Relatively spectral morphisms and applications to K-theory

RELA TIVEL Y SPECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y BOGDAN NICA Abstract. Spectral morphisms betw een Banac h algebras are useful for compari ng their K- theory and their “noncomm utativ e dimensions” as expressed by v arious notions of stable ranks. In practice, one often encoun ters situations where the spectral information is only kno wn ov er a dense subalgebra. W e in vestigat e such relatively sp ectral morphisms. W e prov e a relativ e ve rsi on of the Densit y Theorem regarding isomor phis m in K-theory . W e als o solve Swan ’s problem for the connected stable rank, i n f act f or an entire hierarch y of hi gher connect ed stable ranks that we introduce. 1. Introduction The followi ng useful cr iterion for K- theoretic isomo rphism is known as the Density Theo r em. Initial versions ar e due t o Karoubi [16, p .10 9] a nd Swan [27, S ec.2 .2 & 3.1]; see a lso Connes [6, Appendix 3 ]. The Density Theo rem a s stated b elow is taken from Bost [4, Thm.1.3.1]. Theorem 1.1. L et φ : A → B b e a dense and sp e ctr al morphism b etwe en Ba nach algebr as. Then φ induc es an isomorphism K ∗ ( A ) ≃ K ∗ ( B ) . A morphism φ : A → B is dense if φ has dense image, and sp e ctr al if sp B ( φ ( a )) = sp A ( a ) for all a ∈ A . Thr oughout this pape r, a lgebras a nd their (contin uous) morphisms are ass umed to be unital. While proving his version of the density theorem, Swan remarked [27, p.206] o n the po ssibility that, under the same h yp otheses as in the Density Theorem, one ha s not only isomorphism in K-theory but a lso e quality of stable ranks . That is, Swan ’s pr oblem asks the fo llowing: Question 1. If φ : A → B is a dense and sp ectral mo rphism b etw een Banach a lgebras, are the stable r anks of A and B equal? There a r e many notions of stable rank in the literature, and the stable rank in the a b ove problem s ho uld be interpreted in a generic sense. Broadly s pea k ing, stable ranks ar e nonco m- m utative notions of dimensio n and they are related to stabilization phenomena in K-theory . The stable ra nks that are rea dily interpreted as noncommutativ e dimensions are the Bas s sta- ble ra nk (bsr) and the top ologica l stable r ank (tsr ). As for stabilization in K-theory , the most natural rank to consider is the co nnected stable rank (csr). W e th us view Swan’s problem for the connected stable rank as the suitable companion to the K-theoretic isomor phism descr ibed b y the Density Theore m. Partial r esults on Swan’s problem w ere obtained b y Badea [2]. Mo s t significantly , bsr( A ) = bsr( B ) whenever A is a “smo oth” subalg ebra of a C ∗ -algebra B [2, Thm.1.1, Co r.4.10]. Also, csr( A ) = csr( B ) for A a dense and spectr a l subalgebra of a commut ative Ba na ch algebra B [2, Date : Ma y 30, 2008. K e y wor ds and phr ases. Spectral morphism, Densit y Theorem in K-theory , Swa n’s problem for the connected stable rank. Supported by FQRNT (Le F onds qu ´ eb ´ ecois de la recherc he sur la nature et les te chnologies). 1 2 BOGD AN NICA Thm.4.15]. As for the results co nce r ning the topo logical stable rank, [2, Thm.4.13, Cor.4.14 ], the h yp otheses ar e unnatural. In this pap er we inv estigate a w eaker notion of spectra l morphism. A mo rphism φ : A → B is r elatively sp e ctr al if sp B ( φ ( x )) = s p A ( x ) for all x in some dense suba lgebra X of A . W e do not know examples of relatively sp ectral morphisms that are not spectral. Most likely , examples exist and they are not obvious. If A enjoys s ome form of s p ectra l contin uity then a relatively sp e c tr al mo r phism φ : A → B is in fact sp ectra l (Section 3 .1). The p oint of considering relativ ely spectra l morphisms is that one can get by wit h less sp ectra l information. F or insta nce, when o ne compares tw o co mpletions of a gro up algebra C Γ, it suffices to consider the s p ectra l behavior of finitely-supp orted elements. The generalization of the Densit y Theorem to the rela tiv ely s pec tr al co nt ext reads as follows: Theorem 1.2. L et φ : A → B b e a dense and c ompletely r elatively sp e ctra l morphism b etwe en Banach algebr as. Th en φ induc es an isomorphism K ∗ ( A ) ≃ K ∗ ( B ) . F ollowing a standard pra ctice, we call a mor phism φ : A → B c ompletely relativ ely s pec tr al if each amplified mo rphism M n ( φ ) : M n ( A ) → M n ( B ) is r elatively s pectr al. It is known ([27, Lem.2.1],[4, P rop.A.2.2], [25, Thm.2.1]) that s pec tr alit y is preserved under amplifications. More precisely , if φ : A → B is a dense and spectr al mor phism then each M n ( φ ) : M n ( A ) → M n ( B ) is a dense a nd sp ectral mo rphism. W e hav e b een unable to prov e a similar result for rela tiv ely sp e c tr al morphisms. Nev ertheless , in co ncrete situations o ne often encounters a stro ng for m of relative spectra lit y which propa gates to all matrix levels; see Section 8.2. The surjectivit y part in the Rela tiv e Densit y Theorem w as k nown to La ffor gue ([19, Lem.3.1 .1] and co mmen ts after [18, Cor.0.0 .3]). Next we co nsider Swan’s problem for the connected stable r ank in the context of relatively sp e c tr al morphisms. The answer seems to b e the most satisfacto r y result o n Swan’s pr o blem so far: Theorem 1.3. L et φ : A → B b e a dense and r elatively sp e ctr al morphism b etwe en Banach algebr as. The n csr( A ) = csr( B ) . In fact, we prove more. Homotopy stabiliza tio n phenomena that a re intimately connected to sta bilization in K -theory sug gest the co nsideration o f higher analo g ues of connected stable ranks. W e sho w that a relatively spectr al, dense morphism preserves these higher co nnected stable r anks; see P rop osition 6 .15. The Relative Densit y Theorem can a lso b e considered in a more genera l context. W e int ro duce certain sp ectral K-functors K Ω , ind exed b y op en subsets Ω ⊆ C con taining the orig in. F o r suitable Ω, one recov ers the usual K 0 and K 1 functors. W e pro ve the Relativ e Densit y Theorem for these spectra l K-functors; see Pro po sition 7.9. A q uick pro of for the usual K-theory is given in Prop osition 5.2. Handling Swan’s pro blem for higher connected stable ranks and pro ving the Relative Density Theorem in spe c tral K-theo ry is in fact elemen tary and c a n b e made r ather short. The slo g an is that dense, re la tively spectr a l morphisms b ehav e well with resp ect to homotopy , for homotopy of op en s ubse ts is eq uiv alent to piecewise-affine homotopy ov er a dense subalgebr a. Most effort in the sections devoted to the spectral K - functors and the higher connected stable ranks go es to wards providing a con text for these no tio ns and inv esting them with meaning . The basic mo tiv ation for introducing spectr al K - functors is the need for an a lternate picture of K 0 , in which the subset of idempo tent s is replaced by an o p e n subset. As for the higher connected stable r a nks, they are mea nt to substant iate our claim that, from the K -theoretic pers pective, the connected stable ra nk is the natura l stable r a nk to be considere d in Swan’s problem. W e believe that these tw o no tions, spectral K-functors and higher c o nnected stable ranks, are o f independent interest. RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 3 Although we are mainly int ere s ted in Banach algebras, the results ment ioned a bove are actually true for goo d F r´ echet algebr as. Sectio n 2 con tains the basic fac ts on go o d F r´ echet algebras that are used in this pap er. In Section 3 w e discus s relatively sp ectral morphisms. In Section 4 we show that dense, relatively sp ectral morphisms tra nsfer the proper t y o f b eing a finite alg ebra. Section 5 provides the key homoto py lemma used in proving the Relative Density Theorem and in s ettling Swan’s problem for the higher connected stable r anks. W e discuss the higher connected stable ra nks in Section 6, and the sp ectra l K-functors in Section 7. W e close with s ome a pplications in Section 8. A c kno wledg m en ts. I would like to thank Guolia ng Y u for us e ful discussions, and J´ an ˇ Spakula for a care ful r eading of the pap er. I am g rateful to Qayum K han for explaining me Lemma 6.6. The fact that Lemma 6.6 holds for compact metric spaces was p ointed out to me b y Bruce Hughes. Finally , I thank the referee for noticing a faulty argument in the or iginal version. 2. Good Fr ´ echet algebras F r´ echet algebr a s a ppea r naturally in Noncommutativ e Geometry [7 ]. A motiv ating example is C ∞ ( M ), the F r´ echet algebra of smo oth functions o n a compact manifold M . The context of go o d F r´ echet alg ebras, a context mor e g eneral than that of Ba na ch alg ebras, turns out to b e the most co n venien t for our purp oses. 2.1. Go o d top o logical algebras. The definition b elow uses the terminology o f [4, Appendix]: Definition 2.1. A to p o lo gical algebra A is a go o d top olo gic al algebr a if the gr oup of in vertibles A × is o p en, a nd the inv ersion a 7→ a − 1 is co n tinuous on A × . There a re t wo key facts ab out go o d top olog ical algebr as that we need in what follows: Prop osition 2. 2. If A is a go o d top olo gic al algebr a then M n ( A ) is a go o d top olo gic al algebr a. Prop osition 2. 3. L et A b e a go o d t op olo gic al algebr a. Then: i) sp( a ) is c omp act for al l a ∈ A ; ii) A Ω = { a : sp( a ) ⊆ Ω } is op en in A whenever Ω ⊆ C is an op en set. Prop osition 2 .2 is due to Swan [2 7, Co r.1.2]. The pro of of Pr opo sition 2.3 is easy and we omit it. 2.2. F r ´ ec het algebras. In this pap er , we adopt the following: Definition 2. 4. An algebra A is a F r´ echet algebr a if A is equipped with a coun table family of subm ultiplicative seminorms {k · k k } k ≥ 0 which make A int o a F r´ echet space. W e do not r equire the seminorms to b e unital. One can assume, without lo ss o f generality , that the countable family of se mino r ms in the previous definition is increasing. If A is a F r´ ech et algebr a under the s eminorms {k · k k } k ≥ 0 then M n ( A ) is a F r´ echet alg e bra under the semino rms given by k ( a ij ) k k = P i,j k a ij k k ; this will b e the standard F r´ echet structure on ma tr ix alg ebras in wha t follows. Every F r´ ec het a lgebra can b e realized a s a n inv erse limit of Bana ch algebras; this is the Arens - Mic hael theorem. Indeed, let A b e a F r´ echet algebra under the seminorms {k · k k } k ≥ 0 . Let A k be the B anach algebra obtained b y co mpleting A modulo the v anishing ideal of k · k k . W e obtain an inv er se system of Banach algebra s with dense connecting mo rphisms. Then A is isometrica lly isomorphic to the inv erse limit lim ← − A k ⊆ Q A k . Conv ersely , an inv erse limit of Ba nach a lgebras is a F r´ echet alge br a: the pro duct Q A k of the Banach algebr as A k has a natural F r´ echet algebr a structure given b y the co ordinate nor ms, whic h F r´ echet structure is inher ited b y the closed subalgebra lim ← − A k . 4 BOGD AN NICA It is easily chec ked that lim ← − A k is a sp ectral subalgebra of Q A k , i.e., ( a k ) ∈ lim ← − A k is in vertible in lim ← − A k if and o nly if each a k is inv ertible in A k . Thus sp  ( a k )  = ∪ k sp A k ( a k ) and r  ( a k )  = sup k r A k ( a k ) for ( a k ) ∈ lim ← − A k . In par ticular, if A is a F r´ echet algebra then sp( a ) is nonempty for a ll a ∈ A . W e also hav e the following Prop osition 2. 5. L et A b e a F r´ echet algebr a. If r A (1 − a ) < 1 then a ∈ A × . Viewing a F r´ echet algebra A once again as an in verse limit of Banach algebra s, it is apparent that in version is contin uous on A × . How ever, A × may not b e op e n. A simple example is C ( R ) with the F r´ echet structure giv en by the seminorms k f k k = sup x ∈ [ − k,k ] | f ( x ) | . The in vertible group of C ( R ), consisting of the non-v anishing contin uous functions, is not ope n: if f k is a contin uo us function such that f k = 1 o n [ − k , k ] and f k = 0 outside [ − ( k + 1) , k + 1], then ( f k ) is a sequence of non-inv ertibles conv erging to 1 . Consider, on the other hand, the F r´ echet algebra C ∞ ( M ) of smo oth functions on a co mpact manifold M . The F r´ echet structure on C ∞ ( M ) is given by the norms k f k k = P | α |≤ k k ∂ α f k ∞ , defined using lo cal charts o n M . That C ∞ ( M ) has an op en gro up o f inv er tibles follo ws by viewing C ∞ ( M ) as a s pectr al, contin uously-embedded subalg e br a of the C ∗ -algebra C ( M ). If X , Y are top ologica l spac e s then X ( Y ) deno tes the contin uous maps from Y to X . Let Σ be a compact Ha usdorff space and let A b e a F r´ echet algebra under the seminor ms {k · k k } k ≥ 0 . Then A (Σ) is a F r´ echet alg ebra under the seminorms k f k k := sup p ∈ Σ k f ( p ) k k . As inv ersio n is contin uo us in A × , w e ha ve A (Σ) × = A × (Σ). Since V (Σ) is open in A (Σ) whenever V is an op en subset o f A , we obta in in par ticular that A (Σ) is go o d whenever A is go o d. Prop osition 2.6. L et A b e a go o d F r´ echet algebr a, a ∈ A and Ω ⊆ C an op en neighb orho o d of sp( a ) . Then ther e is a unique morphism O (Ω) → A sending id Ω to a , given by O a ( f ) = f ( a ) := 1 2 π i I f ( λ )( λ − a ) − 1 dλ wher e t he int e gr al is taken ar ound a cycle (finite union of close d p aths) in Ω c ontaining sp( a ) in its interior. F urthermor e, we have sp f ( a ) = f (sp( a )) for e ach f ∈ O (Ω) . The unique morphism indicated by the previo us prop osition is referr e d to as the holomorphic calculus for a . Here O (Ω), the unital alg e br a of functions that are holo morphic in Ω, is e ndowed with the to po logy of unifor m conv erg ence on compacts. 3. Rela tivel y spectral morphisms In this section, we discuss rela tiv ely sp ectral morphisms. The emphasis is on the comparison betw een relatively sp ectral morphisms and sp ectral morphisms. Recall that a mor phism φ : A → B is sp e ctr al if sp B ( φ ( a )) = sp A ( a ) for all a ∈ A ; equiv alently , for a ∈ A we have a ∈ A × ⇔ φ ( a ) ∈ B × . W e ar e conce r ned with the following relative notion: Definition 3 . 1. A mo rphism φ : A → B is sp e ct ra l r elative to a sub algebr a X ⊆ A if sp B ( φ ( x )) = sp A ( x ) for a ll x ∈ X ; equiv a lent ly , for x ∈ X w e hav e x ∈ A × ⇔ φ ( x ) ∈ B × . A morphism φ : A → B is r elatively sp e ctr al if φ is sp ectra l rela tive to some dense subalg e bra of A . If φ : A → B and ψ : B → C are morphisms, then ψ φ is sp ectral rela tive to X if and only if φ is spectra l r e la tive to X and ψ is spectr al r elative to φ ( X ). This shows, in pa r ticular, that the passage from surjectiv e to dense morphisms naturally en tails a passa ge fro m sp ectral to relatively sp ectral mor phisms. It also follows that a morphism φ : A → B is re la tively spectr a l if and only if b oth the dense morphism φ : A → φ ( A ) and the inclusion φ ( A ) ֒ → B a re r e la tively sp e c tr al (where φ ( A ) is the closure of φ ( A ) in B ). In other w or ds, re la tive sp ectrality has t wo asp ects: the dense morphism case and the closed-subalgebra case. W e are interested in the dense mor phism cas e in this pap er . RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 5 In practice, the following criterion fo r r elative spe c tr alit y is useful: Prop osition 3.2. L et φ : A → B b e a dense morp hism b etwe en go o d F r´ echet algebr as. L et X ⊆ A b e a dense sub algebr a. The fol lowing ar e e qu ivalent: i) sp B ( φ ( x )) = sp A ( x ) for al l x ∈ X ii) r B ( φ ( x )) = r A ( x ) for al l x ∈ X . Pr o of. i) ⇒ ii) is trivial. F or ii) ⇒ i), let x ∈ X with φ ( x ) ∈ B × . Let ( x n ) ⊆ X with φ ( x n ) → φ ( x ) − 1 . W e ha ve r B (1 − φ ( x n ) φ ( x )) → 0, that is, r A (1 − x n x ) → 0. In pa rticular, x n x ∈ A × for large n so x is left-inv ertible. A similar argument shows that x is right-in vertible. Thu s x ∈ A × .  Typically , the domain of a dense, rela tively s pectr a l mor phism cannot b e a C ∗ -algebra : Lemma 3.3. L et φ : A → B b e a ∗ -morphism, wher e A is a C ∗ -algebr a and B is a Banach ∗ -algebr a. If φ is dense and sp e ctr al r elative to a dense ∗ -sub algebr a, then φ is an isomorphi sm. Pr o of. Note first that φ is onto, as φ ( A ) is b oth dense and clos e d. Let X b e a dense ∗ -suba lg ebra of A re la tive to which φ is sp ectral. F or x ∈ X w e hav e: k x k 2 A = k xx ∗ k A = r A ( xx ∗ ) = r B ( φ ( xx ∗ )) ≤ k φ ( xx ∗ ) k B ≤ k φ ( x ) k 2 B It follows that k a k A ≤ k φ ( a ) k B ≤ C k a k A for all a ∈ A , where C > 0. Thus φ is an algebr aic isomorphism, and φ ca n b e made into an isometric isomo r phism by re-no rming B .  Let us p oint out t wo disadv antages in working with rela tivel y sp ectral morphisms. First, if φ : A → B is a sp ectral mo rphism b etw een F r´ echet algebra s a nd B is go o d, then A is go o d as well. If φ is o nly relatively sp ectral, then the knowledge that A is go o d has to come from els e w her e; that is why our applications in Section 8 inv olve Banach algebras only . Second, sp ectrality is well-behaved under amplifications: if φ : A → B is a dens e a nd sp ec- tral morphism then each M n ( φ ) : M n ( A ) → M n ( B ) is a dense and sp ectra l morphism ([27, Lem.2.1],[4, Prop.A.2.2], [25, Thm.2.1]). W e have b een unable to prove a similar result for relatively spectra l mor phisms. Question 2. Let φ : A → B be a dense and relatively s pec tr al morphism. Is M n ( φ ) : M n ( A ) → M n ( B ) a relatively spectral morphism for each n ≥ 1 ? W e thus hav e to in tro duce a stronge r pro p er t y that describ es relative sp ectrality at all matrix levels: Definition 3.4. A morphism φ : A → B is c ompletely sp e ctr al r elative t o a sub algebr a X ⊆ A if each M n ( φ ) : M n ( A ) → M n ( B ) is sp ectral rela tive to M n ( X ). A mo rphism φ : A → B is c ompletely r elatively sp e ctr al if each M n ( φ ) : M n ( A ) → M n ( B ) is r elatively spectra l. Example 3.5. Co ns ider the dense inclusion ℓ 1 Γ ֒ → C ∗ r Γ for a finitely-g enerated amenable gr o up Γ. If Γ has p o lynomial growth then ℓ 1 Γ ֒ → C ∗ r Γ is spectra l (Ludwig [20]). On the o ther hand, if Γ co nt ains a free subsemigr oup on tw o gener ators then ℓ 1 Γ ֒ → C ∗ r Γ is not sp ectral (Je nkins [14]). In b etw een these tw o results, say for g r oups o f intermediate growth, it is unknown whether ℓ 1 Γ ֒ → C ∗ r Γ is s pectr al o r no t. T urning to relative s pectr ality , it is easy to s ee that ℓ 1 Γ ֒ → C ∗ r Γ is s p ectra l rela tive to C Γ if Γ has subex p o nential growth. Indeed, w e show that r ℓ 1 Γ ( a ) = r C ∗ r Γ ( a ) for a ∈ C Γ. W e hav e r C ∗ r Γ ( a ) ≤ r ℓ 1 Γ ( a ) from k a k ≤ k a k 1 , so it suffices to prov e that r ℓ 1 Γ ( a ) ≤ r C ∗ r Γ ( a ). If a ∈ C Γ is suppo rted in a ball of radius R , then a n is supp orted in a ba ll of r adius nR . W e then have k a n k 1 ≤ p vol B ( nR ) k a n k 2 ≤ p vol B ( nR ) k a n k . 6 BOGD AN NICA T aking the n -th ro ot and letting n → ∞ , we o bta in r ℓ 1 Γ ( a ) ≤ r C ∗ r Γ ( a ). This exa mple serves as a preview for Exa mple 8.3, where we show that ℓ 1 Γ ֒ → C ∗ r Γ is in fact completely sp ectral rela tiv e to C Γ, a nd for Section 8.3, where we inv estigate the g roups Γ for which ℓ 1 Γ ֒ → C ∗ r Γ is s pectr al r elative to C Γ. In general, it is hard to decide whether a re la tively sp e ctral morphism is sp ectral or not. F or instance, we do not know any examples of relatively sp ectral morphisms that a r e not s pectr al. Under sp ectral contin uity assumptions, how ever, it is eas y to show that relatively sp ectra l morphisms are in fact s p ectra l. W e discuss this p oint in the next subsection. 3.1. Sp ectral con ti n uity . A relatively spectral morphism φ : A → B is descr ib ed by a sp ectral condition o ver a dense subalgebra of A . In the presence of sp ectra l c onti nuit y , this sp ectra l condition can b e then extended to the whole of A , i.e., φ is a spectra l mo rphism. It mig h t seem, at firs t sight, that b oth A and B need to have s pectr a l contin uit y for this to work, but in fact sp e c tr al co nt inuit y fo r A suffices. Spectr a l co n tinuit y ca n b e interpreted in three different wa ys. The s tr ongest form is to view the s pectr um as a map from a go o d F r´ echet a lgebra to the no n- empt y compact subsets of the complex plane, and to req uir e contin uity of the sp ectrum with resp ect to the Hausdor ff distance. Recall, the Hausdor ff distance b etw e e n t wo non-empt y compact subsets o f the complex plane is given b y d H ( C, C ′ ) = inf { ε > 0 : C ⊆ C ′ ε , C ′ ⊆ C ε } , where C ε denotes the o pen ε - neig hborho o d of C . Letting S denote the class o f go o d F r´ echet a lgebras with contin uous sp ectrum in this sense, we hav e: Prop osition 3 . 6. L et φ : A → B b e a r elatively sp e ctr al morphism b etwe en go o d F r´ echet algebr as. If A ∈ S t hen φ is sp e ctr al. Pr o of. Let a ∈ A ; we need to show that sp A ( a ) ⊆ sp B ( φ ( a )). Let ε > 0 and pick x n → a such that sp A ( x n ) = sp B ( φ ( x n )). F or n ≫ 1 we have sp B ( φ ( x n )) ⊆ sp B ( φ ( a )) ε b y Prop osition 2.3 ii). On the other hand, the contin uity of the sp ectrum in A g ives sp A ( a ) ⊆ sp A ( x n ) ε for n ≫ 1. Combinin g these tw o facts, we g et sp A ( a ) ⊆ sp A ( x n ) ε = sp B ( φ ( x n )) ε ⊆ sp B ( φ ( a )) 2 ε for n ≫ 1. As ε is ar bitrary , we are done.  The usefulness of Prop osition 3.6 is limited by the knowledge ab o ut the cla s s S . There are surprisingly few results on S ; w e did not find in the litera ture any results complementing the ones contained in Aupetit’s survey [1]. The following list summarizes the results mentioned b y A up etit: a) if A is a commut ative Banach algebr a then A is in S b) if A is a commutativ e Banach alg e br a and B is a Ba na ch a lgebra in S , then the pro jective tensor pro duct A ⊗ B is in S ([1, Thm.5, p.139 ]) c) if A is a co mm utative Banach algebra then M n ( A ) ∈ S ([1, Cor .1, p.13 9]) d) if A is a Ba nach algebra in S , then every closed subalgebra of A is in S ([1, Thm.3, p.13 8 ]) A r esult of Ka kutani says that B ( H ), the algebra of bounded op erator s on an infinite- dimensional Hilb ert spa ce, is not in S ([1, p.34]). The seco nd form of sp ectra l contin uit y , weak er than the one ab ov e, is adapted to ∗ -algebra s . W e now require con tinuit y of the spectrum (with respect to the Hausdorff distance) on self- adjoint elements o nly . Denoting by S sa the class of go o d F r´ echet ∗ -algebras whose sp ectrum is contin uo us on the self-adjoint elements, we hav e: Prop osition 3.7. L et φ : A → B b e a r elatively sp e ctr al ∗ -morphism b etwe en go o d F r´ echet ∗ -algebr as. If A ∈ S sa then φ is sp e ctr al. Pr o of. Arg uing a s in the pro of of Prop o s ition 3.6, we see that s p A ( a ∗ a ) = sp B ( φ ( a ∗ a )) for all a ∈ A . Let a ∈ A with φ ( a ) ∈ B × . Then φ ( a ∗ a ) , φ ( aa ∗ ) ∈ B × , and the prev ious equalit y of sp e c tr a g ives a ∗ a, aa ∗ ∈ A × . Therefore a ∈ A × .  RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 7 A ccor ding to [1, Co r.4, p.1 43], symmetric Banach ∗ -a lgebras are in S sa . Recall that a Banach ∗ -algebr a is symmetric if every self-adjoint element has real sp ectrum. W e obtain: Corollary 3.8. L et φ : A → B b e a r elatively sp e ctr al ∗ -morphism b et we en Banach ∗ -algebr as. If A is symmetric then φ is sp e ctr al. The third meaning tha t one can give to s pectr a l contin uity is contin uity o f the s pec tr al ra dius. Let R denote the cla ss of go o d F r´ echet algebras with contin uous s p ectra l ra dius. Con tinuit y o f the sp ectrum (with resp ect to the Hausdorff distance) implies contin uity of the sp ectra l radius, that is , S ⊆ R ; the inclusion is strict ([1, p.38]). Kak uta ni’s result, ment ioned ab ov e, a ctually says that B ( H ) is not in R . Prop osition 3.9. L et φ : A → B b e a dense and r elatively sp e ctr al morphism b etwe en go o d F r´ echet algebr as. If A ∈ R t hen φ is sp e ctr al. Pr o of. The pro of is v ery s imila r to tha t of Prop o s ition 3.6. Let a ∈ A ; we show that r A ( a ) ≤ r B ( φ ( a )). F or then Prop o s ition 3.2 a llows us to conclude tha t φ is sp e c tr al. Let ε > 0 a nd pick x n → a suc h that r A ( x n ) = r B ( φ ( x n )). F or n ≫ 1 w e have r B ( φ ( x n )) ≤ r B ( φ ( a )) + ε , i.e., r A ( x n ) ≤ r B ( φ ( a )) + ε . Letting n → ∞ and using the contin uity of the s p ectra l radius in A , we get r A ( a ) ≤ r B ( φ ( a )) + ε . As ε is arbitra ry , we ar e done.  Finally , note that if B satisfies o ne of the a bove forms of sp ectra l contin uity and φ : A → B is sp ectra l, then A satisfies that sp ectra l contin uity as well. 4. Finiteness Recall that an alg ebra A is finite if the left-in vertibles of A are actually in vertible, equiv alently , if the right-in vertibles o f A are actually inv ertible. This terminology is standard in the Banach- and C ∗ -algebra ic setting; in noncomm utative ring theory , this proper t y is called Dedekind- finiteness. An algebra A is stably finite if each matr ix alg e br a M n ( A ) is finite. T r acial C ∗ -algebra s are imp o rtant examples of sta bly finite alg e bras. Stable-finiteness is relev ant in K-theory , for it guarantees the non-v anishing of K 0 . Prop osition 4. 1. L et φ : A → B b e a dense morphism b etwe en go o d top olo gic al algebr as. a) Assume φ is r elatively sp e ctr al. Then A is fin ite if and only if B is finite. b) A ssume φ is c ompletely r elatively sp e ctr al. Then A is stably finite if and only if B is st ably finite. Pr o of. W e prov e a). Part b) is an obvious coro llary . Letting L ( A ) denote the left-inv ertibles of A , note that the density of A × in L ( A ) suffices for finiteness. Indeed, if a ∈ L ( A ) then u n → a for some inv ertibles u n . There is a ′ ∈ A with a ′ a = 1, so a ′ u n → 1. Hence a ′ u n ∈ A × for la rge n , thus a ′ ∈ A × and we conclude a ∈ A × . Let X b e a dense suba lgebra o f A relative to which φ is sp ectral. Assume B is finite. Let a ∈ L ( A ) and let x n → a where x n ∈ X . Then φ ( x n ) → φ ( a ) ∈ L ( B ) = B × so, b y relative sp ectrality , x n is even tually inv ertible. Thu s A × is dense in L ( A ). Assume A is finite. W e cla im that φ ( L ( A )) is dense in L ( B ); a s φ ( L ( A )) = φ ( A × ) ⊆ B × ⊆ L ( B ), it will follow that B × is dense in L ( B ). Let b ∈ L ( B ) with b ′ b = 1. Pic k sequences ( x n ), ( x ′ n ) in X so that φ ( x n ) → b , φ ( x ′ n ) → b ′ . Since φ ( x ′ n x n ) → 1, relative sp ectra lit y g ives that x n is left-inv ertible for lar ge n . This prov es tha t φ ( L ( A )) is dense in L ( B ).  5. A homo topy lemma W e adapt P rop osition A.2.6 of [4] as follows: 8 BOGD AN NICA Lemma 5. 1. L et A , B b e F r´ echet sp ac es and φ : A → B b e a c ontinuous line ar map with dense image. L et U ⊆ A , V ⊆ B b e op en with φ ( U ) ⊆ V . If X is a dense subsp ac e of A such that U ∩ X = φ − 1 ( V ) ∩ X then φ induc es a we ak homotopy e qu ivalenc e b etwe en U a nd V . Pr o of. First, we sho w that φ induces a bijection π 0 ( U ) → π 0 ( V ). W e make rep eated use o f the local con vexit y . F or surjectivit y , pick v ∈ V . As V is open and φ ( X ) is dense, there is v X ∈ V ∩ φ ( X ) such that v and v X can be connected by a seg ment in V . If u X ∈ X is a pre-image of v X , then u X ∈ U . F or injectivity , let u, u ′ ∈ U suc h that φ ( u ) , φ ( u ′ ) are connected in V . As U is op en and X is dense, there a re u X , u ′ X ∈ U ∩ X such that u and u X , resp ectively u ′ and u ′ X , ca n be connected by a segment in U . Tha t is, we may assume that we start with u, u ′ ∈ U ∩ X . As V is op en and φ ( X ) is dense, if φ ( u ) and φ ( u ′ ) can be connected by a path in V then they can b e connected by a piecewise-linea r path p B lying ent irely in V ∩ φ ( X ). T ake pre-images in X fo r the vertices of p B and ex tend to a piecewise-linea r path p A connecting u to u ′ . Since the path p A lies in X and is mapp e d inside V , it follows that p A lies in U . W e conclude that u , u ′ are co nnected in U . Next, let k ≥ 1. W e show that φ induces a bijection π k ( U, a ) → π k ( V , φ ( a )) for each a ∈ U . Up to trans lating U by − a and V by − φ ( a ), we may as sume that a = 0. Fix a basep oint • on S k . The dense linear map φ : A → B induces a dense linear map φ k : A ( S k ) → B ( S k ), which restricts to a dense linea r map φ • k : A ( S k ) • → B ( S k ) • . The • -deco ration stands fo r restr icting to the based maps sending • to 0. With U ( S k ) • , V ( S k ) • , X ( S k ) • playing the roles of U , V , X , we get by the first part that φ • k induces a bijection π 0 ( U ( S k ) • ) → π 0 ( V ( S k ) • ). That means precisely that φ induces a bijection π k ( U, 0) → π k ( V , 0 ). Let us give some details on the prev ious para g raph. That A ( S k ) • , resp ectively B ( S k ) • , is a F r´ echet space follows by viewing it as a closed subspace of the F r´ ec het space A ( S k ), respectively B ( S k ). The map φ k : A ( S k ) → B ( S k ) is given by φ k ( f ) = φ ◦ f , and it is ob viously linea r and co ntin uous. Consequently , the restriction φ • k : A ( S k ) • → B ( S k ) • is linear and conti nuous. The fact that X ( S k ) • is dense in A ( S k ) • is show ed using a partition of unity a r gument. This also prov es the density of φ k and φ • k . Next, U ( S k ) is op en in A ( S k ) (see co mments befor e Prop osition 2 .6) so U ( S k ) • is open in A ( S k ) • ; similar ly , V ( S k ) • is op en in B ( S k ) • . Finally , φ • k  U ( S k ) •  ⊆ V ( S k ) • and U ( S k ) • ∩ X ( S k ) • = ( φ • k ) − 1  V ( S k ) •  ∩ X ( S k ) • are immedi ate to chec k , they b oil down to φ ( U ) ⊆ V and U ∩ X = φ − 1 ( V ) ∩ X respectively .  It is not hard to imagine that, modulo notational complica tions, the idea used to treat π 0 has a higher-dimensional analog ue. The underlying phenomenon is that homotopical considerations for op en subsets of F r´ ech et spa ces can b e carr ied out in a piecewise-affine fashion ov er a dense subspace. Bost’s elega n t approach of upgrading π 0 knowledge to hig her homotopy g roups is, how ever, mor e eco nomical. As we shall see, the ab ov e Lemma immediately yields the Rela tive D ensity Theorem in sp e c tr al K-theor y (Propos ition 7.9) and a positive answer to Swan’s problem for the higher connected stable ranks (Pr op osition 6.15). W e a lso get a quick pro of for the Relative Density Theorem in the Ba na ch algebra ca se: Corollary 5. 2. L et φ : A → B b e a dense and c ompletely r elatively sp e ctr al morphism b etwe en Banach algebr as. Th en φ induc es an isomorphism K ∗ ( A ) ≃ K ∗ ( B ) . Pr o of. Let φ : A → B b e a dense morphism that is sp ectra l relative to a dense s ubalgebra X ⊆ A , that is, A × ∩ X = φ − 1 ( B × ) ∩ X . Lemma 5.1 says that φ induces a bijection π ∗ ( A × ) → π ∗ ( B × ) for ∗ = 0 , 1. Th us, if φ : A → B is a dense and co mpletely r elatively sp ectral morphism then φ induces a bijection π ∗ (GL n ( A )) → π ∗ (GL n ( B )). It follows that φ induces a bijection lim − → π ∗ (GL n ( A )) → lim − → π ∗ (GL n ( B )). Th at is , φ induces an iso mo rphism K ∗ +1 ( A ) ≃ K ∗ +1 ( B ). As the Bott iso morphism K 0 ( A ) ≃ K 2 ( A ) is natural, we c o nclude that φ induces an isomor phism K ∗ ( A ) ≃ K ∗ ( B ).  RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 9 6. Higher connected st able ranks This section is devoted to higher analog ues of the notion of connected stable rank. W e start by defining the higher connected stable ra nks, a nd by estimating them in terms of the top olog ical stable rank. W e compute the higher connected stable ranks of C (Σ) for certain finite CW- complexes Σ. Then w e ex plain the relation betw e en the higher connected stable ranks and certain ho motopy stabilization ranks. In par ticular, we obtain information ab out stabilization in K-theory . Finally , w e give a p ositive a nswer to Swan’s problem for the higher co nnected stable r anks. 6.1. Defining and estimating hig her connected sta ble ranks. Stable ranks for a top o - logical a lgebra A a re usually defined in terms of the sequence of left-genera ting sets Lg n ( A ) = { ( a 1 , . . . , a n ) : Aa 1 + · · · + Aa n = A } ⊆ A n where n ≥ 1 . It is e asy to see that, for a g o o d top olo gical alg ebra A , Lg n ( A ) is o p e n in A n . Recall: Definition 6. 1. Let A b e a go o d top ological algebra. The top olo gic al s t able r ank tsr( A ) is the least n such that Lg n ( A ) is dens e in A n . The c onn e cte d stable r ank csr( A ) is the leas t n such that Lg m ( A ) is connected for a ll m ≥ n . The Bass st able r ank bsr( A ) is the least n such that Lg n +1 ( A ) is r educible to Lg n ( A ) in the following se ns e : if ( a 1 , . . . , a n +1 ) ∈ Lg n +1 ( A ) then ( a 1 + x 1 a n +1 , . . . , a n + x n a n +1 ) ∈ Lg n ( A ) for some x 1 , . . . , x n in A . Remark 6 . 2. The Bass stable rank, a pur ely algebraic notion, is due to Ba ss [3]. The top ologica l stable r ank a nd the connected sta ble rank were in tro duced by Rieffel [23]; implicitly , these tw o stable ra nk s a lso app ear in [8]. Both [23] and [8] a ddress the basic inequalit y cs r − 1 ≤ bsr ≤ tsr. Note that the definition of the connected stable rank that w e adopt in this pap er is not the original definition [2 3, Def.4.7] but ra ther a n equiv alent one [23, Cor.8 .5]. Note a ls o tha t the stable r anks defined ab ove ar e, strictly spea king, left stable ranks. As usua l, a stable rank is declared to b e infinite if there is no integer fulfilling the requirement. W e define higher connected stable r a nks, that is, stable ra nks enco ding the higher connectivity of the left-gener ating sets. As w e shall s e e in the next subsection, suc h higher co nnectivit y prop erties ar e r e le v ant for stabilization phenomena in K-theory . Definition 6 .3. Let A b e a go o d to p o lo gical alg ebra and k ≥ 0. The k -c onn e cte d stable r ank csr k ( A ) is the lea st n such that Lg m ( A ) is k -connected for all m ≥ n . Recall, a space Σ is said to b e k -co nnected if π i (Σ , • ) = 0 for all 0 ≤ i ≤ k (this is indep endent of the choice o f basep o int • in Σ). E quiv alently , Σ is k -connected if each map S i → Σ can be extended to a map I i +1 → Σ, for a ll 0 ≤ i ≤ k . Henceforth, I denotes the unit interv al. This definition yields a hierar chy of connected stable ranks csr 0 ≤ csr 1 ≤ csr 2 ≤ . . . in which the very first one is the usual connected stable rank; indeed, topolo g ical algebra s being lo cally path-connected, connectivity and path-connectivity are equiv alent for o pe n subsets. W e now seek to gener alize the following tw o imp orta nt facts co ncer ning the connected stable rank: i) csr is homoto p y inv ar iant, i.e., csr( A ) = csr( B ) whenever A and B ar e homotop y equiv alent; ii) c s r ≤ tsr + 1 F act i) this is due to Nistor [21]. Recall that tw o morphisms φ 0 , φ 1 : A → B are said to b e homotopic if they are the endpo in ts o f a path of morphisms { φ t } 0 ≤ t ≤ 1 : A → B ; here t 7→ φ t is contin uous in the sense that t 7→ φ t ( a ) is contin uous for ea ch a ∈ A . If there are morphisms 10 BOGD AN NICA α : A → B a nd β : B → A with β α homotopic to id A and αβ ho motopic to id B , then A and B are sa id to b e homo topy e quiv alent. F act ii) ca n be obtained, for instance, by combining csr( A ) ≤ tsr( A ( I )) (a fact implicit in [21]) with tsr( A ( I )) ≤ tsr( A ) + 1 ([23, Cor.7.2]). W e follow this str ategy in the next prop osition. Recall that we use X ( Y ) to deno te the contin uo us maps from Y to X . Prop osition 6. 4. F or e ach k ≥ 0 we have: i) csr k is homotopy invariant. ii) csr k ( A ) ≤ tsr( A ( I k +1 )) . In p articular, csr k ( A ) ≤ tsr( A ) + k + 1 . If Rieffe l’s estimate ( R ) tsr( A ( I 2 )) ≤ tsr( A ) + 1 holds, then in fact csr k ( A ) ≤ tsr( A ) +  1 2 k  + 1 . Pr o of. i) Homotopic morphisms φ 0 , φ 1 : A → B induce homotopic maps φ 0 , φ 1 : L g m ( A ) → Lg m ( B ). Thus, if A and B a re homotopy equiv alent (as a lgebras), then L g m ( A ) and Lg m ( B ) are homo to p y equiv a lent (as top olog ical spaces). ii) W e sho w the slightly b etter estimate csr k ( A ) ≤ bsr( A ( I k +1 )). T o that end w e use the following fact [3, Lem.4.1]: if A → B is o n to and m ≥ bsr( A ) then Lg m ( A ) → Lg m ( B ) is onto. It follows, in par ticular, that bsr( B ) ≤ bsr( A ) whenever B is a quo tient o f A . Let m ≥ bsr( A ( I k +1 )). W e need to show that Lg m ( A ) is k -co nnected. W e fix 0 ≤ i ≤ k a nd we show that every map S i → Lg m ( A ) extends to a map I i +1 → Lg m ( A ). Consider the restriction morphism A ( I i +1 ) ։ A ( S i ). W e hav e m ≥ bsr( A ( I i +1 )), as A ( I i +1 ) is a quo tien t of A ( I k +1 ), so the restriction map Lg m ( A ( I i +1 )) → Lg m ( A ( S i )) is o n to. F o r a compact spac e Σ - more genera lly , for paracompact Σ [2, Lem.3.2 ] - one has a na tural identifica- tion Lg m ( A (Σ)) ≃ (Lg m ( A ))(Σ). W e thus have an o nto map (Lg m ( A ))( I i +1 ) → (Lg m ( A ))( S i ), proving the des ir ed extensio n prop er t y . F rom tsr( A ( I )) ≤ tsr ( A ) + 1 one obtains tsr( A ( I k +1 )) ≤ tsr( A ) + k + 1, hence the first estimate on cs r k in terms o f tsr. If ( R ) holds, then one obtains tsr( A ( I k +1 )) ≤ tsr( A ) +  1 2 k  + 1, hence the second estimate on csr k .  Remark 6.5. The estimate ( R ) a ppea rs as Question 1.8 in [23]. The mo tiv ation comes fr o m the commut ative case, as one has tsr( C (Σ)) =  1 2 dim(Σ)  + 1 whenev er Σ is a compact space. Sudo claims in [26] a pr o of for ( R ); how ever, Prop osition 1 of [2 6] cannot hold. F o r if it is true, then it implies ( † ) tsr( A ( I )) = max { csr( A ) , tsr( A ) } for all C ∗ -algebra s A , by using Nistor’s description for tsr( A ( I )) as the a bsolute connected stable rank of A . P utting A = C ( I k ) in ( † ) lea ds to tsr( C ( I k +1 )) = tsr( C ( I k )) for all k ≥ 1, since csr( C ( I k )) = 1 by the ho motopy in v ariance of the connected stable rank. T his is a contradiction. The reader mig h t enjo y finding element ar y counterexamples to [26, Pro p.1]. 6.2. Higher connected stable ranks for C (Σ) . F rom P r op osition 6.4 we o btain a dimensional upper bo und fo r the higher connected stable ranks : csr k ( C (Σ)) ≤ tsr ( C (Σ × I k +1 )) =  1 2 (dim(Σ) + k + 1)  + 1 That is: (1) csr k ( C (Σ)) ≤  dim(Σ) + k 2  + 1 In general, we ca nnot hav e equality in (1), for the left-hand side is ho motopy inv ariant whereas the right-hand side is not. W e will see, how ever, that in ma n y natural c a ses we do get e q uality in (1). Let us show, for a start, that csr k ( C ) =  1 2 k  + 1. As Lg m ( C ) = C m \ { 0 } has the homo topy t yp e of S 2 m − 1 , whic h is (2 m − 2)-co nnected but not (2 m − 1)-co nnected, w e obtain that csr k ( C ) RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 11 is the lea st n s uc h that 2 n − 2 ≥ k , i.e., csr k ( C ) =  1 2 k  + 1 . It follows that csr k ( C (Σ)) =  1 2 k  + 1 for Σ contractible. The computation of csr k ( C (Σ)) is a question ab out the v anis hing of ce r tain co ho motopy groups asso ciated to Σ. Indeed, if we ident ify Lg m ( C (Σ)) with ( C m \ { 0 } )(Σ), then for i ≥ 0 we hav e: [ S i , Lg m ( C (Σ))] = [ S i , ( C m \ { 0 } )(Σ)] = [Σ × S i , C m \ { 0 } ] = [Σ × S i , S 2 m − 1 ] Thu s csr k ( C (Σ)) is the lea st n suc h that, for all m ≥ n , we hav e [Σ × S i , S 2 m − 1 ] = 0 for 0 ≤ i ≤ k . This explicit for m ulation shows, once a g ain, that the higher connected stable ranks of C (Σ) only dep end on the homotopy t yp e of Σ. W e ca n als o recover (1). Indeed, if Σ ′ is a finite CW-complex then [Σ ′ , S N ] = 0 as so o n as N > dim(Σ ′ ). Thus, if dim(Σ) + k < 2 m − 1 then [Σ × S i , S 2 m − 1 ] = 0 for all 0 ≤ i ≤ k . W e obtain c s r k ( C (Σ)) ≤  1 2 (dim(Σ) + k )  + 1, i.e., (1). Now, for suitable dimensions, no n- trivial c o homology implies non-triv ia l cohomotopy: Lemma 6.6. L et Σ ′ b e an N -dimensional finite CW-c omplex. Then: a) H N (Σ ′ ; Z ) 6 = 0 if and only if [Σ ′ , S N ] 6 = 0 ; b) H N − 1 (Σ ′ ; Z ) 6 = 0 implies [Σ ′ , S N − 1 ] 6 = 0 . Note that pa rt b) is not an equiv a lence, as witnessed by Σ ′ = S N for N ≥ 3. The pr evious lemma also holds for Σ ′ an N -dimensiona l compact metric spa c e, by results from Hurewicz & W allman: Dimension The ory , pp. 1 49-15 0. The cohomolog y is then understoo d in the ˇ Cech sense. Pr o of. Recall that H N (Σ ′ ; Z ) ca n b e identi fied with [Σ ′ , K ( Z , N )]. Realize the K ( Z , N ) CW- complex b y starting with S N as its N - skeleton a nd then adding cells of dimension ≥ N + 2 that kill the higher homo to p y . F or a) we claim that the natural map [Σ ′ , S N ] → [Σ ′ , K ( Z , N )] is bijectiv e. Surjectivit y is clearly a cons e q uence of the Cellular Approximation Theorem. But so is injectivit y: a homoto py in K ( Z , N ) b etw een f 0 , f 1 : Σ ′ → S N can b e assumed cellular, so it maps to the ( N + 1)-skeleton of K ( Z , N ) which is S N . That is, the maps f 0 , f 1 : Σ ′ → S N are ac tually homo to pic over S N . F or b), note that [Σ ′ , S N ] → [Σ ′ , K ( Z , N )] remains surjective if Σ ′ is ( N + 1)-dimensiona l. Again, this follows from the Cellular Approximation Theor em and the fac t that the ( N + 1)- skeleton of K ( Z , N ) is S N . Replacing N b y N − 1 g ives b) as stated.  W e obtain the following sufficient condition for having equality in (1): Prop osition 6. 7. L et Σ b e a d -dimensional finite CW-c omplex. a) L et k ≥ 1 . If H d (Σ; Z ) 6 = 0 then csr k ( C (Σ)) =  1 2 ( d + k )  + 1 . b) L et k = 0 . If H d (Σ; Z ) 6 = 0 for o dd d , or H d − 1 (Σ; Z ) 6 = 0 for even d , then csr( C (Σ)) =  1 2 d  + 1 . Pr o of. a) L e t m =  1 2 ( d + k )  . Then dim(Σ × S i ) = 2 m − 1 for some 0 ≤ i ≤ k , namely i = k − 1 or i = k . Now H 2 m − 1 (Σ × S i ; Z ) ≃ H d (Σ; Z ) ⊗ Z H i ( S i ; Z ) is non-v anishing, so [Σ × S i , S 2 m − 1 ] 6 = 0 . Th us csr k ( C (Σ)) >  1 2 ( d + k )  . b) Let m =  1 2 d  . If d is o dd, then 2 m − 1 = d so H d (Σ; Z ) 6 = 0 implies [Σ , S 2 m − 1 ] 6 = 0. Therefore csr ( C (Σ)) >  1 2 d  . If d is ev en, then 2 m − 1 = d − 1. Here w e use the fact that H d − 1 (Σ; Z ) 6 = 0 implies [Σ , S 2 m − 1 ] 6 = 0 . W e get csr( C (Σ)) >  1 2 d  in this case as well.  Example 6.8 . W e hav e csr k ( C ( T d )) =  1 2 ( d + k )  + 1 for the d -tor us. Example 6.9. F or the d - sphere, Pro po sition 6.7 gives csr k ( C ( S d )) =  1 2 ( d + k )  + 1 for k ≥ 1 , and for k = 0 and o dd d . If k = 0 and d is even, the coho mological cr iterion fails us. Nevertheless, 12 BOGD AN NICA we have [ S d , S d − 1 ] 6 = 0 for d 6 = 2, hence csr k ( C ( S d )) =  1 2 ( d + k )  + 1 in this case, to o. Summarizing, we have csr k ( C ( S d )) =  1 2 ( d + k )  + 1 except for k = 0, d = 2. W e leav e it to the reader to verify that csr( C ( S 2 )) = 1. 6.3. Stabilization in K-theory and higher connected stable ranks. As b efor e , let A b e a g o o d top olo gical alg ebra. The s e q uence { 1 } = GL 0 ( A ) ֒ → A × = GL 1 ( A ) ֒ → GL 2 ( A ) ֒ → . . . induces, for each i ≥ 0, a sequence of (ident ity-based) homo to p y gro ups: ( π i ) π i (GL 0 ( A )) → π i (GL 1 ( A )) → π i (GL 2 ( A )) → . . . Say that m is a stable level fo r ( π i ) if π i (GL m ′ ( A )) → π i (GL m ′ +1 ( A )) is a n isomorphism for a ll m ′ ≥ m . In analogy with k -connectivity , whic h requires v anishing of all homo topy groups up to the k -th o ne, the following notion enco des the s tabilization o f ( π i ) for a ll 0 ≤ i ≤ k : Definition 6.10. Let k ≥ 0. The k -homotopy stabilization ra nk hsr k ( A ) is the least nonnega tiv e in teger which is a stable level for ( π i ), fo r a ll 0 ≤ i ≤ k . Homotopy stabilization ranks a re closely related to hig her co nnected stable r a nks: Prop osition 6. 11. hsr k ( A ) ≤ csr k +1 ( A ) − 1 and csr k ( A ) − 1 ≤ max { hsr k ( A ) , csr ( A ) − 1 } . Roughly sp eaking , we hav e csr k ( A ) − 1 ≤ hsr k ( A ) ≤ csr k +1 ( A ) − 1. This a ctually holds if, say , tsr( A ) = 1 and K 1 ( A ) 6 = 0, for then hsr 0 ( A ) ≥ 1 ≥ csr( A ) − 1. Pr o of. Let m ≥ csr( A ) − 1 . A ccording to [9, Cor.1.6 ] one has a long exact homotopy sequence: · · · → π i +1 (Lg m +1 ( A )) → π i (GL m ( A )) → π i (GL m +1 ( A )) → π i (Lg m +1 ( A )) → · · · · · · → π 1 (Lg m +1 ( A )) → π 0 (GL m ( A )) → π 0 (GL m +1 ( A )) → 0 Let m ≥ csr k +1 ( A ) − 1. Then π i (GL m ( A )) → π i (GL m +1 ( A )) is an iso morphism for 0 ≤ i ≤ k . Hence csr k +1 ( A ) − 1 is a sta ble level for ( π i ), fo r a ll 0 ≤ i ≤ k , so hsr k ( A ) ≤ csr k +1 ( A ) − 1. F or the second inequality , let m ≥ max { hsr k ( A ) , csr ( A ) − 1 } . Then Lg m +1 ( A ) is connected and, for all 1 ≤ i ≤ k , π i (Lg m +1 ( A )) is tr iv ia l since it is sq ueezed b et ween tw o isomo rphisms in the ab ov e exact sequence. Thus Lg m +1 ( A ) is k -connected.  Corollary 6.12. L et Σ b e a fin ite CW-c omplex of dimension 2 d whose top c ohomolo gy gr oup is non-zer o. Then hsr 1 ( C (Σ)) = d + 1 , i.e., the level at which b oth ( π 0 ) and ( π 1 ) b e gin to stabilize is d + 1 . Corollary 6. 13. hsr k ( A ) ≤ tsr( A ) + k + 1 ; if ( R ) holds, then hsr k ( A ) ≤ tsr( A ) +  1 2 k  . Remark 6. 14. The estimate hsr k ( A ) ≤ tsr( A ) + k + 1 is essentially Theorem 6.3 of [9]. Ho wev er, such an estimate is no t our main go al. The purp ose of the a bove discussion is rather to e mphas ize the following ideas: i) the stable rank s that a r e b est suited for controlling stabilizatio n in the ho mo topy sequences ( π ∗ ) are the connected s table ra nks; ii) the connected stable ranks ca n b e estimated by the “dimensional” stable r anks, namely the topo logical s table ra nk a nd the Bass stable ra nk, which are typically easier to compute; iii) Rieffel’s conjectured inequa lit y ( R ) plays a crucial role in providing go o d estimates for the connected sta ble ra nks, hence in estimating homo topy s tabilization. F or Banach algebr a s, the K-theoretic interpretation is the following. The K 1 group and the K 0 group are the limit groups o f the direct seq uences ( π 0 ) and ( π 1 ): K 1 ( A ) = lim − → π 0 (GL n ( A )) , K 0 ( A ) ≃ lim − → π 1 (GL n ( A )) RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 13 Let us stre ss the following asp ect: stabilization for ( π 0 ) is indeed stabilization for K 1 , whereas stabilization for ( π 1 ) is co nstrued as stabiliza tion for K 0 . Therefore: · K 1 ( A ) ≃ π 0 (GL n ( A )) for n ≥ csr 1 ( A ) − 1; · K 1 ( A ) ≃ π 0 (GL n ( A )) and K 0 ( A ) ≃ π 1 (GL n ( A )) for n ≥ csr 2 ( A ) − 1. In particular, r elative to the top olo gical stable rank we hav e · K 1 ( A ) ≃ π 0 (GL n ( A )) for n ≥ tsr( A ) + 1 (co mpare [2 3, Thm.10.1 2 ]); · K 1 ( A ) ≃ π 0 (GL n ( A )) and K 0 ( A ) ≃ π 1 (GL n ( A )) for n ≥ tsr( A ) + 2. If, further more, ( R ) holds , then · K 1 ( A ) ≃ π 0 (GL n ( A )) for n ≥ tsr( A ); · K 1 ( A ) ≃ π 0 (GL n ( A )) and K 0 ( A ) ≃ π 1 (GL n ( A )) for n ≥ tsr( A ) + 1. 6.4. Sw an’s problem for the higher connected s table ranks. Recall that Swan’s problem asks the following: if φ : A → B is a dense and sp ectral morphism, are the stable ra nks of A and B equal? W e show that this is indeed the ca s e for the higher co nnected stable ra nks. The next result improv es a res ult of B a dea [2, Thm.4.15] in several wa ys. First and foremos t, it remo ves the comm utativity assumption. Second, it suffices to know that the morphism is relatively spectra l. Third, it ha ndles the whole hiera r ch y of co nnected s table ra nks. Prop osition 6.15. L et φ : A → B b e a dense and r elatively sp e ctr al morphism, wher e A and B ar e go o d F r´ echet algebr as. Then cs r k ( A ) = csr k ( B ) for al l k ≥ 0 . Pr o of. W e hav e φ (Lg m ( A )) ⊆ Lg m ( B ). W e c la im that Lg m ( A ) ∩ X m = φ − 1 (Lg m ( B )) ∩ X m , where X is a dense subalgebra o f A relative to which φ is sp ectral. L e t ( x i ) ∈ X m with ( φ ( x i )) ∈ Lg m ( B ), so P c i φ ( x i ) = 1 for so me ( c i ) ∈ B m . Density of φ a llows to a pproximate each c i b y some φ ( x ′ i ), where x ′ i ∈ X , so as to get P φ ( x ′ i ) φ ( x i ) = φ  P x ′ i x i  ∈ B × . As P x ′ i x i ∈ X , we infer that P x ′ i x i ∈ A × b y relative spectra lit y . Th us ( x i ) ∈ Lg m ( A ) as desir ed. Lemma 5.1 a pplied to the dense mor phism φ : A m → B m gives that φ in duces a w eak homotopy equiv alence b e tw een Lg m ( A ) a nd Lg m ( B ). In particular, Lg m ( A ) is k -co nnected if and o nly if Lg m ( B ) is k -connected.  Example 6. 16. W e hav e csr k (C ∗ r ( Z d )) =  1 2 ( d + k )  + 1 b y Example 6.8. Since t he dense inclusion ℓ 1 ( Z d ) ֒ → C ∗ r ( Z d ) is sp ectral, it follows that csr k ( ℓ 1 ( Z d )) =  1 2 ( d + k )  + 1 as well. 7. Spectral K-functors By a K-functor we s imply mean a functor from go o d F r´ ec het algebras to abelian gro ups. The notion of K-sc heme w e in tro duce b elow pro vides a general framew ork for constructing K-functors. A K-functor is usually r equired to be stable, homotopy-in v ar iant, half-exact and contin uo us . These prop erties do not concern us here. W e point out, howev er, that the functors induced by K- schemes are stable and homo topy-in v aria nt by co nstruction. Roughly speaking , a K - scheme is a selection of elements in each go o d F r´ echet alg ebra in such a wa y that morphisms and amplifications preser ve the selection. An op en set Ω ⊆ C containing the or igin selects, in every go o d F r´ echet algebra, those elements whose spectr um is contained in Ω. W e can thus asso cia te to eac h Ω a K-functor K Ω ; these K- functor s are called spectra l K-functors. F or suitable choices o f Ω, one recovers the K 0 and K 1 functors. W e inv estigate how K Ω depends on Ω and we show, for instance, that conformally equiv alent do mains yield naturally equiv alent sp ectra l K -functors. Finally , w e prov e the Relative Densit y Theorem for sp e c tr al K -functors. 14 BOGD AN NICA 7.1. K-sc hemes an d induced K-functors. A K -scheme S asso ciates to each go o d F r´ echet algebra A a s ubset A S ⊆ A such that the following axioms a r e sa tisfied: ( S 1 ) 0 ∈ A S ( S 2 ) if a ∈ M p ( A ) S and b ∈ M q ( A ) S then dia g( a, b ) ∈ M p + q ( A ) S ( S 3 ) if φ : A → B is a morphism then φ ( A S ) ⊆ B S Examples will a ppea r shortly . Observe, at t his point, that the set of S -elements in a go o d F r´ echet algebra is inv ar iant under conjuga tio n. A K-scheme S gives rise to a K- functor K S . W e first construct a functor V S with v alues in ab elian mono ids, then we obtain a functor with v alues in ab elian groups via the Grothendieck functor. Let A be a goo d F r´ ec het algebra. The embeddings M n ( A ) ֒ → M n +1 ( A ), given b y a 7→ diag( a, 0), restrict to embeddings M n ( A ) S ֒ → M n +1 ( A ) S . Put V S ( A ) =  G n ≥ 1 M n ( A ) S   ∼ where the equiv alence r elation ∼ is that of even tual homoto p y , that is, a ∈ M p ( A ) S and b ∈ M q ( A ) S are equiv alent if diag( a, 0 n − p ) and diag( b, 0 n − q ) are path-homotopic in M n ( A ) S for some n ≥ p, q . In other words, V S ( A ) = lim − → π 0  M n ( A ) S  as a set. The conjugacy- inv aria nce o f S -elements gives that V S ( A ) is a n ab elian monoid under [ a ] + [ b ] := [diag ( a, b )]. Let φ : A → B be a mo rphism. W e then have a monoid morphism V S ( φ ) : V S ( A ) → V S ( B ) given by V S ( φ )([ a ]) = [ φ ( a ) ]. Here we use φ to deno te ea ch of the a mplified morphisms M n ( A ) → M n ( B ). So far, we have that V S is a monoid-v alued functor on go o d F r´ echet a lgebras. Let K S be obtained by applying the Grothendieck functor to V S . W e conclude: Prop osition 7. 1. K S is a K-functor. The K-functor asso ciated to a K- scheme is modeled after the K 0 functor, whic h arises in this wa y from the idemp otent K-scheme A 7→ Idem( A ). One could introduce a “m ultiplicative” version of K-s cheme, where the axio m 0 ∈ A S is replaced by 1 ∈ A S , a nd suitably define a corre- sp o nding K-functor so that one recov ers the K 1 functor from the invertible K-scheme A 7→ A × . How e ver, the difference b etw een the “additive” axiom 0 ∈ A S and the “mult iplicative” axio m 1 ∈ A S is a unit shift, whic h ma kes suc h a “multiplicativ e” K-sc heme e s sent ially redundan t. Therefore, up to the na tural equiv alence induced b y the unit shift, we may think of the K 1 functor a s ar ising form the shifte d invertible K-scheme A 7→ A × − 1. A morphism of K-schemes f : S → S ′ asso ciates to ea ch g o o d F r´ echet algebra A a c o nt inuous map f : A S → A S ′ such that the following a xioms ar e sa tisfied: ( M S 1 ) f (0) = 0 ( M S 2 ) if a ∈ M p ( A ) S and b ∈ M q ( A ) S then dia g( f ( a ) , f ( b )) = f (diag( a, b )) in M p + q ( A ) S ′ ( M S 3 ) if φ : A → B is a morphism then the following diagram commutes: A S φ − − − − → B S f   y f   y A S ′ φ − − − − → B S ′ Prop osition 7.2. A morphism of K-schemes f : S → S ′ induc es a natur al tr ansformation of K-functors K f : K S → K S ′ . Pr o of. If suffices to show that f : S → S ′ induces a natura l transfor mation V f : V S → V S ′ . Fir st, we show that for an y g o o d F r´ echet alg ebra A there is a mo noid morphism f A : V S ( A ) → V S ′ ( A ). RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 15 Second, we show that for any morphism φ : A → B the following diagram co mm utes: V S ( A ) V S ( φ ) − − − − → V S ( B ) f A   y f B   y V S ′ ( A ) V S ′ ( φ ) − − − − → V S ′ ( B ) Let A b e a g o o d F r´ echet alg ebra. Define f A : V S ( A ) → V S ′ ( A ) b y f A ([ a ]) = [ f ( a )]. Note tha t f A is well-defined: if a ∈ M p ( A ) S and b ∈ M q ( A ) S are even tually homotopic, i.e., dia g( a, 0 n − p ) and diag ( b, 0 n − q ) are path-homotopic in s ome M n ( A ) S , then f (diag( a, 0 n − p )) = diag ( f ( a ) , 0 n − p ) and f (diag( b, 0 n − q )) = diag ( f ( b ) , 0 n − q ) are path-homotopic in M n ( A ) S ′ , i.e., f ( a ) ∈ M p ( A ) S ′ and f ( b ) ∈ M q ( A ) S ′ are even tually homotopic. Clearly f A is a monoid mor phism: we have f A ([0]) = [ f (0)] = [0] by ( M S 1 ), while b y ( M S 2 ) we hav e f A ([ a ] + [ b ]) = f A [diag( a, b )] = [ f (diag( a, b ))] = [diag( f ( a ) , f ( b ))] = [ f ( a )] + [ f ( b )] = f A ([ a ]) + f A ([ b ]) . The commutativit y of the diagra m follows from ( M S 3 ).  Note that, if f : S → S ′ and g : S ′ → S ′′ are mor phisms of K -schemes then g f : S → S ′′ is a morphism o f K-s ch emes and K gf = K g K f . The morphisms of K-schemes f , g : S → S ′ are homotopi c if f , g : A S → A S ′ are homoto pic for any go o d F r´ echet algebr a A . If that is the case, then f and g induce the sa me natura l transformation K f = K g : K S → K S ′ . The K-schemes S , S ′ are homotopy e quivalent if there are mor phisms of K-schemes f : S → S ′ and g : S ′ → S such that g f is homotopic to id S and f g is ho motopic to id S ′ . A K-scheme S is c ontr actible if S is homotopy equiv a lent to the zero K-scheme A 7→ { 0 A } . Prop osition 7.3. If the K-schemes S and S ′ ar e homotopy e quivalent, then the K-functors K S and K S ′ ar e natur al ly e quivalent. In p articular, if the K-scheme S is c ontr actible then K S is the zer o functor. 7.2. Sp ectral K-functors: defining K Ω . Let 0 ∈ Ω ⊆ C be an op en set. The sp e ctr al K- scheme S Ω asso ciates to each go o d F r´ echet algebra A the subset A Ω = { a : sp( a ) ⊆ Ω } . The axioms ar e ea sy to chec k; for instance, ( S 2 ) follows from sp M p + q ( A )  diag( a, b )  = sp M p ( A ) ( a ) ∪ sp M q ( A ) ( b ) while ( S 3 ) follows from the fa c t tha t morphisms ar e non-incr easing on sp ectra. The K-functor ass o ciated to the sp ectral K- s chem e S Ω is denoted by K Ω and is r e ferred to as a s pectr al K - functor. Example 7.4 . W e compute K Ω ( C ). Assume, for simplicit y , that Ω has finitely man y co nnected co mp onents Ω 0 , Ω 1 , . . . , Ω k , wher e Ω 0 is the comp onent of 0. F or 1 ≤ i ≤ k , let # i ( a ) deno te the n umber of eige nv alues, counted with multiplicit y , o f a ∈ M n ( C ) that lie in Ω i . Then the eig e nv alue-co un ting map # : V Ω ( C ) → N k given b y #([ a ]) = (# 1 ( a ) , . . . , # k ( a )) is well-defined. Visibly , # is a surjective morphism o f monoids. W e show # is injective. F or 1 ≤ i ≤ k , pick a basep oint λ i ∈ Ω i and think of λ ( a ) = diag  λ 1 , . . . , λ 1 | {z } # 1 ( a ) , . . . , λ k , . . . , λ k | {z } # k ( a )  16 BOGD AN NICA as a normal form for a ∈ M n ( C ). It s uffices to show that a is event ually ho motopic to λ ( a ). Similar matrices are ev entually homo topic; this is true for any K-sc heme in fact. Thus a is even tua lly homotopic to its Jo rdan no rmal form. Up to a spec trum-pr eserving homotopy , one can assume that the Jordan normal form is in fact diagonal. Finally , a homotopy sends all eigenv a lues in each Ω i to the chosen basepo int λ i , and all eigenv alues in Ω 0 to 0. That is, we can r each a diago nal matr ix diag ( λ ( a ) , 0 , . . . , 0). Therefore V Ω ( C ) is isomo rphic to N k and, co nsequent ly , K Ω ( C ) is isomorphic to Z k . 7.3. Sp ectral K-functors: dep endence on Ω . An op en set Ω ⊆ C containing 0 is thought of as a b ase d op en set. Corre spo ndingly , a holo morphic map f : Ω → Ω ′ sending 0 to 0 is ca lled a b ase d ho lomorphic map. Prop osition 7.5. A b ase d holomorphic map f : Ω → Ω ′ induc es a morphism of K-schemes f ∗ : S Ω → S Ω ′ , henc e a n atur al tr ansformation of K-fun ct ors K f ∗ : K Ω → K Ω ′ . In p articular, a b ase d c onformal e quivalenc e b etwe en Ω and Ω ′ induc es a natur al e quivalenc e b etwe en K Ω and K Ω ′ . Pr o of. Let A b e a go o d F r´ ec het alg e br a. W e ha ve a map f ∗ : A Ω → A Ω ′ given b y f ∗ ( a ) := f ( a ) = O a ( f ), where O a : O (Ω) → A is the ho lomorphic calculus of a ∈ A Ω . W e claim that f ∗ is co nt inuous. Indeed, let a n → a in A Ω . Pic k a (topolog ically tame) cycle γ containing sp( a ) in its interior. Since the set of ele ments whose spectrum is contained in the in terior of γ is op en, we may ass ume without los s of generality that a ll the a n ’s hav e their sp ectrum contained in the in terio r of γ . Then f ∗ ( a n ) − f ∗ ( a ) = 1 2 π i I γ f ( λ )  ( λ − a n ) − 1 − ( λ − a ) − 1  dλ and so f ∗ ( a n ) → f ∗ ( a ) since the in tegr and con verges to 0. W e show f ∗ is a m or phism of K-schemes. Axiom ( M S 1 ) is obvious. F o r ( M S 2 ) we use the uniqueness of holomorphic calculus. If O a : O (Ω) → M p ( A ) is the holomo rphic calculus for a and O b : O (Ω) → M q ( A ) is the ho lomorphic calculus for b , then diag ( O a , O b ) : O (Ω) → M p + q ( A ) given b y g 7→ diag( O a ( g ) , O b ( g )) is a holomorphic calculus for diag( a, b ); thus O diag( a,b ) = diag( O a , O b ), in particular f ∗ (diag( a, b )) = dia g ( f ∗ ( a ) , f ∗ ( b )). F o r ( M S 3 ), we use again the uniqueness of holomorphic calculus. If O a : O (Ω) → A is the holo mo rphic calculus of a ∈ A Ω then φ ◦ O a : O (Ω) → B is a holomor phic calculus for φ ( a ) hence φ ◦ O a = O φ ( a ) , in par ticular φ ( f ∗ ( a )) = f ∗ ( φ ( a )) for a ll a ∈ A Ω . F or the second part, if suffices to check that ( g f ) ∗ = g ∗ f ∗ for based holomorphic maps f : Ω → Ω ′ and g : Ω ′ → Ω ′′ . That is, we need ( g f )( a ) = g ( f ( a )) for a ll a ∈ A Ω . This follows once a gain b y the uniqueness of holomor phic calculus: a s b oth h 7→ ( hf )( a ) and h 7→ h  f ( a )  are holomorphic calculi O (Ω ′ ) → A for f ( a ), w e get ( hf )( a ) = h  f ( a )  for all h ∈ O (Ω ′ ), in particular for g .  Corollary 7. 6. If Ω is c onne ct e d and simply c onne ct e d then K Ω is t he zer o functor. Pr o of. By the co nformal inv ar iance of K Ω and the Riemann Ma pping Theo rem, we may assume that Ω is either the entire complex plane, or the op en unit disk. T o show that K Ω is the zer o functor, it suffices to get π 0 ( A Ω ) = 0 for every go o d F r´ echet algebra A . Indeed, each a ∈ A Ω can b e co nnected to 0 A b y the path t 7→ ta , pa th which lies in A Ω since t Ω ⊆ Ω for 0 ≤ t ≤ 1. In a more sophisticated formulation, S Ω is co n tra c tible.  Prop osition 7.5 shows that the functor K Ω depends o nly on the conformal t yp e of Ω. But more is true, in fact: K Ω depends only on the holomor phic homotopy type of Ω. Roughly sp eaking, the notion of holo morphic homotopy t yp e is obtained from the usual notion of homotopy t yp e b y requiring the maps to b e holomor phic rather than contin uous. RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 17 The ba sed holomor phic ma ps f , g : Ω → Ω ′ are holomorphic al ly homotopic if there is a family of based holomorphic maps { h t } 0 ≤ t ≤ 1 : Ω → Ω ′ such that h 0 = f , h 1 = g and t 7→ h t ∈ O (Ω) is contin uous. If that is the case, then the induced morphisms of K-schemes f ∗ , g ∗ : S Ω → S Ω ′ are homotopic. If there are based holomorphic maps f : Ω → Ω ′ and g : Ω ′ → Ω such that f g is holomorphica lly homo topic to id Ω ′ and g f is holomorphically homotopic to id Ω , then Ω and Ω ′ are sa id to b e holomorph ic-homotopy e quivalent . If that is the ca s e, then the K-schemes S Ω and S Ω ′ are homo to p y equiv a lent in the sense descr ibed in the previous subsection. Therefor e: Prop osition 7.7 . If Ω and Ω ′ ar e h olomorphic-homo topy e quivalent, then K Ω and K Ω ′ ar e natur al ly e qu ivalent. 7.4. Sp ectral K-functors: reco v ering K 0 and K 1 . The K 1 functor can b e o bta ined from the shifted in vertible K-scheme A 7→ A × − 1. Tha t is, K 1 is natura lly equiv alent to K Ω for Ω = C \ {− 1 } . The K 0 functor can b e obtained fro m the idempo ten t K-scheme A 7→ Idem( A ). How e ver, the idemp otent scheme is not a sp ectra l K-scheme. W e now realize the K 0 functor a s a s p ectra l K- functor. Prop osition 7. 8. K 0 = K Ω for Ω = C \ { Re = 1 2 } . Pr o of. Let χ denote the function defined on Ω as χ = 0 on { Re < 1 2 } a nd χ = 1 on { Re > 1 2 } . Consider the ho lomorphic functions { h t } 0 ≤ t ≤ 1 : Ω → Ω defined a s h t = (1 − t )id + tχ . W e claim that { ( h t ) ∗ } 0 ≤ t ≤ 1 is a strong deformation of S Ω to the idempotent K-scheme. It will then follow that K Ω = K 0 . Let A b e a goo d F r´ echet algebra . W e need to show that { ( h t ) ∗ } 0 ≤ t ≤ 1 is a strong deformation from A Ω to the idemp otents of A . Each ( h t ) ∗ : A Ω → A Ω is co nt inuous. The map t 7→ h t is O (Ω)-co n tinuous, so t 7→ ( h t ) ∗ is A -contin uous . At t = 0, ( h 0 ) ∗ = id A Ω . At t = 1, ( h 1 ) ∗ = χ ∗ takes idempo ten t v alues s ince χ 2 = χ . Finally , each ( h t ) ∗ acts identically on idempotents. Indeed, it suffices to show that χ ∗ acts iden tically on idemp otents. Letting e b e an idemp otent, we hav e χ ( e ) = 1 2 π i I γ 1 ( λ − e ) − 1 dλ = 1 2 π i I γ 1  1 − e λ + e λ − 1  dλ = e where γ 1 is a curve ar ound 1 .  That K 0 can b e describ ed in terms o f elements with s p ectrum co nt ained in C \ { Re = 1 2 } is discus s ed in [24, pp. 193 -196]; the ab ov e pro o f is essentially the one g iven there. It is this alternate picture for K 0 that inspired us in defining the K Ω groups. Note that K Ω is natura lly equiv alent to K 0 whenever Ω is the disjoint union of tw o co nnected and simply connected op en subsets o f C . 7.5. Sp ectral K-functors: the Densi t y Theorem. Finally , w e prov e the Relative Density Theorem for spectr al K-functors. In particular, w e obtain the Relativ e Density Theor em for the usual K-theo r y . In the ca se of K 0 , this pro of is more elementary than the pro of o f Corollar y 5.2, which used the Bo tt p erio dicity . Prop osition 7.9. If φ : A → B is a dense and c ompletely r elatively sp e ctr al morphism b etwe en go o d F r´ echet algebr as, then φ induc es an isomorphism K Ω ( A ) ≃ K Ω ( B ) . Pr o of. Let φ : A → B be a dense morphism that is spectral relative to a dens e subalgebra X ⊆ A . O bviously A Ω ∩ X = φ − 1 ( B Ω ) ∩ X , s o φ induces a bijection π 0 ( A Ω ) → π 0 ( B Ω ) by Lemma 5.1. Thus, if φ : A → B is a dense and c o mpletely relatively sp ectral mor phism then φ induces a bijection π 0 (M n ( A ) Ω ) → π 0 (M n ( B ) Ω ) for ea ch n ≥ 1 . It follows that φ induces a bijection lim − → π 0 (M n ( A ) Ω ) → lim − → π 0 (M n ( B ) Ω ). In o ther words, V Ω ( φ ) : V Ω ( A ) → V Ω ( B ) is a bijection. W e conclude that φ induces an isomorphism K Ω ( A ) → K Ω ( B ).  18 BOGD AN NICA 8. Applica tions 8.1. Sp ectral equiv alence. Let A and B be Banach algebra completions of an alg ebra X . Say that A and B ar e sp e ctr al ly e qu ivalent over X if sp A ( x ) = s p B ( x ) for all x ∈ X , a nd c ompletely sp e ctr al ly e qu ivalent over X if M n ( A ) a nd M n ( B ) are spectrally equiv alent ov er M n ( X ) for each n ≥ 1. Prop osition 8.1 . The higher c onne cte d stable r anks and the pr op erty of b eing finite ar e invari- ant under sp e ctr al e quivalenc e. F urthermor e, K-the ory and the pr op erty of b eing stably finite ar e invariant un der c omplete sp e ctr al e quivalenc e. Pr o of. Let A , B b e Banach a lgebra co mpletions of X . Let X be the Banach a lgebra o btained b y completing X under the norm k x k := k x k A + k x k B . Then r X ( x ) = max { r A ( x ) , r B ( x ) } for all x ∈ X . Now, if A a nd B are sp ectra lly equiv alent ov er X then r X ( x ) = r A ( x ) = r B ( x ) for all x ∈ X . W e obtain dense a nd relatively sp ectral mo rphisms X → A , X → B by extending the inclusions X ֒ → A , X ֒ → B . Th us csr k ( A ) = csr k ( X ) = csr k ( B ), and A is finite if and only if X is finite if and only if B is finite. If A and B are completely sp ectrally equiv alent over X then we obtain morphisms X → A , X → B that are completely sp ectral r elative to X . Hence K ∗ ( A ) ≃ K ∗ ( X ) ≃ K ∗ ( B ), and A is stably finite if and o nly if X is stably finite if and only if B is s tably finite.  T wo alg ebras that ar e spectra lly equiv a lent need not b e connected b y a morphism. Co nsider, how ever, the following ∗ -context: A is a Bana ch ∗ -algebra, B is a C ∗ -algebra , and X is a dense ∗ -subalgebr a of A and B . The arg umen t used in the pro of of Lemma 3.3 shows the follo wing: if A a nd B are sp ectrally equiv alent over X , then there is a dense and re la tively spectra l morphism A → B extending the identit y on X . A criterion for complete s p ectra l equiv alence is given in the next section. 8.2. Sub exp onential contr ol. A weight on a group Γ is a map S 7→ ω ( S ) ∈ [1 , ∞ ) on the finite nonempt y subsets of Γ which is non-decreas ing , i.e., S ⊆ S ′ implies ω ( S ) ≤ ω ( S ′ ). A weigh t ω is sub exp onential if ω ( S n ) 1 /n → 1 for all S . Prop osition 8 . 2. L et A Γ and B Γ b e Banach algebr a c ompletions of a gr oup algebr a C Γ . A ssu me that ther e ar e c onst ants C , C ′ > 0 and s u b exp onential weights ω , ω ′ such that k a k B ≤ C ω (supp a ) k a k A , k a k A ≤ C ′ ω ′ (supp a ) k a k B for al l a ∈ C Γ . Then A Γ and B Γ ar e c ompletely sp e ctr al ly e qu ivalent over C Γ . Pr o of. Let k ≥ 1 . F or a ll ( a ij ) ∈ M k ( C Γ) we hav e analo g ous subexp onential estimates in the matrix a lgebras M k ( A Γ) and M k ( B Γ): k ( a ij ) k B ≤ C ω  supp ( a ij )  k ( a ij ) k A , k ( a ij ) k A ≤ C ′ ω ′  supp ( a ij )  k ( a ij ) k B As ω  supp ( a ij ) n  ≤ ω  (supp ( a ij )) n  , w e get r A  ( a ij )  = r B  ( a ij )  for all ( a ij ) ∈ M k ( C Γ).  Example 8.3. Let Γ b e a gr o up of sub exp onential gr owth. W e claim that ℓ 1 Γ ֒ → C ∗ r Γ is completely sp ectral rela tiv e to C Γ. Indeed, let ω be the sub exp onential weigh t on Γ given by ω ( S ) = p vol B ( S ), where B ( S ) deno tes the ball c en tered at the identit y that circumscrib es S . F or all a ∈ C Γ w e hav e k a k ≤ k a k 1 and k a k 1 ≤ ω (supp a ) k a k 2 ≤ ω (supp a ) k a k , i.e., we are in the conditions o f Pr op osition 8.2. W e infer tha t K ∗ ( ℓ 1 Γ) ≃ K ∗ (C ∗ r Γ). Consider, at this p oint, the following Conjecture. F or any discrete countable group Γ, the inclusion ℓ 1 Γ ֒ → C ∗ r Γ induces a n isomor- phism K ∗ ( ℓ 1 Γ) ≃ K ∗ (C ∗ r Γ). RELA TIVEL Y S PECTRAL MORPHISMS AND APPLICA TIONS TO K-THEOR Y 19 Let us refer to this statemen t as the BB C conjecture, for it co nnects the B ost conjecture with the Baum-C o nnes conjecture. W e find the BBC conjecture to b e a natural question on its own, outside of the Bos t-Baum-Connes cont ext. One knows, by com bining r esults of Higson and Ka sparov on the Baum-Connes side with results of Lafforgue o n the Bost side, that the BBC co njecture holds for g roups with the Haa gerup prop er t y . On the other hand, the sp ectral approach allow ed us to v erify the BBC c o njecture for gro ups of subex po nen tial growth. The severe limitations of the sp ectra l appro ach are made evident by this compar is on. Example 8.4. Let Γ b e a finitely generated group satisfying the Rapid Decay prop erty , i.e., there are cons ta n ts C, d > 0 s uc h that k a k ≤ C k a k 2 ,d for all a ∈ C Γ . The weigh ted ℓ 2 -norm k· k 2 ,d is g iven by   P a g g   2 ,d = p P | a g | 2 (1 + | g | ) 2 d , where | · | denotes the word-length. Examples of groups satisfying the Rapid Decay prop erty include free gro ups [13] and, more gener ally , h yp erb olic gr oups [10], gro ups of p o ly no mial g rowth [15], and many other g roups [5], [11], [17], [22]. Consider the w e ight ed ℓ 2 -space ℓ 2 s Γ =  P a g g :   P a g g   2 ,s < ∞  . F or s > d , ℓ 2 s Γ is a Banach subalgebra of C ∗ r Γ ([17, P rop.1.2]). Lafforgue’s goal is the isomo rphism K ∗ ( ℓ 2 s Γ) ≃ K ∗ (C ∗ r Γ), so he shows that ℓ 2 s Γ is a sp ectra l subalg ebra of C ∗ r Γ. Alternatively , one can ado pt the re la tive p ersp ective. Let ω s be the subexp o nential weigh t on Γ given by ω s ( S ) =  1 + R ( S )  s , where R ( S ) is the radius of the ball centered at the identi ty that circumscr ibes S . F or s > d , w e ha ve k a k ≤ C k a k 2 ,s and k a k 2 ,s ≤ ω s (supp a ) k a k 2 ≤ ω s (supp a ) k a k for all a ∈ C Γ. F rom Prop osition 8.2 it follows that K ∗ ( ℓ 2 s Γ) ≃ K ∗ (C ∗ r Γ). 8.3. Σ 1 -groups. W e consider groups Γ for which the inclusion ℓ 1 Γ ֒ → C ∗ r Γ is sp ectral relative to C Γ. Let us refer to such groups Γ as Σ 1 -gr oups . That is, Γ is a Σ 1 -group if r ℓ 1 Γ ( a ) ≤ k a k for all a ∈ C Γ. The Σ 1 condition is the ℓ 1 analogue o f the “ ℓ 2 -sp ectral ra dius prop erty” discussed in [1 2, Def.1.2 ii) & Sec.3]. The major difference is that the “ ℓ 2 -sp ectral r adius pro p e r ty” is satisfied by Rapid Decay gr oups, e.g. by hyper bo lic gr oups, whereas Σ 1 -groups are necessarily amenable: Prop osition 8. 5. W e have: a) Σ 1 is close d under taking sub gr oups; b) Σ 1 is close d under taking dir e ct e d unions; c) Σ 1 -gr oups ar e amenable; d) Σ 1 -gr oups do not c ontain F S 2 , the fr e e semigr oup on two gener ators; e) gr oups of sub ex p onential gr owth ar e Σ 1 -gr oups. Pr o of. a) An embedding Λ ֒ → Γ induces is ometric e mbeddings ℓ 1 Λ ֒ → ℓ 1 Γ, C ∗ r Λ ֒ → C ∗ r Γ. b) Let Γ b e the directed union of the Σ 1 -groups (Γ i ) i ∈ I . If a ∈ C Γ, then a ∈ C Γ i for s ome i and we know that r ℓ 1 Γ i ( a ) = r C ∗ r Γ i ( a ) can also be read a s r ℓ 1 Γ ( a ) = r C ∗ r Γ ( a ). c) Let Γ b e a Σ 1 -group. The inclusion ℓ 1 Γ ֒ → C ∗ r Γ factor s as ℓ 1 Γ ֒ → C ∗ Γ ։ C ∗ r Γ. It follows that C ∗ Γ ։ C ∗ r Γ is sp ectral rela tive to C Γ, so necessar ily C ∗ Γ ։ C ∗ r Γ is an isomo rphism, i.e., Γ is a menable. Here is another argumen t. B y a), it suffices to s how that Γ is amenable in the ca s e Γ is finitely g e nerated, sa y by a finite symmetric set S . F or a ∈ R + Γ we hav e k a n k 1 = k a k n 1 , so r ℓ 1 Γ ( a ) = k a k 1 . W e obtain r C ∗ r Γ ( χ S ) = r ℓ 1 Γ ( χ S ) = k χ S k 1 = # S . By Ke s ten’s criter ion, Γ is amenable. d) Let Γ b e a Σ 1 -group and assume, on the contrary , t hat x, y ∈ Γ generate a free sub- semigroup. If the supp ort of a ∈ C Γ gener a tes a free subsemigro up then k a n k 1 = k a k n 1 , hence r ℓ 1 Γ ( a ) = k a k 1 . T ogether with the Σ 1 condition, this g ives k a k = k a k 1 for those a ∈ C Γ whos e suppo rt generates a free subsemigroup. Consider now a = 1 + ix + ix − 1 ∈ C Γ. On one hand, the supp ort of ( y x ) a genera tes a free subsemigroup, so k a k = k ( y x ) a k = k ( y x ) a k 1 = k a k 1 = 3. On the o ther hand, we have k a k 2 = k aa ∗ k = k 3 + x 2 + x − 2 k ≤ 5, a contradiction. 20 BOGD AN NICA e) This was prov ed in Example 8.3.  Thu s Σ 1 -groups include the subex p o nential g roups a nd ar e included among amenable gr oups without free subsemigroups. Note that there a re examples, first co nstructed by Gr ig orch uk, of amenable groups o f exp o nent ial gr owth that do not contain free subse mig roups. Remark 8.6 . If one considers the groups Γ for which the inclusion ℓ 1 Γ ֒ → C ∗ Γ is sp e c tr al relative to C Γ , then the analo gues of par ts a), b), d), e) of Pr o p o sition 8.5 still ho ld. After L a fforgue, w e sa y that a Banach algebra completion A Γ of C Γ is an unc onditional c ompletion if the norm k · k A of A Γ satisfies k| a |k A = k a k A for all a ∈ C Γ, where | a | denotes the po in twise absolute v alue. The simplest exa mple of unconditional completion is ℓ 1 Γ. F or groups with the Rapid Decay prop erty , ℓ 2 s Γ ( s ≫ 1) is an unconditiona l completion. In light of [18], one is in terested in finding unco nditional completions A Γ having the same K -theory as C ∗ r Γ. One wa y to achiev e K ∗ ( A Γ) ≃ K ∗ (C ∗ r Γ) would b e to hav e A Γ a nd C ∗ r Γ co mpletely sp ectrally equiv alent ov er C Γ. F or amenable gr oups, how ever, one ca nnot do b etter than Σ 1 -groups: Prop osition 8.7. Le t Γ b e amenable . If ther e is an unc onditional c ompletion A Γ with the pr op erty that A Γ and C ∗ r Γ ar e sp e ctr al ly e quivalent over C Γ , t hen in fact ℓ 1 Γ and C ∗ r Γ ar e sp e ctr al over C Γ , i.e., Γ is a Σ 1 -gr oup. Pr o of. W e first sho w that r ℓ 1 Γ ( a ) = r C ∗ r Γ ( a ) whenever a ∈ C Γ is self-adjoin t. The spectra l equiv alence of A Γ and C ∗ r Γ yields k a k = r C ∗ r Γ ( a ) = r A Γ ( a ) ≤ k a k A for all self-adjoint a ∈ C Γ. Since Γ is amenable, we g et k a k 1 = k| a |k ≤ k| a |k A = k a k A for all self-adjoin t a ∈ C Γ, so k a k ≤ k a k 1 ≤ k a k A for all self-adjoint a ∈ C Γ. It follows that r ℓ 1 Γ ( a ) = r C ∗ r Γ ( a ) for all self-adjoint a ∈ C Γ. Next, one needs to adapt the pro of of P rop osition 3 .2 to the following ∗ -version: Let φ : A → B be a dense ∗ -morphism b et ween Banach ∗ -algebra s, and let X ⊆ A b e a dense ∗ -subalgebra . If r B ( φ ( x )) = r A ( x ) for all self-adjoint x ∈ X , then φ : A → B is sp ectra l rela tive to X . Indeed, let x ∈ X with φ ( x ) inv er tible in B ; we show x inv ertible in A . Pick ( x n ) ⊆ X suc h that φ ( x n ) → φ ( x ) − 1 . Then φ (( xx n )( xx n ) ∗ ) → 1, so r A (1 − ( xx n )( xx n ) ∗ ) = r B  φ (1 − ( xx n )( xx n ) ∗ )  → 0 . It follows that ( xx n )( xx n ) ∗ is inv ertible in A for larg e n . Similarly , ( x n x ) ∗ ( x n x ) is inv er tible in A for la rge n . 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