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THE HOMOTOPY LIFTING THEOREM FOR SEMIPROJECTIVE C

-ALGEBRAS

BRUCE BLACKADAR

Abstract

We prove a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semipro- jectiveC-algebras. We also obtain some other results about semiprojectiveC-algebras: a partial lifting theorem with specified quotient, a lifting result for homomorphisms close to a liftable ho- momorphism, and that sufficiently close homomorphisms from a semiprojectiveC-algebra are homotopic.

1. Introduction

It seems obligatory in any exposition of the theory of Absolute Neighborhood Retracts (ANR’s) in topology to refer to the Borsuk Homotopy Extension Theorem as “one of the most important results in the theory of ANR’s” (as well it is).

Theorem1.1 (Borsuk Homotopy Extension Theorem [5], [6, 8.1]).LetX be an ANR,Y a compact metrizable space,Z a closed subspace ofY, t) (0t ≤ 1) a uniformly continuous path of continuous functions fromZ to X(i.e.h(t, z)= φt(z)is a homotopy fromφ0to φ1). Supposeφ0extends to a continuous functionφ¯0from Y to X. Then there is a uniformly continuous pathφ¯t of extensions of theφtto functions fromYtoX(i.e.h(t, y)¯ = ¯φt(y)is a homotopy fromφ¯0toφ¯1).

In particular, any function fromZtoXhomotopic to an extendible func- tion is extendible. The theorem also works for metrizable spaces which are not necessarily compact when phrased in the homotopy language; we have stated it in the version which can potentially be extended to noncommutative C-algebras. The theorem can be regarded as giving a “universal cofibration property” for maps into ANR’s.

There is a direct analog of (compact) ANR’s in the category of (separable) noncommutativeC-algebras: the semiprojectiveC-algebras [2], [1, II.8.3.7].

Many results about ANR’s carry through to semiprojectiveC-algebras with essentially identical proofs (just “turning arrows around”). However, Borsuk’s

Received 5 November 2013.

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proof of the Homotopy Extension Theorem is not one of these: the proof simply does not work in the noncommutative case. The underlying reason is that in a metrizable space, every closed set is aGδ, but this is false in the primitive ideal space of a separable noncommutativeC-algebra in general.

We can, however, by a different argument obtain a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semiprojective C- algebras (Theorem 5.1). In the course of the proof we obtain some other results about semiprojectiveC-algebras which are of interest: a partial lifting the- orem with specified quotient (Theorem 3.1), a lifting result for homomorph- isms close to a liftable homomorphism (Theorem 4.1), and that sufficiently close homomorphisms from a semiprojectiveC-algebra are homotopic (Co- rollary 4.3).

2. The general Chinese Remainder Theorem

We will make use of a general “folklore” result from ring theory, which can be called the Generalized Chinese Remainder Theorem. Although this result should probably be one of the standard isomorphism theorems for rings, it is not covered in most algebra texts, so we give the simple proof. A variant can be found in [10, Prop. 3.1], with the same proof.

Proposition2.1.LetRbe a ring, andIandJ (two-sided)ideals inR. Then the mapφ:a(amodI, amod J )gives an isomorphism fromR/(IJ ) onto the fibered product

P =(R/I )12)(R/J )= {(x, y):xR/I, yR/J, π1(x)=π2(y)}

(R/I )(R/J )

whereπ1:R/IR/(I +J )andπ2:R/JR/(I +J )are the quotient maps.(P is the pullback of(π1, π2).)

Proof. It is obvious thatφ (regarded as a map fromRto P) is a homo- morphism with kernel IJ. We need only show thatφ is surjective. Let (x, y)P. WriteπI andπJ for the quotient maps fromRtoR/I andR/J respectively. Then there is abRwithπJ(b)=y. We have

π1(xπI(b))=π1(x)π1I(b))=π1(x)π2(y)=0

and the kernel ofπ1is exactlyπI(J ), so there is acJwithπI(c)=xπI(b).

Set a = b+ c. Then πI(a) = x and πJ(a) = πJ(b)+πJ(c) = y. Thus φ(a)=(x, y).

In particular, to define a homomorphism from another ring intoR/(IJ ), it suffices to give a compatible pair of homomorphisms intoR/I andR/J.

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There is, of course, a version of this result for finitely many ideals, but it is somewhat complicated to state. The usual Chinese Remainder Theorem is the special case whereI+J =R; the fibered product is then just the full direct sum.

To apply this result toC-algebras, note that ifI andJ are closed ideals in aC-algebra, thenI+J is also closed (see e.g. [1, II.5.1.3]). One can replace

“homomorphism” with “∗-homomorphism” throughout. (When working with C-algebras, we will take “homomorphism” to mean “∗-homomorphism.”) 3. Partial liftings with specified quotient

Recall the definition of semiprojectivity ([2], [1, II.8.3.7]): a separableC- algebraAissemiprojectiveif, wheneverB is aC-algebra, (Jn)an increas- ing sequence of closed (two-sided) ideals ofB, andJ = [∪Jn], then any homomorphism φ:AB/J can be partially lifted to a homomorphism ψ:AB/Jn for some sufficiently largen. But suppose in the above situ- ation, withAsemiprojective, we also have another closed idealI ofBand a homomorphismφ˜ fromAtoB/I such thatφ andφ˜ agree modI +J. Can we partially liftφtoψ so thatψ agrees withφ˜ modI +Jn? The next result shows that this is always possible. For any closed idealKofB, writeπKfor the quotient map toB/K(by slight abuse of notation, this same symbol will be used for the quotient map fromB/LtoB/Kfor any closed idealLcontained inK.)

Theorem 3.1 (Specified Quotient Partial Lifting Theorem). Let A be a semiprojectiveC-algebra,B aC-algebra,(Jn)an increasing sequence of closed ideals ofBwithJ =[∪Jn],Ianother closed ideal ofB, andφ:AB/J andφ:˜ AB/I-homomorphisms withπI+Jφ = πI+J ◦ ˜φ. Then for some sufficiently largenthere is a-homomorphismψ:AB/Jnsuch thatπJψ =φandπI+Jnψ =πI+Jn◦ ˜φ.

Pictorially, we have the diagram in Figure 1 which can be made to commute.

Proof. It is obvious that∪n(I+Jn)is dense inI+J. It is not obvious that∪n(IJn)is dense inIJ, but this can be proved using [1, II.5.1.3]: if xIJ, then

0=inf

n

inf

yJnxy

=inf

n

inf

zIJnxz .

By Proposition 2.1,φandφ˜define a homomorphismφ¯ fromAtoB/(IJ ), which partially lifts to a homomorphismψ¯ fromAtoB/(IJn)for somen by semiprojectivity. The mapψ¯ defines compatible homomorphismsψ:A

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ψ φ˜

φ

...

...

...

... B/(I + Jn)

B/(I + J) B/(I + J2) B/(I + J1)

B/I

B/Jn

B/J A

B/J2 B/J1 B

Figure1.

B/Jnandψ˜:AB/I. Then

˜

ψ =πIψ :AB/(IJn)B/I

=πIπIJψ :AB/(IJn)B/(IJ )B/I

=πIφ :AB/(IJ )B/I

= ˜φ.

Since

πI+Jnψ:AB/JnB/(I +Jn)

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equals

πI+Jn ◦ ˜ψ =πI+Jn ◦ ˜φ:AB/IB/(I +Jn)

we have thatψ is the desired partial lift ofφ.

4. Lifting close homomorphisms

IfAandBareC-algebras andIis a closed ideal ofB, then a homomorphism from Ato B/I need not lift in general to a homomorphism from A to B, even ifAis semiprojective. But supposeφ:AB/I does lift toφ:¯ AB, andψ is another homomorphism fromAto B/I which is close toφ in the point-norm topology. IfAis semiprojective, doesψalso lift toB, and can the lift be chosen close toφ¯in the point-norm topology? The answer is yes in the commutative category [6, 3.1], but the commutative proof does not generalize to the noncommutative case. However, we can by a different argument obtain the same result for general semiprojectiveC-algebras.

Theorem4.1.LetAbe a semiprojectiveC-algebra generated by a finite or countable setG = {x1, x2, . . .}withlimj→∞xj =0ifGis infinite. Then for any >0there is aδ > 0such that, wheneverB is aC-algebra,I a closed ideal ofB,φ andψ-homomorphisms fromAtoB/I withφ(xj)ψ(xj) < δfor allj and such thatφ lifts to a-homomorphismφ:¯ AB (i.e.πI ◦ ¯φ = φ), thenψ also lifts to a-homomorphismψ:¯ AB with ¯ψ(xj)− ¯φ(xj) < for all j.(The δ depends on , A, and the set G of generators, but not on theB,I,φ,ψ.)

Proof. Suppose the result is false. Then there is an >0 andBn,In, and φn,ψnhomomorphisms fromAtoBn/Insuch thatφn(xj)ψn(xj)<1/n for allj,φnlifts toφ¯n:ABn, butψndoes not lift to anyψ¯n:ABnwith ¯φn(xj)− ¯ψn(xj) < for allj. LetB =

nBn, I =

nIn,Jnthe ideal of elements of B vanishing after then’th term, J = [∪Jn] = ⊕nBn. Let

¯

φ:ABbe defined by

¯

φ(x)=¯1(x),φ¯2(x), . . .)

and letφ = πJ ◦ ¯φ:AB/J. There is also a homomorphismφ˜ fromAto B/I ∼=

n(Bn/In)defined by

˜

φ(x)=1(x), ψ2(x), . . .).

We have limn→∞φn(x)ψn(x) = 0 for allx in a dense∗-subalgebra of A(the∗-subalgebra generated byG) and, since theφnandψnare uniformly bounded (all have norm 1), we have limn→∞φn(x)ψn(x) = 0 for all xA. So we have thatφandφ˜ agree modI +J. Thus by Theorem 3.1, for

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somen, there is a liftψ ofφtoB/Jn agreeing withφ˜ modI+Jn. This lift definesψ¯k:ABk for eachk > nliftingψk. Fixm such thatxj < /2 for allj > m. Sinceψ =φmodJ, we have limk→∞ ¯φk(xj)− ¯ψk(xj) =0 for allj. Thus there is ak such that

¯φk(xj)− ¯ψk(xj)<

for 1≤jm. Ifj > m, we have

¯φk(xj)− ¯ψk(xj) ≤ ¯φk(xj) + ¯ψk(xj) ≤2xj< . Thus ¯φk(xj)− ¯ψk(xj)< for allj, a contradiction.

The diagram at the end of Theorem 3.1 summarizes the construction.

As in the commutative case (cf. [8, IV.1.1], [9, 4.1.1]), we obtain that suffi- ciently close homomorphisms from a semiprojectiveC-algebra are homotopic (see [2, 3.6] for a slightly weaker version of this result with a more elementary proof):

Corollary4.2.LetAbe a semiprojectiveC-algebra generated by a finite or countable setG = {x1, x2, . . .}withlimj→∞xj =0ifGis infinite. Then for any >0there is aδ >0such that, wheneverBis aC-algebra,φ0and φ1-homomorphisms fromAtoBwithφ0(xj)φ1(xj)< δfor allj, then there is a point-norm continuous path(φt)(0t ≤1) of∗-homomorphisms fromAtoB connectingφ0andφ1withφt(xj)φ0(xj) < for allj for anyt ∈[0,1].(Theδdepends on,A, and the setGof generators, but not on theB,φ01.)

In fact, for any > 0, a δ that works for Theorem 4.1 also works for Corollary 4.2.

Proof. Chooseδ >0 as in Theorem 4.1 for the given. LetB˜ =C([0,1], B),I = C0((0,1), B)the ideal of elements ofB˜ vanishing at 0 and 1. Then

˜

B/I ∼=BB. Defineφ, ψ:A→ ˜B/Ibyφ(x)=0(x), φ0(x))andψ(x)= 0(x), φ1(x)). Thenφandψsatisfy the hypotheses of Theorem 4.1, andφlifts toB˜ as a constant function, soψ also lifts, and the lift satisfies the conclusion of Theorem 4.1.

Corollary 4.3. Let A be a semiprojective C-algebra generated by a finite or countable setG= {x1, x2, . . .}withlimj→∞xj =0ifGis infinite.

Then there is aδ > 0such that, wheneverB is aC-algebra,φ0andφ1- homomorphisms fromAtoB withφ0(xj)φ1(xj) < δfor allj, thenφ0 andφ1are homotopic.(Theδdepends onAand the setG of generators, but not on theB,φ01.)

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Proof. Fix any >0, say=1, and apply Corollary 4.2.

In the proofs of the commutative versions of these results, a metric is fixed on the space and theδdepends onand the choice of metric. Fixing a set of generators can be regarded as an analog of fixing a metric in our setting.

5. The Homotopy Lifting Theorem

We can now state and prove theC-analog of the Borsuk Homotopy Extension Theorem. When arrows are turned around for theC-algebra setting, extension problems become lifting problems.

Theorem5.1 (Homotopy Lifting Theorem).LetAbe a semiprojectiveC- algebra,BaC-algebra,Ia closed ideal ofB,(φt) (0t ≤1)a point-norm continuous path of-homomorphisms fromAto B/I. Supposeφ0 lifts to a

-homomorphismφ¯0:AB, i.e.πI ◦ ¯φ0 = φ0. Then there is a point-norm continuous path(φ¯t) (0t ≤1)of-homomorphisms fromAtoBbeginning atφ¯0such thatφ¯t is a lifting ofφt for eacht, i.e. the entire homotopy lifts. In particular,φ1lifts to a-homomorphism fromAtoB.

Proof. LetG = {x1, x2, . . .}be a countable set of generators forA, with xj → 0 (by definition, a semiprojective C-algebra is separable, hence countably generated). Fix > 0, say = 1, and fixδ > 0 satisfying the conclusion of Theorem 4.1 for,A,G. Choose a finite partition 0=t0< t1<

t2 <· · ·< tm= 1 such thatφs(xj)φt(xj)< δfor alljwhenevers, t ∈ [ti1, ti] for anyi. There is such a partition since one only needs to consider finitely manyxj, the condition being automatic for anyxj withxj < δ/2;

cf. the last part of the proof of Theorem 4.1.

Begin with [0, t1]. LetB˜ =C([0, t1], B), andJ the ideal ofB˜ consisting of functionsf: [0, t1]→I withf (0)=0. Then

˜

B/J ∼=C([0, t1], B/I )⊕πI B

= {(f, b)C([0, t1], B/I )⊕B :f (0)=πI(b)}.

Define homomorphisms φ, ψ:A → ˜B/J by setting φ(x) = (fx¯0(x)), wherefx(t)= φ0(x)for allt, andψ(x)=(gx¯0(x)), wheregx(t)=φt(x) for allt. We then have

φ(xj)ψ(xj)< δ

for allj. Sinceφlifts to a∗-homomorphism fromAtoB˜ (e.g. by the constant functionφ¯0),ψ also lifts, defining a continuous path of lifts¯t)of theφt for 0≤tt1.

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Now repeat the process on [t1, t2], using the lift φ¯t1 as the starting point, and continue through all the intervals. After a finite number of steps the entire homotopy is lifted.

Corollary5.2.LetAbe a semiprojectiveC-algebra,BaC-algebra,I a closed ideal ofB,φ a-homomorphism fromAtoB/I. Ifφis homotopic to a-homomorphism fromAtoB/I which lifts toB, thenφlifts toB.

This corollary gives an arguably simpler proof than the one in [12] that a contractible semiprojectiveC-algebra is projective, since the zero homo- morphism always lifts from any quotient. (The result in [12] is slightly more general).

6. -open and -closed C-algebras

In this section, allC-algebras will be assumedseparable. We will useC to denote a category of separableC-algebras and∗-homomorphisms, e.g. the category of all separableC-algebras and∗-homomorphisms, the category of separable unitalC-algebras and unital∗-homomorphisms, or the category of separable unital commutativeC-algebras and unital∗-homomorphisms.

IfAandBareC-algebras, denote by Hom(A, B)the set of∗-homomor- phisms fromAtoB, endowed with the point-norm topology. Hom(A, B)is separable and metrizable. IfAandBare unital, let Hom1(A, B)be the set of unital∗-homomorphisms fromAtoB. Then Hom1(A, B)is a clopen subset of Hom(A, B)(since a projection close to the identity in aC-algebra is equal to the identity).

IfA=C(X)andB =C(Y ), then Hom1(A, B)is naturally homeomorphic toXY, the set of continuous functions fromYtoX, endowed with the topology of uniform convergence (with respect to any fixed metric onX, or with respect to the unique uniform structure onXcompatible with its topology).

More generally, ifC is a category ofC-algebras, denote by HomC(A, B) the morphisms inC, with the point-norm topology (i.e. the subspace topology from Hom(A, B)).

IfC is a category ofC-algebras,A, BC, andI is a closed ideal ofB compatible withC (i.e.B/IC and the quotient mapπI is a morphism in C; this is automatic in the three categories above), denote by HomC(A, B, I ) the set ofC-morphisms fromAtoB/I which lift toC-morphisms fromAto B. HomC(A, B, I )is a subset of HomC(A, B/I ).

IfC is the category of separable unital commutativeC-algebras andA= C(X), B = C(Y ), withX,Y compact metrizable spaces,I corresponds to a closed subsetZ ofY andB/I ∼= C(Z); then HomC(A, B, I )is the subset XZY of XZ consisting of maps (continuous functions) fromZ to X which

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extend to maps fromY toX. See the companion article [4] for a discussion of this case.

Examples show that HomC(A, B, I )is neither open nor closed in HomC(A, B/I )in general (see [4] for commutative examples). We seek conditions onA insuring that HomC(A, B, I )is always open or closed in HomC(A, B/I )for anyBandI.

Definition6.1. LetC be a category, andAC.

(i) Ais-open(inC) if, for every pair(B, I )inC, the set HomC(A, B, I ) is open in HomC(A, B/I ).

(ii) Ais-closed(inC) if, for every pair(B, I )inC, the set HomC(A, B, I ) is closed in HomC(A, B/I ).

IfC is the category of all separable C-algebras, we just say Ais -open [-closed].

IfC is the category of separable unital commutativeC-algebras andA= C(X), thenAis-open [-closed] inC if and only ifXise-open [e-closed]

in the sense of [4]. (Theandestand forliftableandextendiblerespectively, the dual notions in the algebra and topology contexts.)

The next result is an immediate corollary of Theorem 4.1:

Corollary 6.2.Every semiprojective C-algebra is both-open and- closed.

Proof. One only needs to observe that if G = {x1, x2, . . .} is a set of generators forAwithxj → 0, and φn,φ ∗-homomorphisms fromAto a C-algebra B, then φnφ in the point-norm topology if and only if, for every > 0, there is annsuch thatφk(xj)φ(xj) < for allj, for all k > n. Apply Theorem 4.1.

IfC is the category of separable unital commutativeC-algebras andA= C(X), then it is shown in [4] thatAis-open inC if and only ifXis an ANR, at least if Ais finitely generated (equivalently, if X is finite-dimensional).

Recall thatAis semiprojective in C if and only if X is an ANR [2]. Thus it is reasonable to conjecture that aC-algebra is -open if and only if it is semiprojective, at least if it is finitely generated.

Although there is no obvious direct proof that an-openC-algebra is- closed, I do not know an example of a C-algebra which is -open but not -closed, and I conjecture that none exist. There are -closed C-algebras which are not-open, as Example 6.3 shows. I do not have a good idea how to characterize-closedC-algebras.

We conclude with some examples ofC-algebras which are not-open.

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Example6.3. (AC-algebra which is-closed but not-open.) LetAbe the universalC-algebra generated by a sequence of projections{p1, p2, . . .}, i.e.Ais the full free product of a countable number of copies ofC. ThenAis not -open: letB=C([0,1]),I =C0((0,1)).B/I ∼=CC. Defineφn:AB/I byφn(pk) = (0,0)ifkn,φn(pk) = (0,1)ifk > n. Thenφnconverges point-norm to the zero homomorphism fromAtoB/I, which obviously lifts toB, but noφnlifts toB. (This shows thatAis not semiprojective, which can also be shown by a direct argument.)

Ais, however,-closed. LetBbe aC-algebra andI a closed ideal ofB.

A sequencen)of homomorphisms fromAto B/I converging point-norm toφdefines a setqk(n) = φn(pk), qk =φ(pk)of projections inB/I such that qk(n)qk for allk. If each φn is liftable toB, i.e. each qk(n) is liftable to a projection inB, it then follows from the semiprojectivity ofCand Theorem 4.1 that eachqk is also liftable to a projection inB, i.e.φis liftable toB.

A similar argument shows that a full free product of a sequence of semipro- jectiveC-algebras is always-closed, although it is not semiprojective unless all but finitely many of theC-algebras are projective; does the latter condition also characterize when the free product is-open? (This seems likely.)

Example 6.4. Let A = C(F), the full group C-algebra of the free group on infinitely many generators, i.e. the universalC-algebra generated by a sequence of unitaries{u1, u2, . . .}. It is known thatAis not semiprojective ([3], [1, II.8.3.16(vii)]). To directly showAis not-open, letSbe the unilateral shift onH = 2, andB = T theC-subalgebra ofL(H)generated by S (the Toeplitz algebra). ThenB containsI = K(H), andB/I ∼=C(T). Lets be the image ofSinB/I. It is well known thats has no normal preimage in B, in fact no normal preimage inL(H), cf. [7]; in particular, it has no unitary preimage inB. Defineφn:AB/I by settingφn(uk) = 1 forknand φn(uk)=s fork > n. Thenφnφ point-norm, whereφ(uk)= 1 for allk, andφlifts toB, but noφnlifts.

An argument similar to the one in Example 6.3, using semiprojectivity of C(T), shows thatAis-closed in the category of separable unitalC-algebras and unital∗-homomorphisms. (More generally, a full unital free product of a sequence of unital semiprojectiveC-algebras is-closed in the unital cat- egory.) However, it seems like a difficult and delicate question whetherAis -closed (in the general category). For a sequence of homomorphisms fromA toB/I defines a convergent sequence(qn)of projections inB/I (the images of the identity ofA) and a sequence of unitaries inqn(B/I )qnfor eachn. The qnand the unitaries must be lifted in a compatible way to obtain a lifting of the limit projection and unitaries.

So: IsA -closed?

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Example 6.5. LetAbe the universal C-algebra generated by a normal elementx of norm≤1. ThenA∼= C0(D\ {(0,0)}), the functions vanishing at(0,0)on the closed unit diskDinR2. To show thatAis not-open, letB, I,S,sbe as in Example 6.4. Defineφn:AB/I by sendingxto 1ns. Then n)converges in the point-norm topology to the zero homomorphism, which obviously lifts toB. But noφnlifts.

Showing thatAis-closed is the same as solving (positively) the following problem: if(yn)is a convergent sequence of normal elements in a quotient B/I with limity, and eachyn lifts to a normal element inB, doesyalso lift to a normal element? This appears to be unknown.

If this argument works, it can be slightly modified to show that the unit- izationC(D) is-closed but not-open. In fact, it seems reasonable that if Xis any ANR, thenC(X)is-closed, but it is-open if and only ifC(X)is semiprojective, i.e. if and only if dim(X)≤1 [11].

Example6.6. Consider theC-algebrascof convergent sequences of com- plex numbers andc0of sequences of complex numbers converging to 0.

To show they are not-open, letB=C([0,1])andIthe ideal of functions which vanish at 1/nfor alln(and hence of course also at 0). ThenB/I ∼= c.

Defineφn:cB/Iby setting [φn(x)](1/k)=αkifk > n, [φn(x)](1/k)=α ifkn, [φn(x)](0) = α, for x = 1, α2, . . .)c with αnα. Then φnφin the point-norm topology, whereφ(x)is the constant function with value α. Then φ lifts to B, but no φn lifts to B sinceB has no non-trivial projections. The restrictions ofφn,φtoc0work the same way.

The question whether cand c0 are-closed is much more involved than in the commutative case. It is relatively easy to show they are-closed in the commutative category (cf. [4]); the commutative case is simpler since

(i) Close projections in a commutativeC-algebra are actually equal.

(ii) A product of two commuting projections is a projection. In particular, if qis a projection in a quotientB/I, withBcommutative, andp1, p2are two projection lifts toB, thenp=p1p2is also a projection lift toBwith pp1,pp2. Nothing like this is true for general noncommutativeB.

A∗-homomorphism fromc0to aC-algebraBis effectively the same thing as a specification of a sequence of mutually orthogonal projections(pk)inB (some of which may be 0): such a sequence defines a homomorphismφby

φ((α1, α2, . . .))=

k=1

αkpk

(the sum converges inBsinceαk→0). For a homomorphism fromctoB, we additionally need a projectionpsuch thatpkpfor alln: the homomorphism

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corresponding to such a set of projections is defined by φ((α1, α2, . . .))=αp+

k=1

kα)pk

whereα=limk→∞αk. Ifn)is a sequence of homomorphisms corresponding to(pk(n), p(n)), andφis another homomorphism corresponding to(pk, p), then φnφin the point-norm topology if and only if limn→∞p(n)k =pkfor each kand limn→∞p(n) =p.

Now supposeBis aC-algebra andIa closed ideal ofB, andφn, φ:cB/Iwithφnφ. Letφncorrespond to(qk(n), q(n))andφto(qk, q). Suppose eachqk(n) lifts to a projection inB. We need to find projections(pk, p)inB with thepkmutually orthogonal,pkpfor allk,πI(pk)=qk for allk, and πI(p) =q. It seems technically difficult, if not impossible, to show that this can be done.

So: Arecandc0-closed?

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DEPARTMENT OF MATHEMATICS/0084 UNIVERSITY OF NEVADA, RENO RENO, NV 89557

USA

E-mail:bruceb@unr.edu

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