A Projective C*-Algebra Related to K-Theory

A PR OJECTIVE C ∗ -ALGEBRA RELA TED TO K -THEOR Y TERR Y A. LORING Abstract. The C ∗ -algebra q C is the smallest of the C ∗ -algebras q A intro- duced b y Cuntz [1] in the con text of K K -theory . An important property of q C is the natural isomorphism K 0 ( A ) ∼ = lim → [ q C , M n ( A )] . Our main result concerns the exp onen tial (boundary) map from K 0 of a quo- tien t B to K 1 of an ideal I . W e sho w if a K 0 elemen t is realized in hom( q C , B ) then its b oundary is r ealized as a unitary in ˜ I . The picture we obtain of the exponent ial map is based on a pro jective C ∗ -algebra P that is univ er sal for a set of r elations slightly w eak er than the relations that define q C . A new, shorter pro of of the semipro ject ivity of q C is describ ed. Smoothing questions related the relations f or q C are addressed. 1. Introduction The simplest nonzero pro j ective C ∗ -algebra is C 0 (0 , 1 ] . A quotient of this is C , the simplest nonzero s e mipro jectiv e C ∗ -algebra . The first is universal for the relation 0 ≤ x ≤ 1 and the seco nd for p ∗ = p 2 = p. When lifting a pro jection fro m a quotient, one m ust either settle for a lift tha t is only a po sitive element or confront some K -theo retical obstruction to finding a lift that is a pr o jection. W e consider noncommutativ e analog s of thes e t wo C ∗ -algebra s. W e use ˜ A to denote the unitization of A, where a unit 1 is to b e added ev en in 1 A exists. F or elements h, x and k o f A, we use the notatio n (1) T ( h, x, k ) =  1 − h x ∗ x k  ∈ M 2 ( ˜ A ) . W e will s how that there is a C ∗ -algebra P with gene r ators h, k and x tha t are universal for the re lations hk = 0 , 0 ≤ T ( h, x, k ) ≤ 1 . Moreov er, P is pro jectiv e. This do es not app ear to be a fa milia r C ∗ -algebra , but it has a familia r quotient. The rela tions hk = 0 , T ( h, x, k ) ∗ = T ( h, x, k ) 2 = T ( h, x, k ) hav e as their universal C ∗ -algebra the semipro jectiv e C ∗ -algebra q C = { f ∈ C 0 ((0 , 1 ] , M 2 ) | f (1) is diagonal } . Key wor ds and phr ases. C*-algebras, semipr o j ectivit y , K-theory , boundary map, pro jectivit y , lifting. 1 2 TERR Y A. LORING A complicated pro of of the semipro jectivity of q C , was given in [2]. Subse- quent pro ofs found with Eilers and Pederson in [3] and [4] worked in the context of nonc o mmu tative CW-complexes. Those pro ofs did not utilize the fact that q C is similar to the noncommutativ e Gra ssmannian G nc 2 , c.f. [5]. The pro of here uses this connectio n. The importance o f q C to K - theo ry is illustrated by the isomorphism K 0 ( A ) ∼ = [ q C , A ⊗ K ] ∼ = lim → [ q C , M n ( A )] . F or e x ample, see [1] and [6]. Our main r esult c oncerns the ex po nential (b oundar y) ma p fro m K 0 of a quotien t B to K 1 of an ideal I . If w e look at K 0 as K 0 ( D ) ∼ = lim → [ q C , M n ( D )] then given 0 → I → A → B → 0 we s how that a K 0 element r ealized in hom( q C , B ) has bo undary in K 1 ( I ) that can be realized a s a unitary in the ˜ I . In the final s e c tion we loo k further into methods for p ertur bing approximate representations of the re lations for q C int o true representations, but this time re- stricting ours elves to using only C ∞ -functional ca lculus. Lemma 1 .1. The C ∗ -algebr a q C = { f ∈ C 0 ((0 , 1] , M 2 ) | f (1 ) is diagonal } is un iversal in the c ate gory of al l C ∗ -algebr as for gener ators h, k and x with r elations h ∗ h + x ∗ x = h, k ∗ k + xx ∗ = k , k x = xh, hk = 0 . (2) The c oncr ete gener ators may b e taken to b e h 0 = t ⊗ e 11 , k 0 = t ⊗ e 22 , x 0 = p t − t 2 ⊗ e 21 . Pr o of. This is almost identical to P rop osition 2 .1 in [2]. T o see these are equiv alent, notice first that the top tw o re la tions imply h a nd k ar e pos itive. Since x ∗ x is po sitive, the relation x ∗ x = h − h 2 implies h ≤ 1 . It also implies k x k ≤ 1 2 . Similar ly k ≤ 1 .  Lemma 1.2. T he C ∗ -algebr a q C is u niversal in the c ate gory of al l C ∗ -algebr as for gener ators h, k , x and r elations hk = 0 , T ( h, x, k ) 2 = T ( h, x, k ) ∗ = T ( h, x, k ) . (3) Pr o of. Since T ( h, x, k ) =  1 − h x ∗ x k  and T ( h, x, k ) 2 =  1 − 2 h + h 2 + x ∗ x x ∗ − hx ∗ + x ∗ k x − xh + k x k 2 + xx ∗  , A PROJECTIVE C ∗ -ALGEBRA RELA TED TO K -THEOR Y 3 if we add hk = 0 we have a set o f rela tions equiv a lent to (2).  2. Internal Ma trix Structure s in C ∗ -Algebras Lemma 2.1. Supp ose A is a C ∗ -algebr a and X 11 , X 21 , X 12 , a nd X 22 ar e close d line ar subsp ac es of A. Supp ose X ∗ ij = X j i and X ij X j k ⊆ X ik and X 11 X 22 = 0 . (1) The subset ˆ X =  X 11 X 12 X 21 X 22  is a C ∗ -sub algebr a of M 2 ( A ) . (2) The sum X 11 + X 21 + X 12 + X 22 is a line ar dir e ct sum and is a C ∗ -sub algebr a of A, isomorphic to ˆ X . (3) Ther e is a homotopy θ t of inje ctive ∗ - homomorphi sms θ t : X 11 + X 21 + X 12 + X 22 → M 2 ( A ) so that θ 0 ( x 11 + x 21 + x 12 + x 22 ) =  x 11 + x 21 + x 12 + x 22 0 0 0  and θ 1 ( x 11 + x 21 + x 12 + x 22 ) =  x 11 x 12 x 21 x 22  . Pr o of. An element x ij of X ij factors as x ij = x ii y x j j with y in A and x j j = | x ∗ ij | 1 4 in X j j and x ii = | x ij | 1 4 in X ii . F r om her e, it is ea sy to sho w that X ij X kl = 0 if j 6 = k and that X ij ∩ X kl = 0 when i 6 = k or j 6 = l . It is clea r tha t ˆ X is a C ∗ -subalgebra of M 2 ( A ) . Let w t be a partial isometry in M 2 with | w t | = e 11 for a ll t and w 0 = e 11 and w 1 = e 21 . Define ψ t : ˆ X → A ⊗ M 2 by ψ t  X x ij ⊗ e ij  = X x ij ⊗ f ( t ) ij where f ( t ) 11 = w ∗ t w t , f ( t ) 12 = w ∗ t f ( t ) 21 = w t , f ( t ) 22 = w t w ∗ t . The fact that X ij X kl = 0 if j 6 = k implies that eac h ψ t is a ∗ -homomo rphism. The image of ψ 0 is ( X 11 + X 21 + X 12 + X 22 ) ⊗ e 11 and so w e see that the direct sum of the X ij is a C ∗ -subalgebra o f A. Now s uppo se ψ t  X x ij ⊗ e ij  = 0 . Then for all r a nd all s we hav e 0 =  x ∗ r s ⊗ f ( t ) 1 r    ψ t   X ij x ij ⊗ e ij      x ∗ r s ⊗ f ( t ) s 1  = x ∗ r s x r s x ∗ r s ⊗ e 11 which implies x r s = 0 . Ther efore ψ t is injective. 4 TERR Y A. LORING If we let γ denote the obvious is o morphism γ : X 11 + X 21 + X 12 + X 22 → ( X 11 + X 21 + X 12 + X 22 ) ⊗ e 11 and ι t the inclus ion of ψ t ( ˆ X ) into M 2 ( A ) then θ t = ι t ◦ ψ t ◦ ψ − 1 0 ◦ γ is the desired path of injective ∗ -ho momorphisms.  Lemma 2 .2. Under the hyp otheses of L emma 2.1, the subset  C1 + X 11 X 12 X 21 C1 + X 22  is a C ∗ -sub algebr a of M 2 ( ˜ A ) , and ρ  α 1 + x 11 x 12 x 21 α 1 + x 22  = α ⊕ β determines a surje ction onto C ⊕ C . Pr o of. This is follows easily from Lemma 2.1.  Lemma 2.3. Supp ose I is an ide al in the C ∗ -algebr a A and h and k in A ar e p ositive elements. Then I ∩ k Ah = k I h Pr o of. The sp ecial case where h = k is r outine, and the general case follows via a 2-by-2 matr ix trick.  3. The Exponential Map in K -Theor y W e chose b as the canonical generator of K 0 ( q C ) = Z , where b is formed as the class of the pro jection P 0 = T ( h 0 , x 0 , k 0 ) min us the class of [ 1 ] . (See (1).) Theorem 3 .1. Supp ose 0 I A π B 0 is a short exact se quenc e of C ∗ -algebr as. If x i s any element o f K 0 ( B ) such that x = ϕ ∗ ( b ) for some ∗ -homomorphism ϕ : q C → B , then ∂ ( x ) = [ u ] in K 1 ( I ) for some unitary u ∈ ˜ I . Pr o of. Let y 0 = q t 1 2 − t 3 2 ⊗ e 21 so that y 0 is a cont ra ction and (4) k 1 8 0 y 0 h 1 8 = x 0 . Orthogo nal po sitive con tractio ns lift to or tho gonal p ositive contractions, so we can find h and k in A with π ( h ) = ϕ ( h 0 ) , π ( k ) = ϕ ( k 0 ) a nd hh = 0 , 0 ≤ h ≤ 1 , 0 ≤ k ≤ 1 . A PROJECTIVE C ∗ -ALGEBRA RELA TED TO K -THEOR Y 5 Now ta ke any y in A with π ( y ) = ϕ ( y 0 ) a nd let x = k 1 8 y h 1 8 and T = T ( h, x, k ) . Then π ( x ) = ϕ ( x 0 ) , (5) ˜ π (2) ( T ) = ˜ ϕ (2) ( P 0 ) , (6) T ∈  C1 + hAh hAk k Ah C1 + k Ak  , (7) ρ ( T ) = 1 ⊕ 0 , and T ∗ = T . Let f ( λ ) = max(min( λ, 1) , 0) and let T ′ = f ( T ) . Then equatio ns (5), (6) a nd (7) hold with T ′ replacing T . This means T ′ = T ( h ′ , x ′ , k ′ ) for s o me h ′ , k ′ and x ′ in A that ar e lifts of h, k and x, and tha t h ′ k ′ = 0 , (8) 0 ≤ T ≤ 1 . This is an int eres ting lifting result that w e will r eturn to b elow. F or now, we turn to the expo nential map. Clearly ∂ ([ 1 ]) = 0 so we need only compute ∂ ◦ ϕ ∗ [ P 0 ] . W e hav e the lifts T and T ′ . W e prefer to work with T ′ . A unita r y tha t repr e s ents this K 1 element is U ′ = e 2 π iT ′ . Since ˜ π (2) ( U ′ ) = ˜ ϕ (2)  e 2 π iP 0  =  1 0 0 1  we k now that U ′ ∈  1 0 0 1  +  I I I I  . By (6) w e know U ′ ∈  C1 + hAh hAk k Ah C1 + k Ak  . Putting these facts together w e disco ver U ′ ∈  1 0 0 1  +  hI h hI k k I h k I k  ⊆  hI h hI k k I h k I k  ∼ . By Lemma 2.1, there is a path of unitaries in ( M 2 ( I )) ∼ from U ′ =  u 11 u 21 u 12 u 22  to  − 1 + u 11 + u 12 + u 21 + u 22 0 0 1  . Thu s ∂ ◦ ϕ ∗ ( b ) = ∂ ◦ ϕ ∗ ( P 0 ) is represented in ˜ I by the unitary u = − 1 + u 11 + u 12 + u 21 + u 22 .  6 TERR Y A. LORING Theorem 3 .2. ([2, Theor em 3.9]) q C is semipr oje ctive. Pr o of. The pr o of of Theo rem 3.1 is easily mo dified to give a new pro of of this result. One needs to assume tha t I is the clo sure of the increasing unio n of idea ls in A. After the lift T is obtained in B /I 1 , one can re place I 1 by I n with there now b eing a hole in the spectrum of T around 1 2 . Replacing the r o le of f by (9) f 1 2 ( λ ) =  0 if λ < 1 2 1 if λ ≥ 1 2 , and following the same constr uc tio n, one finds T ′ that is a pr o jection. The comp o- nent s o f T ′ then provide a lift in B /I n that is a representation of the g enerators of q C .  Corollary 3. 3. Ther e is a universal C ∗ -algebr a P for gener ators h , k and x for which hk = 0 , 0 ≤ T ( h, x, k ) ≤ 1 . The surje ction θ : P → q C that sends gener ators to gener ators is pr oje ctive. Pr o of. Once w e show P exists , the pro of of the pro jectivit y of θ is contained in the pro of of Theorem 3.1. By [4] we need only show that these r elations a r e inv aria nt with respect of in- clusions, are natura l, are clo s ed under products, and are repr esented by a list o f zero elements. (This last r equirement was er roneously missing in [4].) See also [7]. Details ar e left to the reader.  Theorem 3 .4. The C ∗ -algebr a P is pr oje ct ive. Pr o of. Since t 2 ≤ t in C 0 ((0 , 1 ]) , the matr ix T = T ( h, x, k ) satisfies T 2 ≤ T . F ro m this w e deduce x ∗ x ≤ h − h 2 . Similarly , xx ∗ ≤ k − k 2 . By [4 , Lemma 2.2.4] w e can factor x a s x = k 1 8 y h 1 8 for some y in P . The rest o f the pro of is iden tical t o argument b etw een equations (4) and (8).  4. Rela tions In this sec tio n we briefly ex a mine a clas s of rela tions so mewhat more complica ted than ∗ -p olynomials . See [7, 4, 8] fo r different approa ches to re la tions in C ∗ -algebra s, Consider sets of r elations of the form f ( p ( x 1 , . . . , x n )) = 0 , either where p is a self-adjoint ∗ - p o lynomial in n noncommuting v ariables with p ( 0 ) = 0 a nd f ∈ C 0 ( R \ { 0 } ) , or where p is not necessa rily self-adjoint, p ( 0 ) = 0 and f is analy tic on the plane. The p oint to restricting to these rela tio ns is that f ( p ( x 1 , . . . , x n )) makes s e ns e, no matter the norm of the C ∗ -elements x j , and so k f ( p ( x 1 , . . . , x n )) k ≤ δ is a common-sense way to define an approximate re pr esentation. A PROJECTIVE C ∗ -ALGEBRA RELA TED TO K -THEOR Y 7 Certainly a s et R o f relations on x 1 , . . . , x n of this restricted form is inv ar iant with resp ect to inclusion, is natural, and each is satisfied when a ll the indeterminants are set to 0 . Therefor e, R will define a univ ers al C ∗ -algebra if a nd only if it b ounded, meaning for all j we hav e sup  k ˜ x j k   ˜ x 1 , . . . , ˜ x n is a represe ntation of R  < ∞ . W e will als o need to use relations of the form (10) g ( q ( f 1 ( p 1 ( x 1 , . . . , x n )) , . . . , f m ( p m ( x 1 , . . . , x n )))) = 0 where the f k , p k and g , q ar e pairs of contin uous functions and ∗ -p olyno mials subscribing to the ab ove rule. In particular this will a llow us the relation k q ( f 1 ( p 1 ( x 1 , . . . , x n )) , . . . , f m ( p m ( x 1 , . . . , x n ))) k ≤ C. F or an y n - tuple of elements in a C ∗ -algebra A w e define r ( x 1 , . . . , x n ) , a gain in A, by r ( x 1 , . . . , x n ) = f ( q ( f 1 ( p 1 ( x 1 , . . . , x n )) , . . . , f m ( p m ( x 1 , . . . , x n )))) . If x 1 , . . . , x n are is a sub- C ∗ -algebra , then so is r ( x 1 , . . . , x n ) . Th us we a re justified in the no ta tion r instead of the mor e p edantic r A . Also r is natural. It is still the c a se that the universal C ∗ -algebra exists if a nd only if the set of relations is bo unded. Lemma 4 .1. Supp ose r k ( x 1 , . . . , x n ) = 0 for k = 1 , . . . , K form a b ounde d set of r elations of the form (10). Supp ose s ( x 1 , . . . , x n ) = 0 is a r elation of the form (10) that holds t rue in U = C ∗ h x 1 , . . . , x n | r k ( x 1 , . . . , x n ) = 0 ( ∀ k ) i . Then for every ǫ > 0 ther e is a δ > 0 so t hat if y 1 , . . . , y n in a C ∗ -algebr a A satisfy k r k ( y 1 , . . . , y n ) k ≤ δ ( ∀ k ) then k s ( y 1 , . . . , y n ) k ≤ ǫ. Pr o of. This follows from s tandard ar guments involving the quotient of an infinite direct pr o duct by an infinite direct sum.  5. Smoothing Rela tions W e now mo dify the tec hniques from Section 3 for a smo oth version of semipro- jectivit y for q C . The res ult is slightly weak er than [2, Theo rem 1 .10], but comes with a more reasona ble proo f. The result inv olves maps from the gener ators of q C to a dense ∗ -subalgebra A ∞ of a C ∗ -algebra A. The additional hypothesis is that M 2 ( A ∞ ) , a nd n ot just A ∞ , is clo sed under C ∞ functional ca lc ulus on se lf- adjoint elemen ts . This additiona l assumption may be no difficulty in exa mples. The smo oth alge br as of Black adar and Cuntz ar e closed under passing to matrix algebra ([9, Prop ositio n 6 .7]). 8 TERR Y A. LORING Lemma 5 .1. If p ∗ = p is an element of a C ∗ -algebr a A and k p 2 − p k = η < 1 4 then, with f 1 2 as in (9), f 1 2 ( p ) is a pr oje ction in A and    f 1 2 ( p ) − p    ≤ η . Pr o of. This is well-kno wn.  Theorem 5.2 . F or every ǫ > 0 , ther e is a δ > 0 so that if A ∞ is a dense ∗ - sub algebr a of a C ∗ -algebr a A for which b oth A ∞ and M 2 ( A ∞ ) ar e close d under C ∞ functional c alculus on self-ad joint elements, then for any h, k and x in A ∞ for which k h ∗ h + x ∗ x − h k ≤ δ, k k ∗ k + xx ∗ − k k ≤ δ, k k x − xh k ≤ δ, k hk k ≤ δ, ther e exist h k and x in A ∞ so that h ∗ h + x ∗ x − h = 0 , k ∗ k + x x ∗ − k = 0 , k x − x h = 0 , h k = 0 , and   h − h   ≤ ǫ,   k − k   ≤ ǫ, k x − x k ≤ ǫ. Pr o of. Let ǫ be given, with 0 < ǫ < 1 4 . Choose θ > 0 so that k h ′ − h ′′ k ≤ θ, k k ′ − k ′′ k ≤ θ , k x ′ − x ′′ k ≤ θ , k h ′ k ≤ 2 , k k ′ k ≤ 2 , k x ′ k ≤ 2 , implies k ( h ′∗ h ′ + x ′∗ x ′ − h ′ ) − ( h ′′∗ h ′′ + x ′′∗ x ′′ − h ′′ ) k ≤ ǫ 8 , k ( k ′∗ k ′ + x ′ x ′∗ − k ′ ) − ( k ′′∗ k ′′ + x ′′ x ′′∗ − k ′′ ) k ≤ ǫ 8 , k ( k ′ x ′ − x ′ h ′ ) − ( k ′′ x ′′ − x ′′ h ′′ ) k ≤ ǫ 8 , Cho ose g + some r eal-v a lued C ∞ function o n R for which t ≤ 0 = ⇒ g + ( t ) = 0 , t ≥ 0 = ⇒ t − θ 2 ≤ g + ( t ) ≤ t, and let g − ( t ) = g + ( − t ) . Choose q + some r eal-v a lued C ∞ functions on R for which t ≤ 0 = ⇒ q + ( t ) = 0 , t ≥ 0 = ⇒ p t − t 2 − θ 2 ≤ ( q + ( t )) 2 p t − t 2 ≤ p t − t 2 , and let q − ( t ) = q + ( − t ) . A PROJECTIVE C ∗ -ALGEBRA RELA TED TO K -THEOR Y 9 Inside q C , let w e hav e g +  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  = g + ( t ) ⊗ e 11 , and g −  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  = g + ( t ) ⊗ e 22 and q −  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  x 0 q +  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  = ( q + ( t )) 2 p t − t 2 ⊗ e 21 . Therefore     g +  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  − h 0     ≤ θ 2 ,     g −  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  − k 0     ≤ θ 2 ,     q −  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  x 0 q +  1 2 ( h 0 + h ∗ 0 − k 0 − k ∗ 0 )  − x 0     ≤ θ 2 . Of cour se, we also kno w k h 0 k ≤ 1 , k k 0 k ≤ 1 , k x 0 k ≤ 1 2 , Lemma 4.1 tells us ther e is a δ > 0 so that if h , k and x are in a C ∗ -algebra A with k h ∗ h + x ∗ x − h k ≤ δ, k k ∗ k + xx ∗ − k k ≤ δ, k k x − xh k ≤ δ, k hk k ≤ δ then     g +  1 2 ( h + h ∗ − k − k ∗ )  − h     ≤ θ,     g −  1 2 ( h + h ∗ − k − k ∗ )  − k     ≤ θ ,     q −  1 2 ( h + h ∗ − k − k ∗ )  xq +  1 2 ( h + h ∗ − k − k ∗ )  − x     ≤ θ, k h k ≤ 2 , k k k ≤ 2 , k x k ≤ 2 . If nece ssary , replace δ with a smaller num b er to ensure δ < ǫ 2 . Let ˜ h = f +  1 2 ( h + h ∗ − k − k ∗ )  , ˜ k = f −  1 2 ( h + h ∗ − k − k ∗ )  , h 2 = g +  1 2 ( h + h ∗ − k − k ∗ )  , k 2 = g −  1 2 ( h + h ∗ − k − k ∗ )  , 10 TERR Y A. LORING and x 2 = q −  1 2 ( h + h ∗ − k − k ∗ )  xq +  1 2 ( h + h ∗ − k − k ∗ )  . First notice that ˜ h and ˜ k ar e orthog o nal po sitive elemen t of A. Since q +  1 2 ( h + h ∗ − k − k ∗ )  is in the C ∗ -algebra ge nerated by ˜ h, and q −  1 2 ( h + h ∗ − k − k ∗ )  is in the C ∗ -algebra g enerated by ˜ k , we hav e x 2 ∈ ˜ k A ˜ h. Similarly , h 2 ∈ ˜ k A ˜ h and k 2 ∈ ˜ kA ˜ k . Next, observe that h 2 , k 2 and x 2 are in A ∞ , with h 2 and k 2 self-adjoint and k h 2 − h k , k k 2 − k k , k x 2 − x k ≤ θ. Therefore k ( h ∗ 2 h 2 + x ∗ 2 x 2 − h 2 ) − ( h ∗ h + x ∗ x − h ) k ≤ ǫ 8 , k ( k 2 k ∗ 2 + x 2 x ∗ 2 − k 2 ) − ( k k ∗ + xx ∗ − k ) k ≤ ǫ 8 , k ( k 2 x 2 − x 2 h 2 ) − ( k x − xh ) k ≤ ǫ 8 and so   h 2 2 + x ∗ 2 x 2 − h 2   ≤ δ + ǫ 8 ≤ ǫ 4 ,   k 2 2 + x 2 x ∗ 2 − k 2   ≤ δ + ǫ 8 ≤ ǫ 4 , k k 2 x 2 − x 2 h 2 k ≤ δ + ǫ 8 ≤ ǫ 4 . Let T 2 = T ( h 2 , x 2 , k 2 ) ∈ " C1 + ˜ hA ˜ h ˜ hA ˜ k ˜ k A ˜ h ˜ k A ˜ k # . With ρ as in Lemma 2.2 ρ ( T 2 ) = 1 ⊕ 0 . Since k T 2 2 − T 2 k =      − h 2 + h 2 2 + x ∗ 2 x 2 x ∗ 2 k 2 − h 2 x ∗ 2 k 2 x 2 − x 2 h 2 − k 2 + k 2 2 + x x x ∗ 2      we have k T 2 2 − T 2 k ≤ ǫ 2 . Let P = f 1 2 ( T 2 ) and define h, k and x via T ( h, x , k ) = P. As in the pro o f of Theorem 3.1 we see that x 3 , k 3 and x 3 satisfy the r e la tions for q C . Since f 1 2 is smo oth o n interv als containing the spectrum of T 2 , these are elements of A ∞ .  A PROJECTIVE C ∗ -ALGEBRA RELA TED TO K -THEOR Y 11 References [1] J. Cun tz, A new lo ok at K K -theory , K -Theory 1 (1) (1987) 31–51. [2] T. A. Lori ng, Perturbation questions i n the Cunt z pi cture of K -theory , K -Theory 11 (2) (1997) 161–193. [3] S. Ei lers, T. A. Loring, G. K. Pedersen, Stability of anticomm utation relations: an application of noncommut ative CW complexes, J. Reine Angew. Math. 499 (1998) 101–143. [4] T. A. Loring, Lifting solutions to perturbing problems in C ∗ -algebras, V ol. 8 of Fields Institute Monographs, American Mathematical So ciety , Providence, RI, 1997. [5] L. G. Brown, Ext of certain free product C ∗ -algebras, J. Op erator Theory 6 (1) (1981) 135–141. [6] M. D˘ ad˘ arlat, T. A. Loring, K -homology , asymptotic represen tations, and unsusp ended E - theory , J. F unct. Anal. 126 (2) (1994) 367–38 3. [7] D. Hadwin, L. Kaonga, B. Mathes, Noncommut ative contin uous functions, J. Korean Math. Soc. 40 (5) (2003) 789–830. [8] N. C. Phil lips, Inv erse limits of C ∗ -algebras and applications, in: Oper ator algebras and applications, V ol. 1, V ol. 135 of London Math. Soc. Lecture Note Ser., Cam bridge Univ. Press, Camb ri dge, 1988, pp. 127–185. [9] B. Black adar, J. Cun tz, Di fferen tial Banach algebra norms and smo oth subalgebras of C ∗ - algebras, J. Op erator Theory 26 (2) (1991) 255–282 . E-mail addr ess : loring@mat h.unm.edu Dep ar tment of Ma thema tics an d St a tistics, University of New Mexico, Alb uquerque, NM 871 31, USA. URL : http://www. math.unm.edu/ ~loring