r f s P K p B q does not depend on thechoice of ǫ , δ , or f . ii) Assume moreover that A is unital, and that B contains the unit. Say u P M n p A q is an invertible which is δ -in M n p B q and such that } u ´ } ď c .Then there exists an invertible v P M n p B q with } u ´ v } ă ǫ , and theclass r v s P K p B q does not depend on the choice of ǫ , δ , or v .Proof. Let δ ą
0, to be chosen depending on c and ǫ in a moment, andassume that e is δ -in M n p B q so there is b P M n p B q with } b ´ e } ă δ . Then } b ´ b } ď } e }} b ´ e } ` } b }} b ´ e } ` } b ´ e } ď p c ` δ ` q δ. Let χ be the characteristic function of the half-plane t z P C | Re p z q ą { u .Then for suitably small δ (depending only on c and ǫ ), we may apply Lemma2.2 to get that } b ´ χ p b q} ă ǫ {
2. Setting f “ χ p b q and assuming also that δ ă ǫ { } e ´ f } ď } e ´ b } ` } b ´ f } ă ǫ as desired.To see that r f s P K p B q does not depend on the choice of f , let f P M n p B q be another idempotent with } e ´ f } ă ǫ . Then } f ´ f } ă ǫ ă {p c ` q . As } f } ď c `
1, we see that } f ´ f } ă c ` ď } f ´ } , whence [2, Proposition 4.3.2] implies that f and f are similar, and so inparticular define the same K-theory class.For part (ii), let ǫ “ c , let ǫ P p , ǫ s , and let δ “ ǫ . Choose any v P M n p B q with } u ´ v } ă δ . Then } ´ u ´ v } “ } u ´ p u ´ v q} ď } u ´ }} u ´ v } ă cδ “ { . Hence u ´ v is invertible, and so v is invertible too. Moreover, estimating thenorm of p u ´ v q ´ using the series expression p u ´ v q ´ “ ř n “ p ´ u ´ v q n gives that } v ´ u } ď
2, whence } v ´ } “ } v ´ uu ´ } ď c . On the other hand,if v also satisfies } u ´ v } ă ǫ , then } v ´ v } ă ǫ , and so } ´ v ´ v } ď } v ´ }} v ´ v } ă cǫ “ . Hence v ´ v “ e z for some z P M n p B q (see for example [3, II.1.5.3]), andso t ve tz u t Pr , s is a homotopy between v and v passing through invertibles in M n p B q , giving that r v s “ r v s in K p B q .12 efinition 2.5. Let c ą
0, let ǫ P p , c ` q , and let δ ą A be a Banach algebra, and B be a Banach subalgebra of A .1. Say e P M n p A q is an idempotent that is δ -in M n p B q . Then we write t e u B P K p B q for the class of any idempotent f P M n p B q with } e ´ f } ă ǫ .2. Say u P M n p r A q is an invertible that is δ -in M n p r B q . Then we write t u u B P K p B q for the class of any invertible v P M n p r B q with } u ´ v } ă ǫ .The next definition is the key technical point that we need to constructour boundary classes. Definition 2.6.
Let c ą
0, let ǫ P p , c ` q , and let δ ą A be a Banach algebra, let C and D be Banach subalgebras of A ,let u P M n p r A q be an invertible element for some n . An element v P M n p r A q is a p δ, c, C, D q -lift of u if it satisfies the following conditions:(i) } v } ď c and } v ´ } ď c ;(ii) v P δ M n p r D q ;(iii) v ˆ u ´ u ˙ P δ M n p r C q ;(iv) v ˆ ˙ v ´ P δ M n p Č C X D q ;(v) with notation as in Definition 2.5, the K -theory class ! v ˆ ˙ v ´ ) Č C X D ´ „ P K p Č C X D q is actually in the subgroup K p C X D q .We may now use such lifts to construct ‘boundary classes’. Proposition 2.7.
Let c ą , let ǫ P p , c ` q . Then there is δ ą satisfyingthe conclusion of Lemma 2.4, and with the following properties. Let A bea Banach algebra, and let u P M n p r A q be an invertible with } u } ď c and u ´ } ď c . Assume there exist Banach subalgebras C and D of A and a p δ, c, C, D q -lift v of u . Then the K -theory class B v u : “ ! v ˆ ˙ v ´ ) Č C X D ´ „ P K p C X D q has the following properties.(i) If ι is as in Definition 2.1, then ι pB v u q “ in K p C q ‘ K p D q .(ii) If B v u “ , then there is l P N and an invertible x P ǫ M n ` l p r D q such that p u ‘ l q x ´ P ǫ M n p r C q . In particular, if σ is as in Definition 2.1, then σ ptp u ‘ l q x ´ u C , t x u D q “ r u s in K p A q .Proof. Let us first consider ι pB v u q . Note first that as v is δ -in M n p r D q , thereis w P M n p r D q such that } w ´ v } ă δ . In particular, w is invertible for δ suitably small. It follows by definition of the left hand side that ! v ˆ ˙ v ´ ) r D “ ” w ˆ ˙ w ´ ı in K p r D q for all suitably small δ . Hence as elements of K p r D q , ! v ˆ ˙ v ´ ) r D ´ „ “ ” w ˆ ˙ w ´ ı ´ „ . However, as w is in M n p r D q , ” w ˆ ˙ w ´ ı “ „ in K p r D q , so theabove is the zero class in K p r D q , hence also in K p D q .On the other hand, our assumption that v ˆ u ´ u ˙ is δ -in M n p r C q im-plies similarly that for all δ suitably small, we have B v u “ ! v ˆ ˙ v ´ ) r C ´ „ “ ! ˆ u u ´ ˙ v ´ v ˆ ˙ v ´ v ˆ u ´ u ˙ ) r C ´ „ , which is zero as a class in K p C q . We have shown that the image of B v u inboth K p C q and K p D q is zero, whence ι pB v u q “ δ n ’, it is implicitthat this is a positive number, depending only on c and δ , and that tends tozero when δ tends to zero as long as c stays in a bounded set.Now let us assume that B v u “
0. This implies that there exists l P N andan invertible element w of M n ` l p Č C X D q such that ››› w ´ v ˆ ˙ v ´ ‘ l ¯ w ´ ´ ˆ ˙ ‘ l ››› ă δ for some δ ą
0. Write v “ ˆ v v v v ˙ , and let v : “ ¨˚˚˝ v v
00 1 l v v
00 0 0 1 l ˛‹‹‚ P δ M n ` l ` n ` l p r D q (writing the matrix size as n ` l ` n ` l is meant to help understand the sizeof the various blocks) and if w “ ¨˝ w w w w w w w w w ˛‚ P M n ` n ` l p Č C X D q let w : “ ¨˚˚˝ w w w l w w w w w w ˛‹‹‚ P M n ` l ` n ` l p Č C X D q . Then in M p n ` l q`p n ` l q p r C q we have ››› w v ˆ ˙ v ´ w ´ ˆ ˙ ››› ă δ for some δ . This implies that for δ suitably small there exist invertible x, y P M n ` l p r D q and δ such that ››› w v ´ ˆ x y ˙ ››› ă δ . v ˆ u ´ u ˙ P δ M n p r C q . Write u : “ u ‘ l P M n ` l p r A q . Then v ˆ u ´ u ˙ P δ M p n ` l q`p n ` l q p r C q . and thus as w is in M p n ` l q p r C q , we have that w v ˆ u ´ u ˙ P δ M p n ` l q p r C q for some δ . Hence in particular, xu ´ is invertible for δ suitably small,is δ -in M n ` l p r C q , and has norm bounded above by some absolute constantdepending only on c . We now have that for δ suitably small (depending onlyon ǫ and c ), u x ´ is ǫ -in M n ` l p r C q and that x is ǫ -in M n ` l p r D q , completingthe proof. Definition 2.8.
With notation as in Proposition 2.7, we call B v p u q P K p C X D q the boundary class associated to the data p u, v, C, D q . Our main goal in this section is to show that approximate ideal structures inDefinition 1.1 can be used to build lifts as in Definition 2.6, and thus allowus to build boundary classes.It would be possible to get analogous results for general Banach algebras,but it would make the statements and proofs more technical. As our ap-plications are all to the K -theory of C ˚ -algebras, at this stage we thereforespecialise to that case.First, it will be convenient to give a technical variation of Definition 1.1. Definition 3.1.
Let A be a C ˚ -algebra, let X Ď A be a subspace, and let δ ą
0. Then a δ -ideal structure for X is a triple p h, C, D q h in the multiplier algebra of A , and C ˚ -subalgebras C and D of A such that(i) }r h, x s} ď δ } x } for all x P X ;(ii) hx and p ´ h q x are δ } x } -in C and D respectively for all x P X ;(iii) h p ´ h q x and h p ´ h q x are δ } x } -in C X D for all x P X .We say that A has an approximate ideal structure over a class C of pairs of C ˚ -subalgebras if for any δ ą X of A thereexists a δ -ideal structure p h, C, D q of X with p C, D q in C . Remark . The conditions on multiplying into the intersection in (iii) fromDefinition 3.1 might look odd for two reasons. First, they are asymmetric in h and 1 ´ h : this is a red herring, however, as it would be essentially the sameto require that h p ´ h q x and h p ´ h q x are both δ } x } -in C X D . Second,there are two conditions for C X D , and only one each for C and D . Thisseems ultimately attributable to the fact that one needs two polynomials togenerate C p , q as a C ˚ -algebra, but only one each for C p , s and C r , q .We need to show that admitting an approximate ideal structure boot-straps up to a stronger version of itself (following a suggestion of AaronTikuisis and Wilhelm Winter). Lemma 3.3.
Say A is a C ˚ -algebra, X is a finite-dimensional subspaceof A , and N ě . Then there exists a finite-dimensional subspace X of A containing X , such that for any δ ą there exists δ ą such thatif p h, C, D q is a δ -ideal structure for X , then p h, C, D q also satisfies thefollowing properties:(i) }r h, x s} ď δ } x } for all x P X ;(ii) for all n P t , ..., N u , h n x (respectively, h n p ´ h q x , and h n p ´ h q x ) is δ } x } -in C (respectively D , and C X D ) for all x P X .Proof. Take a basis of X consisting of contractions, and write each of theseas a sum of four positive contractions. Let X be the space of spanned byall these positive contractions, say t a , ..., a n u . Ley X be spanned by all m th roots of all of a , ..., a n for m P t , ..., N ` u . Clearly if δ ď δ , then as Y contains X , we have the almost commutation property in the statement.17et us now look at h n x for x P X . It suffices to look at h n a for some a Pt a , ..., a n u . Then using the almost commutation property, we have that h n a is close to p ha { n q n , so for δ suitably small we get what we want. Similarly,if a P t a , ..., a n u , if we write g “ h ´
1, then h n p ´ h q “ p ` g q n p´ g q a “ ´ n ÿ k “ ˆ nk ˙ g k ` a, and again using the almost commutation property, this is close to n ÿ k “ ˆ nk ˙ p ga {p k ` q q k ` , so we get the right property for δ suitably small. The corresponding propertyfor the intersection is similar, once we realise that for all n ě h n p ´ h q can be written as a polynomial in h p ´ h q and h p ´ h q (proof by inductionon n , for example): we leave the details of this to the reader.The next lemma discusses how approximate ideal structures behave undertensor products. If X is a subspace of a C ˚ -algebra A , recall that we write X b B for the norm closure of the subspace of A b B generated by elementarytensors x b b with x P X and b P B . Lemma 3.4.
Say A is a C ˚ -algebra, and X is a finite-dimensional subspaceof A . Then there exists a constant M X ą depending only on X such thatif p h, C, D q is a δ -ideal structure for X , and if B is any C ˚ -algebra, then p h b , C b B, D b B q is an M X δ -ideal structure for X b B .Proof. Let x , ..., x n be a basis for X consisting of unit vectors, and let φ , ..., φ n P A ˚ be linear functionals dual to this basis, so φ i p x j q “ δ ij (here δ ij is the Kronecker δ function). Let M “ max ni “ } φ i } . We claim that M X : “ nM has the property required by the lemma. Note first that any a P X b B can be written a “ n ÿ i “ x i b b i for some unique b , ..., b n P B , and that we have for each i } b i } “ }p φ i b id qp a q} ď } φ i }} a } ď M } a } .
18o see property (i), note that for any a “ ř ni “ x i b b i P X b B we have }r h b , a s} ď n ÿ i “ }r h, x i s b b i } ď n ÿ i “ δ } x i }} b i } ď δnM } a } . To see properties (ii) and (iii), let us look at ha for some a P X b B ; the casesof p ´ h q a , h p ´ h q a , and h p ´ h q a are similar. For each each i P t , ..., n u choose c i P C with } hx i ´ c i } ă δ . Then if a “ ř ni “ x i b b i P X b B is asabove and if c “ ř ni “ c i b b i P C b B we have }p h b q a ´ c } ď n ÿ i “ }p hx i ´ c i q b b i } ď n ÿ i “ δ } b i } ď δnM } a } , which completes the proof. Corollary 3.5.
Say A and B are C ˚ -algebras and X is a finite-dimensionalsubspace of A b B . Then for any δ ą there exists a finite-dimensionalsubspace Y of A and δ ą such that if p h, C, D q is a δ -ideal structure for Y , then p h b B , C b B, D b B q is a δ -ideal structure for X .Proof. As the unit sphere of X is compact, there is a finite dimensionalsubspace Y of A such that for any x in the unit sphere of X there exists y inthe unit sphere of Y b B such that } y ´ x } ă δ {
2. Let M Y be as in Lemma3.4, and let δ “ δ {p M Y q . Lemma 3.4 implies that if p h, C, D q is a δ -idealstructure for Y then p h b , C b B, D b B q is a δ -ideal structure for X .For the remainder of this section, we will apply Lemma 3.4 to tensorproducts M n p A q “ A b M n p C q without further comment. We will also abusenotation, writing things like ‘ hu ’ for an element u P M n p A q , when we reallymean ‘ p h b n q u ’.The next proposition is the key technical result of this section. It saysthat we can use approximate ideal structures to build boundary classes as inDefinition 2.8. For the statement, recall the notion of a p ǫ, c, C, D q -lift fromDefinition 2.6 above. Proposition 3.6.
Let A be a C ˚ -algebra and let κ P K p A q be a K -class.Then there exist n and an invertible element u P M n p r A q , c ą , and a finite-dimensional subspace X of A such that for any ǫ ą there exists δ ą suchthat the following hold. i) The class r u s equals κ .(ii) If p h, C, D q is a δ -ideal structure of X , and if a “ h ` p ´ h q u and b “ h ` u ´ p ´ h q then v : “ ˆ a ˙ ˆ ´ b ˙ ˆ a ˙ ˆ ´
11 0 ˙ is an p ǫ, c, C, D q -lift for u . First we have an ancillary lemma.
Lemma 3.7.
Let A be a C ˚ -algebra and let u be an invertible element of r A such that u “ ` y and u ´ “ ` z with y, z elements of A with normsbounded by some c ą . Let δ ą and let h be a positive contraction in M p A q such that }r h, x s} ď δ } x } for all x P t y, z u . Define a : “ h ` p ´ h q u and b : “ h ` u ´ p ´ h q . Then ba ´ and ab ´ are both within p c ` c q δ of p y ` z q h p ´ h q .Proof. Using that y and z commute, we have that r a, b s “ p ´ h q yz p ´ h q ´ z p ´ h q y “ rp ´ h q , z s y p ´ h q ` z p ´ h qr y, p ´ h qs“ r z, h s y p ´ h q ` z p ´ h qr h, y s , whence }r a, b s} ď c δ . Hence it suffices to show that ab ´ cδ of h p ´ h qp y ` z q . Using that yz “ ´ y ´ z , we see that ab ´ “ p ´ h q yh ` hz p ´ h q and using that }r y, h s} ď δ } y } and }r z, h s} ď δ } z } , we are done. Proof of Proposition 3.6.
Let u P M n p r A q be any invertible element such that r u s “ κ . Using that GL n p C q is connected, up to a homotopy we may assumethat u and u ´ are of the form 1 ` y and 1 ` z respectively with y, z P M n p A q .Let X be the subspace of A spanned by all matrix entries of all monomialsof degree between one and three with entries from t y, z u . Let X be as inLemma 3.3 for this X and N “
4. Let then ǫ ą δ ą p h, C, D q be an δ -idealstructure for X .Throughout the proof, anything called ‘ δ n ’ is a constant depending on X , δ and max t} y } , } z }u , and with the property that δ n tends to zero as δ tendsto zero (assuming the other inputs are held constant). Note that Lemma 3.4implies that there is δ such that p h, M n p C q , M n p D qq is a δ -ideal structureof M n p A q for all n . We check the properties from Definition 2.6. Property(i) is clear from the formula for v (which implies a similar formula for v ´ ).For property (ii), one computes v “ ˆ a p ´ ba q ab ´ ´ ba b ˙ “ ˆ a b ˙ ` ˆ a p ´ ba q ab ´ ´ ba ˙ . (5)As a “ ` p ´ h q y and b “ ` z p ´ h q , we have that a and b are both δ -in M n p r D q for some δ . Hence also ˆ a b ˙ is δ -in M n p r D q . On the otherhand, Lemmas 3.7 and 3.3 and the choice of X imply that 1 ´ ba and 1 ´ ab are δ -in M n p r D q for some δ . It follows from this and that a is δ -in M n p r D q that ˆ a p ´ ba q ab ´ ´ ba ˙ is δ -in M n p r D q for some δ .For part (iii), we compute v ˆ u ´ u ´ ˙ “ ˆ au ´ bu ˙ ` ˆ a p ´ ba q u ´ p ab ´ q u p ´ ba q u ´ ˙ . (6)We have that au ´ “ ` hz and that } bu ´ p ` yh q} ă δ for some δ . Hencethe first term in line (6) is δ -in M n p r C q for some δ . For the second term,using Lemma 3.7 we have that up to some δ , p ´ ba q u ´ and p ab ´ q u equal p y ` z q h p ´ h qp ` z q and p y ` z q h p ´ h qp ` y q . On the other hand } a p ´ ba q u ´ ´ p ` hz qp y ` z q h p ´ h q} ă δ for some δ . The claim follows from all of this and the choice of X .For parts (iv) and (v), note that v ´ “ ˆ ´
11 0 ˙ ˆ ´ a ˙ ˆ b ˙ ˆ ´ a ˙ “ ˆ b ´ baab ´ a p ´ ba q ˙ “ ˆ b a ˙ ` ˆ ´ baab ´ a p ´ ba q ˙ . v ˆ ˙ v ´ ´ ˆ ˙ equals ˆ ab ´ ˙ ` ˆ a p ´ ba q b p ´ ba q b ˙ ` ˆ a p ´ ba q ˙ ` ˆ a p ´ ba q p ´ ba q ˙ . (7)Now, using Lemma 3.7 and the fact that h almost commutes with y and z , every term appearing is within some δ of something of the form p ´ h q hp C p h q q C p y, z q , where p C is a polynomial of degree at most 3 in h (possiblywith a constant term), q C is a noncommutative polynomial of degree atmost 3 with no constant term, and moreover the coefficients in p C and q D are universally bounded. Hence by choice of X , all the terms are δ -in M n p C X D q , for some δ . This completes the proof.We are now ready for the proof of Theorem 1.2 from the introduction. Theorem 3.8.
Say that A admits an approximate ideal structure over a set C such that for all p C, D q P C , the C ˚ -algebras C , D , and C X D have trivial K -theory. Then A has trivial K -theory.Proof. It suffices to show that K p A q “ K p SA q “
0. For K p A q , let α P K p A q be an arbitrary class. Then using Proposition 3.6 we may build aboundary class B v p u q P K p C X D q . As K p C X D q “
0, this class B v p u q is zero.Hence by Proposition 2.7 it is in the image of σ : K p C q ‘ K p D q Ñ K p A q .However, K p C q “ K p D q “ K p SA q is almost the same. Indeed, Corollary 3.5 impliesthat SA admits an approximate ideal structure over the set tp SC, SD q |p
C, D q P C u , and we have that SC , SD , and SC X SD “ S p C X D q all havetrivial K -theory.We remark that Theorem 1.2 can be used to simplify the proof of the maintheorem of [16], in particular obviating the need for filtrations and controlled K -theory in the proof, and replacing the material of [16, Section 7] entirely. In this section we collect together some technical results on boundary classesthat are needed for the proof of Theorem 1.4 on the K¨unneth formula. We22tate results for Banach algebras when it makes no difference to the proof,and C ˚ -algebras when the proof is simpler in that case.The first result corresponds to exactness at position (I) in line (2) fromthe introduction. For the statement, recall the notion of a p δ, c, C, D q -liftfrom Definition 2.6, and the map ι : K p C X D q Ñ K p C q ‘ K p D q fromDefinition 2.1. Proposition 4.1.
Let A be a Banach algebra and let C and D be Banachsubalgebras of A . Assume that p, q P M n p Č C X D q are idempotents such that r p s ´ r q s P K p C X D q , and so that ι pr p s ´ r q sq “ .Then there exist k P N , an invertible element u of M n ` k p r A q , an invertibleelement v of M p n ` k q p r A q , and c ą such that for any δ ą , v and v ´ are p δ, c, C, D q -lifts of u and u ´ respectively, and such that B v u “ r p s ´ r q s and B v ´ p u ´ q “ r q s ´ r p s .Proof. As ι pr p s ´ r q sq “
0, there exist natural numbers l ď k and invertibleelements u C P M n ` k p r C q , u D P M n ` k p r D q such that u C p p ‘ l q u ´ C “ q ‘ l “ u D p p ‘ l q u ´ D . Define u : “ p ´ p ‘ l q u ´ C ` p p ‘ l q u ´ D P M n ` k p r A q . Direct checks that we leave to the reader show that u is invertible with inverse u ´ “ u C p ´ p ‘ l q ` u D p p ‘ l q . Define now v : “ ˆ p p ‘ l q u ´ D p ‘ l ´ ´ q ‘ l u D p p ‘ l q ˙ P M p n ` k q p r D q . Note that v is invertible: indeed, direct computations show that v ´ : “ ˆ u D p p ‘ l q ´ q ‘ l ´ p n ‘ l p p ‘ l q u ´ D ˙ . We also compute that v ˆ u ´ u ˙ “ ˆ p ‘ l p ´ p ‘ l q u ´ C u C p ´ p ‘ l q q ‘ l ˙ , which is an element of M p n ` k q p r C q , so at this point we have properties (i),(ii), and (iii) from Definition 2.6. 23o complete the proof, we compute using the formulas above for v and v ´ that v ˆ ˙ v ´ “ ˆ p ‘ l
00 1 ´ q ‘ l ˙ , which is in M p n ` k q p Č C X D q . Moreover, as a class in K p Č C X D q , ” v ˆ ˙ v ´ ı ´ „ “ r p s ´ r q s , so in particular this class is in K p C X D q , completing the proof that v satisfiesthe conditions from Definition 2.6, and that B v p u q “ r p s ´ r q s .The computations with v ´ and u ´ replacing v and u are similar: weleave them to the reader.The proof of the next lemma consists entirely of direct checks; we leavethese to the reader. Lemma 4.2.
Let A be a Banach algebra, let c ą , and let ǫ P p , c ` q . Let δ ą satisfy the conclusion of Proposition 2.7. Assume that for i P t , ..., m u ,there is an invertible element u i P M n i p r A q such that } u i } ď c and } u ´ i } ď c ,and let C and D be Banach subalgebras of A such that for each i there is a p δ, c, C, D q -lift v i of u i . Let s P M p n `¨¨¨` n m q be the self-inverse permutationmatrix defined by the following diagram in the sizes of the matrix blocks n (cid:15) (cid:15) n * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ n } } ④④④④④④④④ n ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ¨ ¨ ¨ ¨ ¨ ¨ n m s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ n m (cid:15) (cid:15) n O O n = = ④④④④④④④④ ¨ ¨ ¨ n m ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ n j j ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ n h h ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ¨ ¨ ¨ n m O O and define v ‘ ¨ ¨ ¨ ‘ v m : “ s p v ‘ ¨ ¨ ¨ ‘ v m q s Then v : “ v ‘ ¨ ¨ ¨ ‘ v m is a p δ, c, C, D q -lift of u : “ u ‘ ¨ ¨ ¨ ‘ u m , and B v u “ n ÿ i “ B v i p u i q in K p C X D q . We conclude this section with a technical result on inverses that we willneed later. 24 emma 4.3.
Assume that the assumptions of Proposition 3.6 are satisfied.Then on shrinking δ , we may assume that v ´ is also an p ǫ, c, C, D q -lift of u ´ , and moreover that B v p u q “ ´B v ´ p u ´ q as elements of K p C X D q .Proof. Checking that v ´ “ ˆ ´
11 0 ˙ ˆ ´ a ˙ ˆ b ˙ ˆ ´ a ˙ satisfies the properties from Definition 2.6 with respect to u ´ is essentiallythe same as checking the corresponding properties for v and u in the proofof Proposition 3.6. We leave the details to the reader.It remains to establish the formula B v p u q “ ´B v ´ p u ´ q . For t P r , s ,define v t : “ ˆ ta ˙ ˆ ´ tb ˙ ˆ ta ˙ ˆ ´
11 0 ˙ . Analogous computations to those we used to establish to property (iii) in theproof of Proposition 3.6 show that v ´ t v ˆ ˙ v ´ v t is in M n p Č C X D q upto an error we can make as small as we like depending on δ (with c and X fixed), and that the difference v ´ t v ˆ ˙ v ´ v t ´ v ´ t ˆ ˙ v t is in M n p C X D q , again up to an error that we can make as small as we likeby making δ small (and keeping c and X fixed). Hence for all t P r , s weget that the classes ! v ´ t v ˆ ˙ v ´ v t ) Č C X D ´ ! v ´ t ˆ ˙ v t ) Č C X D of K p C X D q are well-defined. They are moreover all the same, as theelements defining them are homotopic. However, the above equals δ v p u q when t “
0, and equals ´ δ v ´ p u ´ q when t “
1, so we are done.25
Approximate ideal structures and the sum-mation map
In this section, we prove a technical result, based very closely on [25, Lemma2.9], and corresponding to exactness at position (III) in line (2) from theintroduction.The precise statement is a little involved, but roughly it says that givena finite-dimensional subspace X of A there is δ ą p h, C, D q isa δ -ideal structure for X as in Definition 3.1, then the maps σ and ι fromDefinition 2.1 have the following exactness property: if p κ, λ q P K p C q ‘ K p D q is such that σ p κ, λ q “ and the subspace X contains a ‘reason’ forthis element being zero , then p κ, λ q is in the image of ι .This result is weak: it seems the quantifiers are in the wrong order for itto be useful, meaning that one would like to be able to choose X based on C and D , but the statement of the result is the other way around. Nonetheless,the result is useful, and plays a crucial role in the proof of the injectivity halfof theorem 1.4.For the proof of the result, we need a condition that is closely relatedto the so-called ‘CIA property’ as used in the definition of ‘nuclear Mayer-Vietoris pairs’ in [25, Definition 4.8]. For the statement, let us say thata function f : p ,
8q Ñ p , is a decay function if f p t q Ñ t Ñ ?? from theintroduction. Definition 5.1.
Let p C, D q be a pair of C ˚ -subalgebras of a C ˚ -algebra A ,and let f be a decay function. Then p C, D q is f -uniform if for all C ˚ -algebras B and δ ą
0, if c P C b B and d P D b B satisfy } c ´ d } ď δ , then thereexists x P p C X D q b B with } x ´ c } ď f p δ q and } x ´ d } ă f p δ q .Let A be a C ˚ -algebra and C a set of pairs p C, D q of C ˚ -subalgebras of A . Then A admits a uniform approximate ideal structure over C if it admitsan approximate ideal structure over C , and if in addition there is a decayfunction f such that all pairs in C are f -uniform.The following example and non-example might help illuminate the defi-nition. We give some more interesting examples in Appendix B. Example 5.2. If p C, D q is a pair of C ˚ - ideals in A , then p C, D q is f -uniformwhere f p t q “ t . To see this, say that c P C b B and d P D b B satisfy } c ´ d } ď δ . Let p h i q be an approximate unit for C .26e claim first that for each i , p h i b B q d is in p C X D q b B . Indeed, let ǫ ą
0, and let d be an element of the algebraic tensor product D d B such that } d ´ d } ă ǫ . Then }p h i b B q d ´p h i b B q d } ă ǫ , and p h i b B q d P p C X D qd B .As ǫ was arbitrary, p h i b B q d is in p C X D q b B .Choose i large enough so that }p h i b B q c ´ c } ă δ , and set x “ p h i b B q d .Then x is in p C X D q b B by the claim, and } x ´ c } ď }p h i b B q d ´ p h i b B q c } ` }p h i b B q c ´ c } ă δ and } x ´ d } ď }p h i b B q d ´ p h i b B q c } ` } c ´ p h i b B q c } ` } c ´ d } ă δ, completing the argument that p C, D q is f -uniform.On the other hand, the following non-example shows that f -uniformityis quite a strong condition: while it is automatic for ideals by the above, itcan fail badly for very simple examples of hereditary subalgebras. Example 5.3.
Let A “ K be the compact operators on H “ ℓ p N q . Chooseprojections p and q on H whose ranges have trivial intersection, but suchthat there are sequences p x n q and p y n q of unit vectors in the ranges of p and q respectively with } x n ´ y n } Ñ C “ p K p and D “ q K q , so C and D are hereditarysubalgebras of K . As range p p q X range p q q “ t u , we have that C X D “ t u :indeed, any self-adjoint element of C X D is a self-adjoint compact operatorwith all its eigenvectors contained in range p p q X range p q q . On the otherhand, if p n and q n are the rank-one projections onto the spans of x n and y n respectively, then p n P C and q n P D for all n , and } p n ´ q n } Ñ
0. It followsthat the pair p C, D q is not f -uniform for any decay function f .The following lemma is immediate from the associativity of the minimal C ˚ -algebra tensor product. Lemma 5.4.
Say A and B are C ˚ -algebras and f is a decay function. If p C, D q is an f -uniform pair for A , then p C b B, D b B q is an f -uniform pairfor A b B . We need two preliminary lemmas before we get to the main result. Recallfirst that if u is an invertible element of a unital ring, then we have the‘Whitehead formula’ ˆ u u ´ ˙ “ ˆ u ˙ ˆ ´ u ´ ˙ ˆ u ˙ ˆ ´
11 0 ˙ . (8)27his implies that invertible elements of the form ˆ u u ´ ˙ are equal tozero in K -theory for purely ‘algebraic’ reasons (compare [22, Lemma 2.5 andLemma 3.1]). The following lemma can thus be thought of as saying thatany invertible element u of a C ˚ -algebra that is zero in K for ‘topologicalreasons’ (i.e. is homotopic to the identity) is also zero in K for ‘algebraic’reasons, up to an arbitrarily good approximation .For the statement of the lemma, recall the notion of being δ -in a subspaceof a C ˚ -algebra from Definition 2.3. Lemma 5.5.
Let c, ǫ ą . Then there exists δ ą with the following prop-erty. Let X be a subspace of a C ˚ -algebra A and let t u t u t Pr , s be a homotopyof invertibles in M n p r A q such that:(i) u “ n ;(ii) for each t , both u t and u ´ t are δ -in t ` x P M n p r A q | x P M n p X qu ;(iii) for each t , } u t } ď c and } u ´ t } ď c .Then there exists m P N and invertible elements a P δ t ` x P M mn p r A q | x P M mn p X qu and b P δ t ` x P M p m ` q n p r A q | x P M p m ` q n p X qu such that a , b , a ´ and b ´ all have norm at most c , and such that the difference ˆ u
00 1 p m ` q n ˙ ´ ¨˚˚˝ n a a ´
00 0 0 1 n ˛‹‹‚ˆ b b ´ ˙ in M p m ` q n p r A q has norm at most ǫ .Proof. Let δ ą “ t ă ... ă t m “ r , s with the property that for any i , } u t i ` ´ u t i } ă δ .Define a : “ ¨˚˚˚˝ u ´ t . . . u ´ t . . . . . . u ´ t m ˛‹‹‹‚ P M mn p r A q . This cannot be exactly true – otherwise the algebraic and topological K groups of a C ˚ -algebra would always be the same. : “ ¨˚˚˚˝ u t . . . u t . . . . . . u t m ˛‹‹‹‚ P M p m ` q n p r A q . Then we have that ˆ u
00 1 p m ` q n ˙ ´ ¨˚˚˝ n a a ´
00 0 0 1 n ˛‹‹‚ˆ b b ´ ˙ equals ¨˚˚˚˚˚˝ p m ` q n ¨ ¨ ¨
00 1 ´ u t u ´ t ¨ ¨ ¨
00 0 1 ´ u t u ´ t ¨ ¨ ¨ ¨ ¨ ¨ ´ u ´ t m ˛‹‹‹‹‹‚ . Recalling that u t m “
1, the latter element has norm bounded above bymax i } ´ u t i ` u ´ t i } “ max i } u t i ´ u t i ` }} u ´ t i } ă δc, which we can make as small as we like by decreasing the size of δ .The next lemma uses decompositions and the identity in line (8) to splitup an element of the form ˆ a a ´ ˙ using approximate ideal structures asin Definition 3.1. Lemma 5.6.
Say A is a C ˚ -algebra and X a finite-dimensional subspaceof A . Then there is a finite-dimensional subspace Y of A such that for any ǫ ą there exists δ ą so that the following holds. Assume that a P M n p r A q is an invertible element such that a and a ´ have norm at most c , and are δ -in the set t ` x P M n p r A q | x P M n p X qu . Assume that p h, C, D q is a δ -ideal structure for Y . Then there are homotopies t v Ct u t Pr , s and t v Dt u t Pr , s ofinvertible elements such that:(i) for each t , v Ct P ǫ t ` c | c P M n p C qu and v Dt P ǫ t ` d | d P M n p D qu ;(ii) ˆ a a ´ ˙ “ v C v D ; iii) v C “ v D “ n ;(iv) for each t the norms of v Ct and v Dt are both at most p ` c q .Proof. Let Y be the subspace of A spanned by all monomials of degreebetween one and four with entries from X . Let Y be as in Lemma 3.3 forthis Y and N “
4. Let then ǫ ą δ ą p h, C, D q be a δ -ideal structure for X . Write a “ ` x and a ´ “ ` y with x, y P δ M n p X q . Consider theproduct decomposition ˆ a a ´ ˙ “ ˆ a ˙ ˆ ´ a ´ ˙ ˆ a ˙ ˆ ´
11 0 ˙ . (9)Set x C : “ ` hx and x D : “ p ´ h q x , so that x C ` x D “ a . Similarly, set y C : “ ` hy and y D “ p ´ h q y , so that y C ` y D “ a ´ . For any element z of a C ˚ -algebra, set X p z q : “ ˆ z ˙ and Y p z q : “ ˆ z ˙ . Then using that X p z ` z q “ X p z q X p z q and similarly for Y , the productin line (9) equals X p x D q X p x C q Y p´ y C q Y p´ y D q X p x C q X p x D q ˆ ´
11 0 ˙ . Rewriting further, this equals the product of v C : “ X p x D q X p x C q Y p´ y C q X p x C q ˆ ´
11 0 ˙ X p´ x D q , and v D : “ X p x D q ˆ ´ ˙ X p´ x C q Y p´ y D q X p x C q X p x D q ˆ ´
11 0 ˙ . We claim this v C and v D have the properties required of v C and v D in thestatement. The norm estimates are clear, as is the equation ˆ a a ´ ˙ “ C v D . For the remainder of the proof, any constant called δ n depends onlyon c , X , and δ , and tends to zero as δ tends to zero (with the other inputsheld constant).We first claim that v C is ǫ -in the set t ` c | c P M n p C qu for δ suitablysmall. Using Lemma 3.4, h commutes with x and y up to some error δ .Using this, plus the fact that xy “ yx “ ´ y ´ x , one computes that X p x C q Y p´ y C q X p x C q ˆ ´
11 0 ˙ is within some δ of an element of the form ˆ ˙ ` ˆ h h ˙ Z , where all entries of Z are products of a noncommutative polynomial in x and y of degree at most two and with no constant term, with a polynomial in h of degree at most two. Hence up to error some δ , we have that v C agreeswith X p x D q ´ ˆ ˙ ` ˆ h h ˙ Z ¯ X p´ x D q , and that up to some δ , this is the same as ˆ ˙ ` ˆ h h ˙ Z , where every entry of Z is a product of a noncommutative polynomial in x and y of degree at most four and with no constant term, with a polynomialin h of degree at most four. The claim follows from this, and the choice of X . The computations showing that v D is ǫ -in the set t ` d | d P M n p D qu for δ suitably small are similar. Indeed, we first we note that Y p´ y D q “ ˆ ˙ ` ˆ ´ h
00 1 ´ h ˙ ˆ ´ y ˙ , whence X p´ x C q Y p´ y D q X p x C q is within δ of an element of the form ˆ ˙ ` ˆ ´ h
00 1 ´ h ˙ Z Z is a product of a noncommutative polynomial in x and y of degree at most two and with no constant term, with a polynomialin h of degree at most two. Hence X p´ x C q Y p´ y D q X p x C q X p x D q is within δ of an element of the form ˆ ˙ ` ˆ ´ h
00 1 ´ h ˙ Z , where every entry of Z is a product of a noncommutative polynomial in x and y of degree at most three and with no constant term, with a polynomialin h of degree at most three. The same is true therefore of ˆ ´ ˙ X p´ x C q Y p´ y D q X p x C q X p x D q ˆ ´
11 0 ˙ . We thus get that v D is within δ of an element of the form ˆ ˙ ` ˆ ´ h
00 1 ´ h ˙ Z , where every entry of Z is a product of a noncommutative polynomial in x and y of degree at most four and with no constant term, with a polynomialin h of degree at most four.To construct homotopies with the required properties, define x Ct : “ `p ´ t q hx , x Dt : “ p ´ t qp ´ h q x , y Ct : “ ` p ´ t q hy , and y Dt ; “ p ´ t qp ´ h q y .Define moreover v Ct : “ X p x Dt q X p x Ct q Y p´ y Ct q X p x Ct q ˆ ´
11 0 ˙ X p´ x Dt q and v Dt : “ X p x Dt q ˆ ´ ˙ X p´ x Ct q Y p´ y Dt q X p x Ct q X p x Dt q ˆ ´
11 0 ˙ . Using precisely analogous computations to those we have already done, onesees that these elements have the claimed properties: we leave the remainingdetails to the reader.Here is the key technical result of this section.32 roposition 5.7.
Let A be a C ˚ -algebra, let f be a decay function, let ǫ ą ,let c ą , and let X be a finite-dimensional subspace of A . Then there existsa finite-dimensional subspace Y of A and δ ą with the following property.Assume that for some n P N there is a homotopy t u t u t Pr , s of invertibleelements in M n p r A q with u “ n , and such that each u t and u ´ t are δ -inthe set t ` x P M n p r A q | x P M n p X qu , and have norm at most some c .Then if p h, C, D q is a δ -ideal structure for Y with p C, D q f -uniform then thefollowing holds.Say l ă n and u C P M n ´ l p r C q and u D P M n ´ l p r D q are invertible, such thatthey and their inverses have norm at most c , and such that } u ´ u C u D ‘ l } ă δ . Then there exists k P N and an invertible element x P M k p Č C X D q suchthat if r x s P K p C X D q is the corresponding class, then with notation as inDefinition 2.1, ι r x s “ pr u C s , r u D sq P K p C q ‘ K p D q . Proof.
Applying Lemma 5.5 to the homotopy t u t u we get m P N and in-vertible elements a P δ t ` x P M n p r A q | x P M mn p X qu an b P δ t ` x P M p m ` q n p r A q | x P M n p X qu such that ˆ u
00 1 p m ` q n ˙ ´ ¨˚˚˝ n a a ´
00 0 0 1 n ˛‹‹‚ˆ b b ´ ˙ has norm at most δ . Let Y a and Y b have the properties in Lemma 9 withrespect to a and b , and let Y : “ Y a ` Y b , a finite dimensional subspace of A .Let then c and ǫ be given, and let δ be fixed, to be determined by the rest ofthe proof. Let p h, C, D q be a δ -ideal structure of Y with p C, D q f -uniform.As usual, throughout the proof any constant called δ n depends on f , c , Y ,and δ , and tends to zero as δ tends to zero. Applying (a very slight variationof) Lemma 9 to ˆ a a ´ ˙ and ˆ b b ´ ˙ , we get elements v C,at and v D,at ,and v C,bt and v D,bt for t P r , s satisfying the conditions there for some δ .33oreover, if we write v C,a : “ v C,a and similarly for the other terms, then ¨˚˚˝ n a a ´
00 0 0 1 n ˛‹‹‚ˆ b b ´ ˙ “ v D,a v C,a v C,b v D,b “ v D,a v C,a v C,b p v D,a q ´ looooooooooomooooooooooon “ : v C v D,a v D,b looomooon “ : v D . Note that v C and v D are δ -in M n p r C q and M n p r D q respectively, that theydefine the trivial class in K p C q and K p D q respectively, and that they andtheir inverses have norm at most p ` c q .Let u C and u D have the properties in the statement. Replacing u C and u D by their block sums with 1 l , we may (for notational simplicity) assumethat l “
0. Now, we have that u C u D and v C v D are within some δ of eachother. Hence 1 ´ v ´ C u C and 1 ´ v D u ´ D are within some δ of each other.Applying our f -uniformity assumption, there exists an element y in somematrix algebra over C X D that is within some δ of both. Set x “ ` y .Then x is an invertible element of some matrix algebra over Č C X D (as longas δ is suitably small) that is close to both v ´ C u C and to v D u ´ D . Hence forsuitably small δ , we have that as classes in K p C qr x s “ r v ´ C u C s “ r u C s , where the second equality follows as v C represents the trivial class in K p C q .Similarly, in K p D q , r x s “ r v D u ´ D s “ r u ´ D s . It follows from the last two displayed lines that ι r x s “ pr u C s , r u D sq as required. In this section we recall some facts about the product map ˆ : K ˚ p A q b K ˚ p B q Ñ K ˚ p A b B q K b K ’ formula that we use.For each n and m , fix an identification M n p C q b M m p C q – M nm p C q thatis compatible with the usual top-left corner inclusions M n p C q Ñ M n ` p C q as m and n vary. Use this to identify M n p A q b M m p B q with M nm p A b B q forany C ˚ -algebras A and B . Any two such identifications differ by an innerautomorphism, so the choice does not matter on the level of K -theory. Wewill use these identifications without comment from now on.We recall a basic lemma that is useful for setting up products in thenon-unital case: see [19, Lemma 4.7.2] for a proof. Lemma 6.1.
For a non-unital C ˚ -algebra A , let ǫ A : r A Ñ C denote thecanonical quotient map. For non-unital C ˚ -algebras A and B , define φ to bethe ˚ -homomorphism p ǫ A b id B q ‘ p id A b ǫ B q : r A b r B Ñ A ‘ B. (where we have identified A b C with A and similarly for B to make senseof this). Then the map K ˚ p A b B q Ñ K ˚ p r A b r B q induced by the canonical inclusion A b B Ñ r A b r B is an isomorphism ontoKernel p φ ˚ q .Similarly, if A is unital and B is non-unital and ψ : “ id b ǫ B : A b r B Ñ A ,then the map K ˚ p A b B q Ñ K ˚ p A b r B q induced by the canonical inclusion A b B Ñ A b r B is an isomorphism ontoKernel p ψ ˚ q . A precisely analogous statement holds if A is non-unital and B is unital. Definition 6.2.
Let A and B be unital C ˚ -algebras, and let p P M n p A q and q P M n p B q be idempotents. Then the product of the corresponding K -theoryclasses r p s P K p A q and r q s P K p B q is defined to be r p s ˆ r q s : “ r p b q s P K p A b B q . A and B are unital, let u P M n p A q be invertible and p P M m p B q be an idempotent, and define u b p : “ u b p ` b p ´ p q P M nm p A b B q . Note that u b p is invertible, with inverse u ´ b p . The product of r u s P K p A q and r p s P K p B q is defined to be r u s ˆ r p s : “ r u b p s P K p A b B q . One checks that these formulas defined on generators extend to well-definedhomomorphisms ˆ : K p A q b K p B q Ñ K p A b B q and ˆ : K p A q b K p B q Ñ K p A b B q . Assume now that A and B are non-unital. Then one checks that foreither p i, j q “ p , q , or p i, j q “ p , q , the canonical composition K i p A q b K j p B q Ñ K i p r A q b K j p r B q ˆ Ñ K i ` j p r A b r B q takes image in the subgroup Kernel p φ ˚ q of the right hand side, where φ is asin Lemma 6.1. Using the identification Kernel p φ ˚ q – K i ` j p A b B q of Lemma6.1, we thus get a general product map ˆ : K i p A q b K j p B q Ñ K i ` j p A b B q if p i, j q P tp , q , p , qu . This all works analogously if just one of A or B isnon-unital, using the other part of Lemma 6.1.For the next definition, for any C ˚ -algebra, let β ´ : K ˚ p S A q Ñ K ˚ p A q be the inverse of the Bott periodicity isomorphism. Definition 6.3.
Let A and B be C ˚ -algebras. Define K p A q b K p B q : “ ` K p A q b K p B q ˘ ‘ ` K p SA q b K p SB q ˘ . Define a ‘product’ map π : K p A q b K p B q Ñ K p A b B q
36o be the composition ` K p A q b K p B q ˘ ‘ ` K p SA q b K p SB q ˘ ˆ‘ˆ ÝÑ K p A b B q ‘ K p S p A b B qq id ‘ β ´ ÝÑ K p A b B q ‘ K p A b B q add ÝÑ K p A b B q We define K p A q b K p B q : “ ` K p A q b K p B q ˘ ‘ ` K p SA q b K p SB q ˘ and π : K p A q b K p B q Ñ K p A b B q completely analogously.The product map is natural with respect to suspensions and Bott peri-odicity. Hence the map π above identifies with the usual product map ` K p A q b K p B q ˘ ‘ ` K p A q b K p B q ˘ Ñ K p A b B q under the usual canonical identifications relating suspensions to dimensionshifts in K -theory, and similarly in the K case.We need a tensor product lemma. Recall that if C, D are C ˚ -subalgebrasof a C ˚ -algebra A , and if B is another C ˚ -algebra, then there is a naturalinclusion p C X D q b B Ď p C b B q X p D b B q . This inclusion need not be an equality above in general: see for example [20].However, f -uniform pairs as in Definition 5.1 behave well in this setting. Lemma 6.4.
Let p C, D q be an f -uniform pair of C ˚ -algebras of some C ˚ -algebra A for some decay function f . Then the natural inclusion p C X D q b B Ď p C b B q X p D b B q is the identity.Proof. The assumption of f -uniformity directly implies that the image of theinclusion is dense. The image is a C ˚ -subalgebra, however, so closed.37EREThe next lemma is the key technical result of this section. Morally, it canbe thought of as saying that if notation is as in Proposition 2.7 and if p anidempotent in some matrix algebra over B , then the diagram K p A q b K p B q B v / / ˆ (cid:15) (cid:15) K p C X D q ˆ (cid:15) (cid:15) K p A b B q B v b p / / K pp C X D q b B q makes some sort of sense, and commutes, when one inputs the class r u sbr p s P K p A q b K p B q . Lemma 6.5.
Let A be a unital C ˚ -algebra, let c ą , and let ǫ P p , c ` q .Then there exists δ ą satisfying the assumptions of Proposition 2.7, andwith the following additional property. Assume that u P M n p A q is invertibleand that v P M n p A q is a p δ, c, C, D q -lift for u as in the conclusion of Propo-sition 2.7. Let B be a C ˚ -algebra, and let p P M m p B q be an idempotent with } p } ď c .Then (with notation as in Definition 6.2) v b p is a p ǫ, c, C, D q -lift for u b p , and we have B v p u q ˆ r p s “ B v b p p u b p q as classes in K pp C X D q b B q .Proof. We leave it to the reader to check that v b p is a p ǫ, c, C, D q -lift of u b p for suitably small δ ą ǫ and c ). Computing, wesee that B v b p p u b p q“ !` v b p ` b p ´ p q ˘ ˆ ˙ ` v ´ b p ` b p ´ p q ˘) p Č C X D qb B ´ „ “ ! v ˆ ˙ v ´ b p ` ˆ ˙ b p ´ p q ) p Č C X D qb B ´ „ . Using that the two terms inside the curved brackets are orthogonal, we have ! v ˆ ˙ v ´ b p ` ˆ ˙ b p ´ p q ) p Č C X D qb B “ ! v ˆ ˙ v ´ b p ) p Č C X D qb B ` ” ˆ ˙ b p ´ p q ı . ” ˆ ˙ b p ´ p q ı ´ „ “ ´ ” ˆ ˙ b p ı , we get that B v b p p u b p q “ ! v ˆ ˙ v ´ b p ) p Č C X D qb B ´ ” ˆ ˙ b p ı “ ˜! v ˆ ˙ v ´ ) Č C X D ´ „ ¸ ˆ r p s , which is exactly B v p u q ˆ r p s as claimed.We also need compatibility results for the maps ι and σ of Definition 2.1and the maps π of Definition 6.3. These are recorded by the following lemma. Lemma 6.6.
Let C and D be an excisive pair of C ˚ -subalgebras of a C ˚ -algebra A , and let B be a C ˚ -algebra. Then for i P t , u , the diagrams K p C X D q b i K p B q ι b id / / π (cid:15) (cid:15) K p C q b i K p B q ‘ K p D q b i K p B q π (cid:15) (cid:15) K i pp C X D q b B q ι / / K p C b B q ‘ K p D b B q and K p C q b i K p B q ‘ K p D q b i K p B q σ b id / / π (cid:15) (cid:15) K p A q b i K p B q π (cid:15) (cid:15) K i p C b B q ‘ K i p D b B q σ / / K i p A b B q commute (where we have the canonical identification of Lemma 6.4 amongstothers to make sense of this).Proof. This follows directly from naturality of the product maps and Bottmaps in K -theory. For a C ˚ -algebra A , let β ´ : K ˚ p S A q Ñ K ˚ p A q
39e the inverse Bott isomorphism. It will be convenient to have a model for β ´ based on an asymptotic family. In this section, we recall some facts aboutasymptotic families and their action on K -theory (in the ‘naive’, rather than E -theoretic, picture). We then discuss how the inverse Bott map can berepresented by an asymptotic family with good properties.Recall (see for example [13, Definition 1.3]) that an asymptotic family between C ˚ -algebras A and B is a collection of maps t α t : A Ñ B u t Pr , such that:(i) for each a P A , the map t ÞÑ α t p a q is continuous and bounded;(ii) for all a , a P A and z , z P C , the quantities α t p a a q ´ α t p a q α t p a q , α t p a ˚ q ´ α t p a q ˚ and α t p z a ` z a q ´ z α t p a q ´ z α t p a q all tend to zero as t tends to infinity.An asymptotic family t α t : A Ñ B u t Pr , canonically defines a map α ˚ : K ˚ p A q Ñ K ˚ p B q . One way to define α ˚ uses the composition productin E -theory and the identification of E ˚ p C , A q with K ˚ p A q . However, thereis also a more naive and direct way. This is certainly very well-known, butwe are not sure exactly where to point in the literature for a description, sowe describe it here for the reader’s convenience.Assume for simplicity that A and B are not unital (this is the only case wewill need), and that t α t : A Ñ B u is an asymptotic family. We extend t α t u to unitisations and matrix algebras just as we would for a ˚ -homomorphism.Note that as A and B are not unital, the extended asymptotic morphism onunitisations takes units to units.If e P M n p r A q is an idempotent, then } α t p e q ´ α t p e q} Ñ t Ñ 8 . Henceif χ is the characterisitic function of the half-plane t z P C | Re p z q ą { u then χ p α t p e qq (defined using the holomorphic functional calculus) is a well-defined idempotent in M n p B q for all t suitably large. If r e s ´ r f s is a formaldifference of idempotents in M n p r A q defining a class in K p A q , then one seesthat for all t suitably large the formal difference r χ p α t p e qqs ´ r χ p α t p f qqs P K p r B q
40s in the kernel of the natural map K p r B q Ñ K p C q induced by the canonicalquotient r B Ñ C . We define α ˚ pr e s ´ r f sq : “ r χ p α t p e qqs ´ r χ p α t p f qqs for anysuitably large t . The choice of t does not matter, as for any t ě t , the path t χ p α s p e qqu s Pr t,t s is a homotopy of idempotents, and similarly for f .Similarly (and more straightforwardly), if u P M n p r A q is invertible, thenas the extension of α t to unitisations is unital, for all suitably large t , α t p u q P M n p r B q is invertible, and we get a well-defined class α ˚ r u s : “ r α t p u qs for anysuitably large t . In this way, we get a well-defined homomorphism α ˚ : K ˚ p A q Ñ K ˚ p B q . We also need to discuss the tensor product of an asymptotic family and a ˚ -homomorphism. First, we describe how an asymptotic family is essentiallythe same thing as a ˚ -homomorphism A Ñ C b pr , , B q{ C pr , , B q . Moreprecisely, given an asymptotic family t α t : A Ñ B u , we can define α : A Ñ C b pr , , B q C pr , , B q , a ÞÑ r t ÞÑ α t p a qs . Conversely, the Bartle-Graves selection theorem implies the existence of acontinuous section s : C b pr , , B q{ C pr , , B q Ñ C b pr , , B q . Thengiven a homomorphism α : A Ñ C b pr , , B q{ C pr , , B q we can definean asymptotic family t α t : A Ñ B u by the formula α t p a q : “ s p α p a qqp t q . If s and s are two different choices of section and t α t u and t α t u the correspondingasymptotic families, then α t p a q ´ α t p a q Ñ t Ñ 8 (compare for example[13, pages 4-5]). In particular, this implies that the induced maps α ˚ and α on K -theory on the same.We may use this correspondence to define the tensor product of an asymp-totic family and a ˚ -homomorphism. Say t α t : A Ñ B u is an asymptoticfamily, and φ : C Ñ D a ˚ -homomorphism with D nuclear. As in [13,Proposition 4.3], we get a natural ˚ -homomorphism C b pr , , B q C pr , , B q b D Ñ C b pr , , B b D q C pr , , B b D q , where we have used nuclearity of D to see that the spatial tensor product ¨ b D agrees with the maximal tensor product ¨ b max D . Hence we get a ˚ -homomorphism A b C α b φ ÝÑ C b pr , , B q C pr , , B q b D Ñ C b pr , , B b D q C pr , , B b D q . (10)41 efinition 7.1. We let t α t b φ : A b C Ñ B b D u be any choice of asymptoticfamily corresponding to the ˚ -homomorphism in line (10).‘The’ asymptotic family t α t b φ u is unfortunately not canonically deter-mined by t α t u and φ . Nonetheless, any such choice will satisfy p α t b φ qp a b c q ´ α t p c q b φ p c q Ñ t Ñ 8 on elementary tensors, and any two such choices will induce the same map K ˚ p A b C q Ñ K ˚ p B b D q .The following lemma is the main technical result of this section. It saysthat asymptotic families are compatible with boundary classes as in Defi-nition 2.8. For the statement, recall the definition of a p δ, c, C, D q -lift fromDefinition 2.6. Lemma 7.2.
Let c, ǫ ą . Then there is δ ą with the following property.Let t α t : A Ñ B u be an asymptotic family between non-unital C ˚ -algebras,and let p C A , D A q be a pair of C ˚ -subalgebras of A and p C B , D B q a pair of C ˚ -subalgebras of B such that for all c P C A and d P D A , d p α t p c q , C B q and d p α t p d q , D B q tend to zero as t tends to infinity. Assume that u P M n p r A q is an invertibleelement with } u } ď c and } u ´ } ď c , and let v be a p δ { , c { , C A , D A q -liftof u . Then for all suitably large t , α t p v q P M n p r B q is a p δ, c, C B , D B q -lift of α t p u q , and moreover B α t p v q p α t p u qq “ α ˚ pB v p u qq in K p C B X D B q for all suitably large t .Proof. We use the same notation t α t u for the canonical extensions to matrixalgebras and unitisations. Note first that as the extension of t α t u to uniti-sations is unital, and as α t is asymptotically multiplicaitve, α t p u q and α t p v q are invertible for all suitably large t .We first claim that asymptotic families are ‘asymptotically contractive’in the following sense: for any a P A and any ǫ ą } α t p a q} ă } a } ` ǫ for all suitably large t . Indeed, let α : A Ñ C b pr , , B q C pr , , B q , a ÞÑ r t ÞÑ α t p a qs
42e the corresponding ˚ -homomorphism. As α is a ˚ -homomorphism, it iscontractive. Hence by definition of the quotient norm, for any ǫ ą b P C pr , , B q such thatsup t Pr , } α t p a q ´ b p t q} ă } α p a q} ` ǫ ď } a } ` ǫ. As } b p t q} Ñ t Ñ 8 , the claim follows.Now, from the claim and the fact that for all d P D A , d p α t p d q , D B q tendsto zero as t tends to infinity, that we have that α t p v q is δ -in M n p Ă D B q forall suitably large t . Similarly, and using also the asymptotic multiplicativityand unitality of t α t u , we get that α t p v q ˆ α t p u q ´ α t p u q ˙ P δ M n p Ă C B q for all suitably large t . The remaining conditions from Definition 2.6 followsimilarly.To see that B α t p v q p α t p u qq “ α ˚ pB v p u qq for t large enough, note that forsuitably large t , the former is represented by ! α t p v q ˆ ˙ α t p v q ´ ) Č C B X D B ´ „ . (11)For the latter, one starts by choosing an idempotent f P M n p Č C A X D A q suitably close to v ˆ ˙ v ´ as in Lemma 2.4 so that ! v ˆ ˙ v ´ ) Č C A X D A “ r f s in K p Č C A X D A q . Then α ˚ pB v p u qq is represented by χ p α t p f qq ´ „ (12)for t suitably large, where χ is as usual the characteristic function of t z P C | Re p z q ą { u . Now, as } α t p f q} is uniformly bounded in t and as } α t p f q ´ α t p f q} Ñ
0, we may apply Lemma 2.2 to conclude that } α t p f q ´ χ p α t p f qq} Ñ
0. On the other hand, by making δ suitably small and t large, and using43he ‘asymptotic contractiveness’ claim at the start of the proof, we can make α t p f q as close as we like to α t p v q ˆ ˙ α t p v q ´ . Comparing lines (11) and (12), the proof is complete.We need the fact that Bott periodicity is induced by an appropriateasymptotic morphism. The following lemma is well-known.
Lemma 7.3.
For any C ˚ -algebra A there is an associated asymptotic family α t : S A A b K with the following properties:(i) the map α ˚ induced on K -theory by t α t u is the inverse Bott map β ´ ;(ii) if B is a C ˚ -subalgebra of A and t α At u and t α Bt u are the asymptoticfamilies associated to A and B respectively, then for all b P S B , α At p b q´ α Bt p b q Ñ as t Ñ 8 ;(iii) for any finite-dimensional subspace X of A and any element of S X , sup t d p α t p x q , X b K q | x P S X, } x } ď u tends to zero as t tends to infinity;(iv) if we fix an inductive limit description K “ Ť n “ M n p C q , then for all t and all a P S A , α t p a q has image in the ˚ -subalgebra Ť n “ M n p A q of A b K .Proof. There are several different ways to do this. We sketch one from [11]based on the representation theory of the Heisenberg group. As in [11, Sec-tion 4], one may canonically construct a continuous field of C ˚ -algebras over r , s with the fibre at 0 equal to S C , and all other fibres equal to K . Asexplained in [8, Appendix 2.B] or [9, pages 101-2], such a deformation (non-canonically) gives rise to an asymptotic family t α t : S C Ñ A b K u , andthis family induces the map on K -theory described in general in [11, Section3], and which is shown in [11, Theorem 4.5] to be the inverse of the Bottperiodicity isomorphism. 44his gives us our asymptotic family t α t u for the case A “ C . In thegeneral case, we may take t α At u to be a choice of asymptotic family t α t b id A : S C b A Ñ K b A u as in Definition 7.1.Note that the construction of t α At u is not canonical at two places: goingfrom a deformation to an asymptotic family, and taking the tensor product.However, any two asymptotic families t α t u , t α t u constructed from differentchoices will satisfy α t p a q ´ α t p a q Ñ t Ñ 8 for all a P S A . It followsthat the asymptotic families so constructed satisfy (i), (ii), and (iii).To make it also satisfy (iv), let t k t u t Pr , be a continuous family of positivecontractions in Ť M n p A q Ď K such that for all k P K , k t kk t ´ k Ñ Ñ 8 .For each a P A , choose a homeomorphism s a : r ,
8q Ñ r , such that α At p a q ´ p b k s a p t q q α t p a qp b k s a p t q q Ñ t Ñ 8 . Replacing α t with the map a ÞÑ p b k s a p t q q α At p a qp b k s a p t q q , we get the result. In this section, we prove the surjectivity half of Theorem 1.4.
Theorem 8.1.
Let A be a C ˚ -algebra, and say A admits a uniform idealstructure over a class C such that for each p C, D q P C , C , D , and C X D satisfy the K¨unneth formula. Then for any C ˚ -algebra B with free abelian K -theory, the product map ˆ : K ˚ p A q b K ˚ p B q Ñ K ˚ p A b B q is surjective.Proof. It suffices to show that the product maps π : K p A q b K p B q Ñ K p A b B q and π : K p A q b K p B q Ñ K p A b B q of Definition 6.3 are surjective for any B with K ˚ p B q free. Replacing B withits suspension, it moreover suffices to show that the second of the maps aboveis surjective. Let then κ be an arbitrary class in K p A b B q .45et X Ď A b B and u P M n p r A q be as in Proposition 3.6 for this κ . UsingCorollary 3.5 and Lemma 5.4, for any δ ą f -uniform δ -idealstructure of the form p h b , C b B, D b B q for X . Fix such an ideal structurefor a very small δ ą v P δ M n p Č A b B q withthe properties stated there, for some constant δ that tends to zero as δ tendsto zero. We may use v to construct an element B v u P K pp C X D q b B q as inProposition 2.7 (here we use the identification p C X D q b B “ C b B X D b B of Lemma 6.4), and have that if ι : K pp C X D q b B q Ñ K p C b B q ‘ K p D b B q is the map from Definition 2.1, then ι pB v u q “ π for C X D is surjective, we may lift B v u to an element λ of K p C X D q b K p B q . With notation as in Definition 6.3,Lemma 6.6 gives that the diagram K p C X D q b K p B q ι b id / / π (cid:15) (cid:15) K p C q b K p B q ‘ K p D q b K p B q π (cid:15) (cid:15) K pp C X D q b B q ι / / K p C b B q ‘ K p D b B q commutates. Hence π pp ι b id qp λ qq “ ι p π p λ qq “ ι pB v u q “ . Using that the product maps for C and D are injective, this gives us that p ι b id qp λ q “ λ “ k ÿ i “ λ i b µ i ` m ÿ i “ k ` λ i b µ i for some k ď m , where λ i P K p C X D q for i ď k , λ i P K p S p C X D qq for i ą k , and similarly µ i P K p B q for i ď k and µ i P K p SB q for i ą k . As K ˚ p B q is free, we may assume moreover that the set t µ , ..., µ m u generates afree direct summand of K p B q ‘ K p SB q . We then have that p ι b id qp λ q “ m ÿ i “ ι p λ i q b µ i “ , ι p λ i q “ i by assumption that the collection t µ , ..., µ m u generates a free direct summand of K p B q ‘ K p SB q . Applying Lemma 4.1to each λ i separately gives us l , ..., l m P N and invertible elements w , ..., w m with w i P M l i p r A q i ď kM l i p Ă SA q i ą k and corresponding lifts v , ..., v m with v i P M l i p r A q i ď kM l i p Ă SA q i ą k such that B v i p w i q “ λ i and B v ´ i p w ´ i q “ ´ λ i . It will be important that thereis c ą i ď k , each v i is an p ǫ, c, C, D q -lift of u i for any ǫ ą i ą k , with SC and SD in place of C and D .Now, write µ i “ r p i s ´ r q i s for projections p i and q i in matrix algebrasover r B for i ď k , and over Ą SB for i ą k . Let t α t : S p A b B q A b B b K u be an asymptotic family inducing the inverse Bott map as in Lemma 7.3.With notation as in Definition 6.2, let us define u : “ u ‘ p w ´ b p q ‘ p w b q q ‘ ¨ ¨ ¨ ‘ p w ´ k b p k q ‘ p w k b q k q‘ α t p w ´ k ` b p k ` q ‘ α t p w k ` b q k ` q ‘ ¨ ¨ ¨ ‘ α t p w ´ m b p m q ‘ α t p w m b q m q , and with notation also as in Lemma 4.2 define v : “ v ‘ p v ´ b p q ‘ p v b q q ‘ ¨ ¨ ¨ ‘ p v ´ k b p k q ‘ p v k b q k q α t p v ´ k ` b p k ` q ‘ α t p v k ` b q k ` q ‘ ¨ ¨ ¨ ‘ α t p v ´ m b p m q ‘ α t p v m b q m q which we can think of as elements of M n t p r A b r B q and M n t p r A b r B q respectivelyfor some n t P N depending on t (recall from Lemma 7.3 that each α t : S p A b B q Ñ A b B b K takes image in M m t p A b B q for some m t P N depending on t ). Then Lemma 4.2 gives that as long as our original δ wassufficiently small, we have B v p u q “ B v p u q ` k ÿ i “ B v ´ i b p i p w ´ i b p i q ` k ÿ i “ B v i b q i p w i b q i q` m ÿ i “ k ` B α t p v ´ i b p i q α t p w ´ i b p i q ` m ÿ i “ k ` B α t p v i b q i q α t p w i b q i q .
47n the other hand, Lemmas 6.5, 7.2, and 7.3 give that for suitably large t this equals B v p u q ` k ÿ i “ B v ´ i b p i p w ´ i b p i q ` k ÿ i “ B v i b q i p w i b q i q` m ÿ i “ k ` α ˚ pB v ´ i b p i p w ´ i b p i qq ` m ÿ i “ k ` α ˚ pB v i b q i p w i b q i qq“ B v p u q ` k ÿ i “ B v ´ i p w ´ i q ˆ r p i s ` k ÿ i “ B v i p w i q ˆ r q i s` m ÿ i “ k ` α ˚ pB v ´ i p w ´ i q ˆ r p i sq ` m ÿ i “ k ` α ˚ pB v i p w i q ˆ r q i sq“ B v p u q ` k ÿ i “ p´ λ i q ˆ r p i s ` k ÿ i “ λ i ˆ r q i s` m ÿ i “ k ` β ´ pp´ λ i ˆ r p i sq ` m ÿ i “ k ` β ´ p λ i ˆ r q i sq“ B v p u q ´ k ÿ i “ λ i ˆ µ i ´ m ÿ i “ k ` β ´ p λ i ˆ µ i q“ B v p u q ´ π p λ q , and this last line is zero.We have just shown that B v p u q “
0. Noting that r u s defines a classin K p A b r B q by Lemma 6.1, it follows at this point from Proposition 2.7that (as long as the original δ ą ν P K p C b r B q ‘ K p D b r B q such that σ p ν q “ r u s . Moreover, if we define ξ : “ m ÿ i “ r w i s b r p i s ` m ÿ i “ r w ´ i s b r q i s “ m ÿ i “ r w i s b µ i P K p A q b K p B q then we have by definition of u that σ p ν q “ r u s “ r u s ´ π p ξ q . Using surjectivity of the product maps for C and D , and with notation as inDefinition 6.3, we may lift ν to some ζ P K p C q b K p B q ‘ K p D q b K p B q .48emma 6.6 gives commutativity of the diagram K p C q b K p B q ‘ K p D q b K p B q σ b id / / π (cid:15) (cid:15) K p A q b K p B q π (cid:15) (cid:15) K p C b B q ‘ K p D b B q σ / / K p A b B q , which implies that r u s “ π p ξ q` σ p ν q “ π p ξ q` σ p π p ζ qq “ π p ξ q` π pp σ b id qp ζ qq “ π p ξ `p σ b id qp ζ qq , so we have that r u s is in the image of the map π , and are done. Finally, in this section we complete the main part of the paper by provingthe injectivity half of Theorem 1.4.
Theorem 9.1.
Let A be a C ˚ -algebra, and say A admits a uniform approxi-mate ideal structure over a class C such that for each p C, D q P C , C , D , and C X D satisfy the K¨unneth formula. Then for any C ˚ -algebra B with freeabelian K -theory, the product map ˆ : K ˚ p A q b K ˚ p B q Ñ K ˚ p A b B q is injective.Proof. With notation as in Definition 6.3, it suffices to show that the maps π : K p A q b K p B q Ñ K p A b B q and π : K p A q b K p B q Ñ K p A b B q defined there are injective for any B with K ˚ p B q free abelian. On replacing B with its suspension, it suffices just to show injectivity in the K case.Consider then an element κ P K p A q b K p B q such that π p κ q “
0. Wewill show that κ “
0. Fix a very small δ ą
0, to be determined by the restof the proof.We may assume κ is of the form κ “ k ÿ i “ κ i b pr p i s ´ r q i sq ` m ÿ i “ k ` κ i b pr p i s ´ r q i sq , n P N , each κ i is an element of K p A q for i ď k or of K p SA q for i ą k , and each pair p i , q i consists of projections in M n p r B q for i ă k or in M n p Ą SB q for i ą k , so that the difference is in M n p B q or M n p SB q asappropriate, and so that the collection pr p i s ´ r q i sq ni “ constitutes part of abasis for the free abelian group K p B q ‘ K p SB q . Using Proposition 3.6,we may assume that for i ď k there is a finite-dimensional subspace X ,i of A and invertible u i P M n p r A q with the properties stated there for κ i and δ ;and similarly for each i ą k , a finite-dimensional subspace X ,i of SA andinvertible u i P M n p Ă SA q with the properties stated in Proposition 3.6 withrespect to κ i and δ .With notation ‘ b ’ as in Definition 6.2, and with t α t : S p r A b r B q Ñ r A b r B b K u an asymptotic family for r A b r B as in Lemma 7.3 that realizes the inverseBott periodicity isomorphism, define u t : “ k à i “ u i b p i ‘ k à i “ u ´ i b q i ` m à i “ k ` α t p u i b p i q ‘ m à i “ k ` α t p u ´ i b q i q . (13)Then for all t suitably large, r u t s defines a class in K p r A b r B q , which we mayconsider as a class in K p A b B q thanks to Lemma 6.1.By definition of π , there is t P r , such that π p κ q “ r u t s for all t ě t ,and so that the map r t ,
8q Ñ ď n “ M n p A q , t ÞÑ u t is a continuous path of invertibles. As r u t s “ π p κ q “
0, we may assumemoreover that there exist l, p P N and a homotopy t w s u s Pr , s of invertibleelements in M p p r A b r B q such that w ‘ l “ u t , such that w “ p , suchthat each w s and w ´ s are in t ` x P M p p r A b r B q | x P M p p A b r B qu . Let X be a finite-dimensional subspace of A b r B such that all w s and w ´ s are δ -in t ` x P M p p r A b r B q | x P M p p X qu . Using part (iii) of Lemma 7.3, there ismoreover a finite-dimensional subspace X of A b r B such that for all t ě t there exists n t P N such that u t and u ´ t are δ -in M n t p X q .Now, using Corollary 3.5 and Lemma 5.4, for any δ ą p h, C, D q such that p b h, SC, SD q is an f -uniform δ -ideal structure of50 , and such that p h b , C b r B, D b r B q is an f -uniform δ -ideal structure ofboth X and X . If δ is small enough, Proposition 3.6 and Lemma 4.3 thenlet us build for each i an invertible element v i P M n p r A q ď i ď kM n p Ă SA q k ` ď i ď m such that for some c ą δ that tends to zero as δ tends to zero, we havethat v i and v ´ i is a p δ , c, C, D q lift of u i and u ´ i respectively for 1 ď i ď k ,and so that v i and v ´ i are a p δ , c, SC, SD q lift for u i and u ´ i respectivelyfor k ` ď i ď m . With notation as in Lemma 4.2, define also v : “ k ð i “ p v i b p i q ‘ n ð i “ k p v ´ i b q i q and v S : “ m ð i “ k ` p v i b p i q ‘ m ð i “ k ` p v ´ i b q i q , which are elements of matrix algebras over r A b r B and Ă SA b Ą SB respectively.Define also u : “ k à i “ u i b p i ‘ k à i “ u ´ i b q i and u S : “ m à i “ k ` u i b p i ‘ m à i “ k ` u ´ i b q i . Then as long as δ ą B v u P K pp C X D q b r B q and B v S p u S q P K p S p C X D q b Ą SB q .Now, with β ´ the inverse Bott periodicity map, the element p id ‘ β ´ qpB v u, B v S p u S qq P K pp C X D q b r B q is necessarily zero. Indeed, using Lemmas 7.2 and 4.2, this element is repre-sented by B v u ` B α t v S p α t p u S qqq “ B v ‘ α t p v S q p u ‘ α t p u S qq for suitably large t . With notation as in line (13), this equals B v ‘ α t p v S q p u t q .Now, we can drag a homotopy between u t and 1 through the constructionof Proposition 3.6 to produce a homotopy between this element and 1 (thisuses our choice of p h, C, D q , and the fact that there is a homotopy throughinvertibles between u t and 1 that is close to p n t ` M n t p X qqYp n t ` M n t p X qq for some appropriate n t P N ). 51emmas 6.5 and 4.2 give then that π ˜ m ÿ i “ B v i p u i q b pr p i s ´ r q i sq ¸ “ p id ‘ β ´ qpB v u, B v S p u S qq whence the class π ˜ m ÿ i “ B v i p u i q b pr p i s ´ r q i sq ¸ P K pp C X D q b r B q is zero also. Hence by injectivity of the product map for C X D , we have that m ÿ i “ B v i p u i q b pr p i s ´ r q i sq is zero in K p C X D q b K p B q . Using the assumption that the collection pr p i s´r q i sq mi “ forms part of a basis for K p B q‘ K p SB q , we get that B v i p u i q “ K p C X D q ‘ K p S p C X D qq for each i . Hence Proposition 2.7 gives us j, l P N and invertible elements s i P M j ` l p r D q ď i ď kM j ` l p Ą SD q k ` ď i ď m such that for each i P t , ..., k u we have that p u i ‘ l q s ´ i is in M j ` l p r C q , andsuch that for each i P t k ` , ..., m u we have that p u i ‘ l q s ´ i is in M j ` l p Ă SC q .Applying the same reasoning with the roles of u i and u ´ i interchanged, wesimilarly get invertible elements r i P M j ` l p r C q ď i ď kM j ` l p Ă SC q k ` ď i ď m such that for each i P t , ..., k u , we have that p u ´ i ‘ l q r ´ i is in M j ` l p r C q ,and for each i P t k ` , ..., m u , we have that p u ´ i ‘ l q r ´ i is in M j ` l p Ă SC q .Now, consider the class λ P ` K p C q b K p B q ˘ ‘ ` K p D q b K p B q ˘ definedby λ “ p λ C , λ D q where λ C : “ m ÿ i “ rp u i ‘ l q s ´ i s b r p i s ` m ÿ i “ rp u ´ i ‘ l q r ´ i s b r q i s λ D : “ m ÿ i “ r s i s b r p i s ` m ÿ i “ r r i s b r q i s , and note that κ “ σ p λ q . The image of λ under the product map ˆ : K p C q b K p B q ‘ K p D q b K p B qÑ ` K p C b B q ‘ K p SC b SB q ˘ ‘ ` K p D b B q ‘ K p SD b SB q ˘ is represented by the invertible element x : “ ˜ m à i “ ` p u i ‘ l q s ´ i b p i ˘ ‘ m à i “ ` p u ´ i ‘ l q r ´ i b q i ˘ , m à i “ ` s i b p i ˘ ‘ m à i “ ` r i b q i ˘¸ . We have that π p λ q equals the image of the class above under the mapid ‘ β ´ : ` K p C b B q ‘ K p SC b SB q ˘ ‘ ` K p D b B q ‘ K p SD b SB q ˘ Ñ K p C b B q ‘ K p D b B q , which, with notation as in Lemma 7.3, is represented concretely by the in-vertible element p id ‘ α t qp x q for all suitably large t . On the other hand,using almost multiplicativity of the asymptotic family t α t u and comparingthis with the formula for u t in line (13), we see that u t can be made arbitrarilyclose to p id ‘ α t q ˜´ m à i “ ` p u i ‘ l q s ´ i b p i ˘ ‘ m à i “ ` p u ´ i ‘ l q r ´ i b q i ˘¯ ¨ ´ m à i “ ` s i b p i ˘ ‘ m à i “ ` r i b q i ˘¯¸ by increasing t , and up to taking block sum with 1 q for some q depending on t . Now, for each fixed t there is n t P N such that u t is homotopic to theidentity through invertibles that are δ -in t ` x P M n t p A b r B q | x P M n t p X q Y M n t p X qu t u s u s Pr t ,t s and t w s ‘ n t ´ p u s Pr , s andour assumption on p h, C, D q . We are thus in a position to apply Proposition5.7 to conclude that there exists a class µ P K pp C X D q b r B q such that ι p µ q “ π p λ q . Using surjectivity of the product map for C X D , we may lift µ to some element ν of K p C X D q b K p r B q . Using Lemma 6.6, we have that π p λ q “ ι p µ q “ ι p π p ν qq “ π p ι p ν qq . Hence by injectivity of the product maps for C and D , this forces λ “ ι p ν q .Finally, we have that κ “ σ p λ q and so κ “ σ p λ q “ σ p ι p ν qq . However, σ ˝ ι is clearly the zero map on K -theory, so we are done. A Nuclear dimension
In this appendix, we give examples of weak approximate ideal structurescoming from nuclear dimension one. See [35] for background on the theoryof nuclear dimension.For the statement of the next result, if A is a C ˚ -algebra, let A denotethe quotient ś N A {‘ N A of the product of countably many copies of A by thedirect sum. If p B n q is a sequence of C ˚ -suablgebras of A , we let B denotethe C ˚ -subalgebra ś N B n { ‘ N B n of A .The following fact was told to me by Wilhelm Winter . Proposition A.1.
Let A be a separable unital C ˚ -algebra of nuclear dimen-sion one. Then there exist(i) a positive contraction h P A X A , and(ii) sequences p C n q , p D n q of C ˚ -subalgebras of A such that:1. each C n and each D n is a quotient of a cone over a finite-dimensional C ˚ -algebra, Professor Winter probably knows a better proof! Not really necessary, but the statement would be a little fiddlier otherwise. . for all a P A , ha P C , p ´ h q a P D ,Proof. Using [35, Theorem 3.2] (and that A is separable) there exists a se-quence p ψ n , φ n , F n q where:(i) each F n is a finite-dimensional C ˚ -algebra that decomposes as a directsum F n “ F p q n ‘ F p q n ;(ii) each ψ n is a ccp map A Ñ F n such that the induced diagonal map ψ : A Ñ F is order zero;(iii) each φ n is a map F n Ñ A such that the restriction φ p i q n of φ n to F p i q n isccp and order zero;(iv) for each a P A , φ n ψ n p a q Ñ a as n Ñ 8 .Let φ : F Ñ A , and φ p i q : F p i q8 Ñ A denote the induced maps, let κ p i q : F Ñ F p i q8 denote the canonical quotient, and consider the composition θ p i q : “ φ p i q ˝ κ p i q ˝ ψ : A Ñ A . Each θ p i q is then ccp and order zero, and we have moreover that θ p q ` θ p q : A Ñ A agrees with the canonical diagonal inclusion.Now, let M i : “ M p C ˚ p θ p i q p A qqq be the multiplier algebra of the C ˚ -subalgebra C ˚ p θ p i q p A qq of A generated by θ p i q p A q . Using [34, Theorem 2.3]if we set h i : “ θ p i q p q , then h i is a positive contraction in C ˚ p θ p i q p A qq X A ,and there exists a unital ˚ -homomorphism π p i q : A Ñ M i X t h i u such that θ p i q p a q “ h i π p i q p a q for all a P A . As 1 “ θ p q p q ` θ p q p q “ h ` h , we will switch notation andwrite h : “ h , so 1 ´ h “ h , so for all a P A , a “ hπ p q p a q ` p ´ h q π p q p a q . (14)Note in particular that h commutes with both θ p q p A q (as h “ h and h commutes with this collection), and with θ p q p A q (as 1 ´ h “ h , and h Unitality follows from the proof in the given reference, but does not appear explicitlyin the statement. h commutes with θ p q p A q ` θ p q p A q Ě A , so in particular h is in A X A .Now, let us think of π p i q : A Ñ M i as having image in the double dual p A q ˚˚ by postcomposing with the canonical embedding M i Ñ p A q ˚˚ . Letus replace π p i q with the map a ÞÑ χ r , szt i u p h q π p i q p a q ` χ t i u p h q a. (15)Then the equation in line (14) still holds for all a P A . Let B be the unital C ˚ -algebra generated by h , A , π p q p A q and π p q p A q , and note that h is centralin B . For each λ P r , s in the spectrum of h in C ˚ p h, q , let I λ be the C ˚ -ideal in B generated by the corresponding maximal ideal in C ˚ p h, q (with I λ “ B if λ is not in the spectrum of h ). Then in B { I λ , the equation in line(14) descends to a “ λπ p q p a q ` p ´ λ q π p q p a q . If λ P p , q and a “ u P A is unitary, this writes the image of u in B { I λ as a convex combination of two elements in the unit ball; as unitaries arealways extreme points in the unit ball of a C ˚ -algebra [3, Theorem II.3.2.17],this is impossible unless π p q p u q “ π p q p u q “ u modulo I λ for all λ P p , q .As the unitaries span any unital C ˚ -algebra [3, Proposition II.3.2.12], thisforces π p q p a q “ π p q p a q “ a modulo I λ for all a P A and all λ P p , q . Onthe other hand, if λ “
0, we clearly get π p q p a q “ a modulo I for all a P A ,while π p q p a q “ a modulo I follows from the replacement we made in line(15). Similarly, if λ “
0, we also get that π p q p a q “ a and π p q p a q “ a modulo I . Putting this together, we have that the postcomposition of either π p q or π p q with the natural diagonal ˚ -homomorphismΦ : B Ñ ź λ P spectrum p h q B { I λ agrees with the natural map A Ñ ś λ Pr , s B { I λ induced by the inclusion A Ñ B . However, as C ˚ p h, q is contained in the center of B , the map Φ isinjective by [10, Theorem 7.4.2]. Hence we get that both π p q and π p q agreewith the diagonal inclusion A Ñ A , and thus have the equations θ p q p a q “ ha and θ p q p a q “ p ´ h q a for all a P A . 56o complete the proof, therefore, we need to find sequences p C n q and p D n q of C ˚ -subalgebras of A with the right properties. For each n and each i P t , u , consider φ p i q n : F p i q n Ñ A . As this is order zero, [34, Corollary3.1] gives a ˚ -homomorphism ρ p i q n : C p , s b F p i q n Ñ A such that φ p i q n p b q “ ρ p i q n p x b b q for all b P A , where x P C p , s is the identity function. Set C n : “ ρ p q n p C p , s b F p q n q and D n “ ρ p q n p C p , s b F p q n q , which contain theimages of φ p q n and φ p q n respectively. It is straightforward to check that p C n q and p D n q have the right properties.The next corollary follows by lifting the element h P A to a positivecontraction p h n q P ś n A : we leave the details to the reader. Corollary A.2.
Let A be a separable C ˚ -algebra of nuclear dimension one,and let C be the class of pairs p C, D q of C ˚ -subalgebras of A such that eachof C and D is isomorphic to a quotient of a cone over a finite dimensional C ˚ -algebra. Then A has a weak approximate ideal structure over C .Remark A.3 . Based on the above it is natural to ask: if A admits a weakapproximate ideal structure over a class C as in Definition 1.5, can one use anadditional argument to show that A admits an approximate ideal structureover C ? We do not believe this is true due to the following example ; wewarn the reader that we did not check the details of what follows. It seemsby adapting Proposition A.1 that one can show that if A has nuclear dimen-sion one and real rank zero, then it has a weak approximate ideal structureover the class C of pairs of its finite dimensional C ˚ -subalgebras. In particu-lar, this would apply to any Kirchberg algebra (see [5, Theorem G] and [27,Proposition 4.1.1]). However, if A admits an approximate ideal structureover a class of pairs of finite-dimensional C ˚ -algebras, then a mild elabora-tion of Proposition 3.6 below shows that K p A q is torsion free. As there areKirchberg algebras with non-trivial torsion K group (see [27, Section 4.3]),this (if correct!) would show that admitting a weak approximate ideal struc-ture over C and admitting an approximate ideal structure over C are not thesame. Inspired by a suggestion of Ian Putnam. Finite dynamical complexity
In this appendix, we give examples of excisive decompositions coming fromdecompositions of groupoids as introduced in [16]. Our conventions ongroupoids will be as in [16, Appendix A] and [26, Section 2.3].The following is a slight variant of [16, Definition A.4].
Definition B.1.
Let G be a locally compact, Hausdorff, ´etale groupoid, let H be an open subgroupoid of G , and let C be a set of open subgroupoids of G . We say that H is decomposable over C if for any open, relatively compactsubset K of H there exists an open cover H p q “ U Y U of the unit spaceof H such that for each i P t , u the subgroupoid of H generated by t h P K | s p h q P U i u is contained in an element of C .The first technical result of this section is as follows. See Definitions 1.1and 1.3 for terminology. Proposition B.2.
Say G is a second countable, locally compact, Hausdorff´etale groupoid that that decomposes over a set D of open subgroupoids of G .Then the reduced groupoid C ˚ -algebra C ˚ r p G q admits an approximate idealstructure over the class of pairs C : “ tp C ˚ r p H q , C ˚ r p H qq | H , H P D u . Moreover, if every groupoid in D is clopen, then C ˚ r p G q admits a uniformapproximate ideal structure over the class C above. The proof will proceed via some lemmas. First we give the existence ofapproximate ideal structures.
Lemma B.3.
Say G is a locally compact, Hausdorff ´etale groupoid that de-composes over a set D of subgroupoids of G in the sense of Definition B.1.Then the reduced groupoid C ˚ -algebra C ˚ r p G q admits an approximate idealstructure over the set C : “ tp C ˚ r p H q , C ˚ r p H qq | H , H P D u .Proof. Let X be a finite-dimensional subspace of C ˚ r p G q . Up to an approx-imation, we may assume that there is an open relatively compact subset K of G such that every element of X is an element of C c p G q supported in K .58sing (a slight variation on) [16, Lemma A.12], for any ǫ ą
0, there is anopen cover G p q “ U Y U of the base space of G and a pair of continuouscompactly supported functions t φ , φ : G p q Ñ r , su with the followingproperties.(i) each φ i is supported in U i ;(ii) for each i P t , u , the set t k P K | r p k q P U i u generates an opensubgroupoid of G that is contained in some element H i of D ;(iii) for each x P G p q , φ p x q ` φ p x q “ k P K , φ p r p k qq ` φ p r p k qq “ k P K and i P t , u , | φ i p s p k qq ´ φ i p r p k qq| ă ǫ .We claim that for any δ ą
0, there exists ǫ suitably small such that if φ and φ are chosen as above, then p h, C, D q “ p φ , C ˚ r p H q , C ˚ r p H qq is a δ -approximate ideal structure.Indeed, the fact that }r h, a s ď δ } a } for all a P X follows from condition(iv) above and [17, Lemma 8.20]. We have moreover that for any a P X , ha “ φ a , and this is supported in t k P K | r p k q P U u by condition (i),whence is in C ˚ r p H q by condition (ii). On the other hand, p ´ h q a “ φ a for any a P X by condition (iii), whence p ´ h q a is in C ˚ r p H q by the sameargument.The next lemma is presumably well-known. Lemma B.4.
Let G be a locally compact, Hausdorff, ´etale groupoid, and let H Ď G be a clopen subgroupoid. Then the restriction map E : C c p G q Ñ C c p H q extends to a conditional expectation E : C ˚ r p G q Ñ C ˚ r p H q .Proof. For x P H p q , let π x : C ˚ r p H q Ñ B p ℓ p H x qq be the associated regularrepresentation defined by p π x p b q ξ qp h q : “ ÿ k P H x b p hk ´ q ξ p h q as in [26, Section 2.3.4]. Let ξ, η P ℓ p H x q , and consider x ξ, π x p E p a qq η y ℓ p H x q “ ÿ h,k P H x E p a qp hk ´ q η p k q ξ p h q “ ÿ h,k P G x a p hk ´ q ˜ η p k q ˜ ξ p h q ξ P ℓ p G x q is the function defined by extending ξ by zero on G x z H x ,and the second equality uses that H is a subgroupoid to deduce that if h, k P H , then hk ´ is in H . Hence if π Gx is the corresponding representationof G on ℓ p G x q , we have x ξ, π x p E p a qq η y ℓ p H x q “ x ˜ ξ, π Gx p a q ˜ η y , and so } E p a q} “ sup } ξ }“} η }“ |x ξ, π x p E p a qq η y ℓ p H x q | “ sup } ξ }“} η }“ |x ˜ ξ, π Gx p a q ˜ η y| ď } a } . Hence E is contractive, and so in particular extends to an idempotent linearcontraction E : C ˚ r p G q Ñ C ˚ r p H q . This extended map is necessarily a con-traction by a classical theorem of Tomiyama: see for example [6, Theorem1.5.10]. Lemma B.5.
Say G is a locally compact, Hausdorff, ´etale groupoid. Thenthe set of pairs of C ˚ -subalgebras of C ˚ r p G q of the form p C ˚ r p H q , C ˚ r p H qq with H , H Ď G both open subgroupoids, and at least one of them also closed, isstrongly excisive as in Definition 1.3.Proof. Say B is an arbitrary C ˚ -algebra, and consider c P C ˚ r p H q b B and d P C ˚ r p H q b B . Say without loss of generality that H is closed, andlet E : C ˚ r p G q Ñ C ˚ r p H q be the conditional expectation of Lemma B.4.As E is just defined on C c p G q by restriction of functions, it follows that E takes C ˚ r p H q into itself, and therefore into C ˚ r p H q X C ˚ r p H q . Henceby functoriality of tensor product maps, we see that E b id restricted to C ˚ r p H q b B is a map E b id : C ˚ r p H q b B Ñ p C ˚ r p H q X C ˚ r p H qq b B and in particular p E b id qp d q is in p C ˚ r p H q X C ˚ r p H qq b B . On the otherhand, as E b id is contractive (see for example [6, Theorem 3.5.3]) and takes C ˚ r p H q to itself, so we get that } c ´ p E b id qp d q} “ }p E b id qp c ´ d q} ď } c ´ d } and } d ´ p E b id qp d q} ď } c ´ d } ` } c ´ p E b id qp d q} ď } c ´ d } so we are done. 60roposition B.2 now follows directly from Lemmas B.3, B.4, and B.5.We spend the rest of this appendix deriving some consequences of Propo-sition B.2. Corollary B.6.
Say G is an ample second countable, locally compact, Haus-dorff ´etale groupoid. Let K be the class of clopen subgroupoids of G , suchthat for any H P K , and any clopen subgroupoid K of H , C ˚ r p K q satisfies theK¨unneth formula. Then K is closed under decomposability.Proof. Say H is a clopen subgroupoid of G that decomposes over K . Then C ˚ r p H q strongly excisively decomposes over the class tp C ˚ r p K q , C ˚ r p K qq | K , K P K u by Proposition B.2, and so C ˚ r p H q satisfies K¨unneth by Theorem1.4. The same argument also applies to any clopen subgroupoid of H : indeed,any clopen subgroupoid of H is easily seen to also decompose over K (comparethe proof of [16, Lemma 3.16]).We will finish with an example that is closely related to the notion offinite dynamical complexity for groupoids introduced in [16, Definition A.4] Definition B.7.
Say G is an ample, locally compact, Hausdorff ´etale groupoidwith finite dynamical complexity. Let C be the class of compact open sub-groupoids of G , and let D be the smallest class of clopen subgroupoids of G containing C and closed under decomposability. Then G has strong finitedynamical complexity if G itself is contained in D . Theorem B.8.
Say G is a principal, locally compact, Hausdorff ´etale groupoidwith strong finite dynamical complexity. Then C ˚ r p G q satisfies the K¨unnethformula. This result is not new: groupoids as in the statement are amenable by[16, Theorem A.9], and therefore their C ˚ -algebras satisfy the UCT by aresult of Tu [32, Proposition 10.7] (at least in the second countable case).Nonetheless, it seems interesting to give a relatively direct proof based onthe internal structure of the C ˚ -algebra. Proof.
Let K be as in Corollary B.6, and let C be the class of compact opensubgroupoids of C . Then for any H P K , the reduced C ˚ -algebra C ˚ r p H q is principal and proper, so Morita equivalent to the continuous functions C p H p q { H q on the orbit space by [23, Example 2.5 and Theorem 2.8] (thesecond countability assumptions in that paper are not necessary in the ´etale61ase [12]). Hence C ˚ r p H q satisfies the K¨unneth formula. As C is closed undertaking clopen subgroupoids, K contains C .Hence if D is as in Definition B.7, then K contains D by Corollary B.6.However, strong finite dynamical complexity implies that G itself is in D , sowe are done. Example B.9.
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