Projection lifting from the corona algebra using UCT

J. Math. Anal. Appl. 442 (2016) 259–266

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Projection lifting from the corona algebra using UCT Hyun Ho Lee Department of Mathematics, University of Ulsan, Ulsan, 44610, South Korea

a r t i c l e

i n f o

Article history: Received 8 December 2015 Available online 19 April 2016 Submitted by H. Lin Keywords: Corona algebra Projection lifting Kirchberg algebras C(X)-algebras

a b s t r a c t In this paper we consider the projection lifting from the corona algebra of C(X) ⊗ B where X is a compact, Hausdorff, finite dimensional space and B is either the algebra of compact operators on a separable infinite dimensional Hilbert space or a stable Kirchberg algebra whose multiplier algebra has real rank zero. Assuming a geometric condition on the projection, we suggest a new approach to the projection lifting using major classification results of continuous fields of C ∗ -algebras. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In this paper we study the projection lifting problem from the corona algebra of C(X) ⊗ B to the multiplier algebra. For a simple C ∗ -algebra B itself, the projection lifting from a quotient algebra of B can be observed in the literature of Elliott’s classification program. But little has been known for the projection lifting from the quotient algebra of a non-simple C ∗ -algebra C(X) ⊗ B. Recently, in the case B = K the algebra of compact operators on an infinite dimensional Hilbert space and some connected spaces X of covering dimension one, L. Brown and the author completely characterized the necessary and sufficient conditions for the lifting using the classical Brown–Douglas–Fillmore’s essential codimension in [4]. Theorem 1.1. Let K be the algebra of compact operators on a separable infinite dimensional Hilbert space, and X be [0, 1], [0, ∞), (−∞, ∞) or [0, 1]/{0, 1}. Let p be a projection in the corona algebra of C(X) ⊗ K, which is represented by a family of projection valued functions (p0 , . . . , pn ) defined on closed intervals Xi ’s such that Xi−1 ∩ Xi = {xi } for i = 1, . . . , n. Let ki be essential codimensions from pi and pi−1 for i = 1, . . . , n. Then p is liftable if and only if there exist li ’s such that (1) li − li−1 = ki for i > 0 and l0 − ln = −k0 in the circle case. (2) If for some x in Xi , pi (x) has finite rank, then li ≥ rank(pi (x)). E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2016.04.037 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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(3) If for some x in Xi , 1 − pi (x) has finite rank, then li ≤ rank(1 − pi (x)). (4) If either end point of Xi is infinite, then li = 0. In a similar fashion the author uses the idea of representing a projection by locally projection valued functions for the case that B a σ-unital, purely infinite, simple C ∗ -algebra such that K1 (B) = 0 while keeping the same X to solve the lifting problem in [8]. Theorem 1.2. Let B be a σ-unital, non-unital, purely infinite simple C ∗ -algebra such that K1 (B) = 0, and X be [0, 1], [0, ∞), (−∞, ∞) or [0, 1]/{0, 1}. Let p be a projection in the corona algebra of C(X) ⊗ B, which is represented by a family of multiplier projection valued functions (p0 , . . . , pn ) defined on closed intervals Xi ’s such that Xi−1 ∩ Xi = {xi } for i = 1, . . . , n. If the generalized essential codimension [pi (xi ) : pi−1 (xi )] vanishes for all i = 1, . . . , n, then p is liftable. We note that the obstruction for the projection lifting is of K-theoretical nature for both cases. In addition, it seems natural to expect that K-theory of B and the topology of X are intertwined and both of them are needed to solve the lifting problem. Though for more general X it is possible to represent any projection in the corona algebra of C(X) ⊗ B in terms of local projection valued functions, it is not promising to try to define local K-theoretical invariants associated to such a representation. Nonetheless it is still useful for a projection p to be locally projection valued. (See Definition 3.1.) To solve the projection lifting problem for more general finite dimensional space X, we take a different approach based on the extension theory of C ∗ -algebras. A strategy for lifting a projection is as follows: Suppose p is a projection in the corona algebra Q(C(X) ⊗ B). Then we can define an injective extension τ : C → Q(C(X) ⊗ B) by sending 1 to p. If we can show that τ is trivial up to the strong unitary equivalence, then p is liftable. Due to the nature of lifting problem, we only solve the projection lifting problem conditionally (see Corollary 3.10, Corollary 3.11). However, as far as we know this is a new approach to the lifting problem, thus we hope the technique introduced in this paper to be worthwhile for other kinds of lifting problems. The author is grateful to Huaxin Lin for sharing his keen insight, and Etienne Blanchard for a number of helpful discussions. He also would like to thank Ulrich Penning for correcting the proof of Theorem 3.9. 2. UCT and KT In this section we briefly recall the Universal Coefficient Theorem and the Künneth Theorem from [10], which we use frequently in the next section. An extension of A by B is a short exact sequence of C ∗ -algebras 0

B

E

A

0

Two such extensions are strongly unitarily equivalent if there is a unitary u ∈ M(B) in the multiplier algebra of B such that the following diagram is commutative, 0

B Ad u

0

B

E1

A

0

A

0

ϕ

E2

where the isomorphism ϕ: E1 → E2 is induced by Ad u, since Ei ⊂ M(B). The Busby invariants associated to the extensions are strongly equivalent, say τ1 ∼u τ2 if τ2 (a) = π(u)τ1 (a)π(u)∗ , where π: M(B) → M(B)/B is the natural quotient map. When B is stable we can define the so called a BDF-sum of two extensions so

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that this additive structure is well-defined on the set Ext(A, B) of strong equivalence classes of extensions of A by B. We say the extension τ is trivial if τ lifts to a ∗-homomorphism from A to M(B). The classes of trivial extensions forms a subsemigroup and we define Ext(A, B) as the quotient of Ext(A, B) by the subsemigroup of trivial extensions. Given an extension τ : A → M(B)/B, or 0

B

D

A

0

we can define a map γ: Ext(A, B) → Homodd (K∗ (A), K∗ (B)) by letting γ(τ ) be the connecting maps (δ0 , δ1 ) in the associated six-term exact sequence K0 (B)

K0 (D)

K0 (A)

δ1

δ0

K1 (A)

K1 (D)

K1 (B)

If γ(τ ) = 0, then the six-term exact sequence degenerates into two short exact sequences of the form 0

Ki (B)

Ki (D)

Ki (A)

0

thus defines an element κ(τ ) in the graded extension group Ext1Z,ev (K∗ (A), K∗ (B)) := Ext1Z (K0 (A), K0 (B))⊕ Ext1Z (K1 (A), K1 (B)). Rosenberg and Schochet have shown that γ is surjective and κ has a partial inverse δ in the following form. Theorem 2.1 (Universal Coefficient Theorem (UCT)). Let A, B be separable C ∗ -algebras, with A in the bootstrap category. Then there is a short exact sequence Ext1Z,ev (K∗ (A), K∗ (B))

0

γ

Ext(A, B)

Homodd (K∗ (A), K∗ (B))

0

They also proved a version of Künneth theorem from algebraic topology for the tensor product of C -algebras. As usual, we denote the minimal tensor product of A and B by A ⊗ B. ∗

Theorem 2.2 (Künneth theorem for tensor products). Let A, B be C ∗ -algebras, with A the bootstrap category. Then there is a short exact sequence 0

K∗ (A) ⊗ K∗ (B)

α

K∗ (A ⊗ B)

σ

TorZodd (K∗ (A), K∗ (B))

0

In particular, if K∗ (A) or K∗ (B) is torsion free, α is an isomorphism. 3. Projection lifting from the corona algebra of C(X) ⊗ D Throughout the paper we always assume B to be a simple, stable C ∗ -algebra and the multiplier algebra M(B) has real rank zero. We let I = C(X) ⊗ B and π: M(I) → Q(I) be the quotient homomorphism. Note that Q(I) is a C(X)-algebra in a natural way. For a closed subset V ⊂ X let rV : Q(I) → Q(I)(V ) ∼ = Q(I(V )) be the restriction homomorphism. By [1, Cor. 3.4] we can identify M(I) with C(X, M(B)s ), where M(B)s is the multiplier algebra equipped with the strict topology. Definition 3.1. We will call the extension of C by C(X) ⊗ B given by p as above locally projection valued, if every point x ∈ X has a closed neighborhood V , such that there exists a projection f ∈ M(C(V ) ⊗ B) with π(f ) = rV (p).

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The following Lemma is shown in the proof of [8, Thm. 3.2]. But since it is of fundamental importance for all of our theorems, we restate it here for the reader’s convenience. Lemma 3.2. Let B be a stable C ∗ -algebra such that M(B) has real rank zero. Let X be a compact Hausdorff space. Let I = C(X) ⊗ B and let p ∈ Q(I) be a projection. Then the extension associated to p is locally projection valued. Proof. Fix a point x0 ∈ X. Choose a preimage f  ∈ M(I) of p. Without loss of generality we may assume that f  is self-adjoint and 0 ≤ f  ≤ 1. Since M(B) has real rank 0, we can apply the technique used in the proof of [5, Lem. 3.14] to see that there is T ∈ M(B), such that T − f  (x0 ) ∈ D and the spectrum of T ˜ 0) has a gap around 12 . Therefore f˜ = f  + (T − f  (x0 ))1C(X) satisfies π(f˜) = p and the spectrum of f(x 1 2 2 ˜ ˜ ˜ has a gap around 2 . Let r(t) = t − t and note that r(f ) = f − f ∈ ker(π) = C(X) ⊗ B. In particular, ˜ 0 )) has a gap in its spectrum around 1 . By norm continuity x → r(f˜(x)) is norm continuous and r(f(x 4 of the continuous functional calculus, we obtain a closed neighborhood x0 ∈ V ⊂ X, such that r(f˜(x)) has a gap in its spectrum around 14 for all x ∈ V . This implies that there is an > 0 and an interval J = ( 12 − , 12 + ) not contained in σ(f˜(x)) for all x ∈ V . Let χI be the characteristic function of the interval  ˜  I and let f = χ(1/2,1] ( f V ). This is a projection in M(C(V ) ⊗ B) with π(f ) = rV (p). 2 3.1. The compact case Pimsner, Popa and Voiculescu have considered extensions of C(X) ⊗ K for the first time and obtained a Weyl–von Neumann type theorem. We rephrase their result in the following form. Note that the original theorem is much more general. We recall that the extension τ is absorbing if it is strongly equivalent to τ ⊕ σ, for any trivial extension σ. Theorem 3.3. (See [9, Proposition 2.9].) Let X be a separable, finite dimensional, compact, Hausdorff space and I be C(X) ⊗ K, and A be a unital separable nuclear C ∗ -algebra. Suppose τ : A → M(I)/I is an essential extension. If τ is homogeneous (i.e. τx is injective for every x ∈ X), then τ is absorbing. Here τx = evx ◦τ where evx : Q(I) → Q(K) is the evaluation map at x. Theorem 3.4. Let X be a separable, finite dimensional, compact, Hausdorff space such that K1 (C(X)) = 0. Let p ∈ Q(C(X) ⊗ K) be a projection such that p(x) := evx (p) = 0 for all x ∈ X. Then p is liftable to a projection in M(C(X) ⊗ K). Proof. Consider the extension τ defined by p ∈ Q(C(X) ⊗ K). Note that τx (λ) = λ p(x) for all x ∈ X. Thus, p(x) = 0 implies that τx is injective. By Theorem 3.3, the extension τ is absorbing. If we apply the UCT to the extension of A by B B = C(X) ⊗ K

0

E

A = C ∗ (p)

0

we get the short exact sequence 0

Ext1Z,ev (K∗ (A), K∗ (B))

Ext(A, B)

Homodd (K∗ (A), K∗ (B))

0

Note that K1 (A) = 0 and K1 (B) = 0 by our assumptions on X. It follows that Homodd (K∗ (A), K∗ (B)) = 0, and 0

K0 (B)

K0 (E)

K0 (A) = Z

0

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splits. Moreover, 0

K1 (B)

K1 (E)

K1 (A)

0

becomes trivial. Thus Ext1Z,ev (K∗ (A), K∗ (B)) = 0. Consequently Ext(A, B) = 0. Hence τ ⊕ σ0 ∼u τ1 ⊕ σ1 where τ1 is a trivial absorbing extension and σ0 , σ1 are trivial extensions. Since τ is absorbing, it follows that τ ∼u τ1 . 2 3.2. Kirchberg algebra case Now we turn to the case that B is a Kirchberg algebra, which is a simple, purely infinite, nuclear C ∗ -algebra that satisfies the UCT. We begin with an absorption result for a separable C(X)-algebra whose fibers are purely infinite simple nuclear C ∗ -algebras due to E. Blanchard. Proposition 3.5. (See [2, Prop. 2.5].) Let A be a separable σ-unital continuous field of purely infinite simple nuclear C ∗ -algebras over a compact Hausdorff space X. Let D be a separable C(X)-subalgebra of M(A ⊗ K) such that the identity representation idD : D → M(A ⊗ K) is a continuous field of faithful representations and there is a unital C(X)-embedding of the C(X)-algebra C(X) ⊗O∞ in the commutant of D in M(A ⊗K). If φ: D → M(A ⊗ K) is a unital morphism of C(X)-algebras which is zero on the intersection D ∩ (A ⊗ K), there exists a sequence of unitaries un ∈ M(A ⊗ K) such that for all d ∈ D, (d ⊕ φ(d)) − u∗n dun ∈ A ⊗ K

and

lim (d ⊕ φ(d)) − u∗n dun  = 0.

n→∞

Remark 3.6. We can summarize idD ⊕φ ≈u idD as the conclusion of Proposition 3.5. In general, we say that two representations ψ and φ from a C ∗ -algebra A to a multiplier algebra of a stable C ∗ -algebra B are approximately unitarily equivalent if there is a sequence of unitaries {Un } in the M(B) such that for all a∈A (i) ψ(a) − Un φ(a)Un∗ ∈ B, (ii) ψ(a) − Un φ(a)Un∗  → 0 as n → ∞. Lemma 3.7. Let A be a separable σ-unital continuous field of purely infinite simple nuclear C ∗ -algebras over a compact Hausdorff space X. Let D be a separable C(X)-subalgebra of M(A ⊗ K) that contains A ⊗ K as an ideal. Suppose that D has the following properties: (a) The identity representation D → M(A ⊗ K) is a continuous field of faithful representations. (b) There is a unital C(X)-embedding of the C(X)-algebra C(X) ⊗O∞ in the commutant of D in M(A ⊗K). Consider the associated extension τ given by 0

A⊗K

D

ψ

C

0

Then τ is absorbing. Proof. Let ι: D → M(A ⊗ K) be the embedding. Then we have π ◦ ι = τ ◦ ψ. We will identify D with the image of ι and drop ι from the notation. Let σ: C → Q(A ⊗ K) be the Busby invariant of a trivial extension with lift α: C → M(A ⊗ K), i.e. the following diagram commutes:

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M(A ⊗ K) α

C

σ

π

Q(A ⊗ K)

Note that φ = α ◦ ψ is zero on the intersection D ∩ (A ⊗ K). By Proposition 3.5, there exists a unitary u ∈ M(A ⊗ K) such that udu∗ − (d ⊕ φ(d)) ∈ A ⊗ K for all d ∈ D. Let c ∈ C and choose d ∈ D such that ψ(d) = c. Then π(d) = τ (c). Thus, π(u)τ (c)π(u)∗ = π(u)π(d)π(u)∗ = π(udu∗ ) = π(d ⊕ φ(d)) = (τ ⊕ σ)(c).

2

Lemma 3.8. Suppose K∗ (B) is torsion free. If K1 (C(X)) = 0 and K1 (B) = 0, or if K∗ (B) = 0, then K1 (C(X) ⊗ B) = 0. Proof. Since K∗ (B) is torsion free, the Künneth theorem implies that α: K∗ (A) ⊗ K∗ (B) → K∗ (A ⊗ B) is an isomorphism. In particular (K1 (A) ⊗ K0 (B)) ⊕ (K0 (A) ⊗ K1 (B)) ∼ = K1 (A ⊗ B). 2 Theorem 3.9. Let X be a compact Hausdorff space and let B be a stable Kirchberg algebra, such that K1 (C(X) ⊗ B) = 0. Let p ∈ Q(C(X) ⊗ B) be a projection such that p(x) := evx (p) = 0 for all x ∈ X. Let D ⊂ M(C(X) ⊗ B) be the subalgebra that fits into the extension sequence 0

C(X) ⊗ B

D

C

0

with Busby invariant τ : C → Q(C(X) ⊗ B) induced by p. Suppose that there is a unital C(X)-embedding of the C(X)-algebra C(X) ⊗ O∞ in the commutant of D in M(C(X) ⊗ B), then p is liftable. ˜ ⊂ M(C(X) ⊗ B) be the Proof. First observe that D is not a C(X)-subalgebra of M(C(X) ⊗ B). Let D extension 0

C(X) ⊗ B

˜ D

C(X)

0

with Busby invariant τ˜: C(X) → Q(C(X) ⊗ B) given by τ˜(g) = π(g ⊗ 1B ) p. We have evx (˜ τ (g)) = evx (π(g ⊗ 1B )p) = g(x)p(x). Since p(x) = 0 we obtain that τ˜(g) = 0 implies g(x) = 0 for all x ∈ X, i.e. τ˜ is injective. ˜ embeds as a C(X)-algebra into M(C(X) ⊗ B). In fact, It follows that D ˜ = {f ∈ M(C(X) ⊗ B) | π(f ) = τ˜(g) for some g ∈ C(X)} . D ˜ 0 (X \ {x0 })D ˜ be the fiber of the C(X)-algebra D ˜ with ˜ x = D/C ˜ at x0 . Let f ∈ D Fix x0 ∈ X. Let D 0 ˜ x → M(B) given by [h] → h(x0 ) is faithful, we have to prove f ∈ C0 (X \ {x0 })D. ˜ f (x0 ) = 0. To see that D 0 Since τ˜ is injective there is a unique g ∈ C(X), such that π(f ) = τ˜(g). Let πB : M(B) → Q(B) be the quotient homomorphism. We have g(x0 )p(x0 ) = evx0 (˜ τ (g)) = evx0 (π(f )) = πB (f (x0 )) = 0, therefore ˜ be a preimage of p. Observe that π(f −g ·f0 ) = 0 and f (x0 ) −g(x0 )f0 (x0 ) = 0, g ∈ C0 (X \{x0 }). Let f0 ∈ D ˜ But since g · f0 ∈ C0 (X \ {x0 })D ˜ as well, we have i.e. f − g · f0 ∈ C0 (X \ {x0 }) ⊗ B ⊂ C0 (X \ {x0 })D. ˜ Thus, the identity representation D ˜ → M(C(X) ⊗ B) is a continuous field of faithful f ∈ C0 (X \ {x0 })D. ˜ representations and D is a continuous C(X)-algebra. ˜ in M(C(X) ⊗ B) agrees with the one of D. Since C(X) is central in M(C(X) ⊗ B), the commutant of D ˜ satisfies the conditions of Lemma 3.7 and τ˜ is absorbing. Let σ: C → Q(C(X) ⊗ B) be a trivial Thus, D extension with lift α: C → M(C(X) ⊗ B). Then we can extend α to α ˜ : C(X) → M(C(X) ⊗ B) given by

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α ˜ (g) = g · α(1). Since τ˜ is absorbing, we have τ˜ ⊕ σ ˜ ∼u τ˜. If we restrict both sides to constant functions we obtain τ ⊕ σ ∼u τ , i.e. τ is absorbing. By the UCT we have Ext(C, C(X) ⊗ B) = 0. In particular, there are trivial extensions σ1 and σ2 , such that τ ⊕ σ1 ∼u σ2 . But then τ ∼u σ2 , which implies that p is liftable. 2 Next we show some applications of Theorem 3.9 by realizing the condition that there is a unital C(X)-embedding of C(X) ⊗ O∞ in the commutant of D. Proposition 3.10. Let p ∈ Q(C(X) ⊗ B) be a projection where B is O2 ⊗ K with K1 (C(X)) = 0. (Note that K1 (B) = 0 in this case.) Let f be the preimage of p in M(C(X) ⊗ B). If the C ∗ -algebra generated by f in M(C(X) ⊗ B) has no nonzero intersection with C(X) ⊗ B, f [f, g]f = 0 for all g ∈ C(X) ⊗ B, and evx (f ) is full for all x ∈ X, then p is liftable. Proof. Note that K0 (B) = 0 for B = O2 ⊗K, and K1 (C(X) ⊗B) = 0 with no restriction on X. Thus the first condition of Theorem 3.9 is satisfied. Let D1 be the C ∗ -algebra generated by the preimage f ∈ M(C(X)⊗B). We show that D1 contains C(X) ⊗ O∞ . Note that f (C(X) ⊗ B)f is a continuous C(X)-algebra whose fiber is fx Bfx where evx (f ) = fx . Since fx Bfx is a full hereditary subalgebra of B, it is stably isomorphic to B by [3], thus isomorphic to B. Then by [6, Theorem 1.3] it is trivial so that it is isomorphic to C(X) ⊗B. Then the condition that f [f, g]f = 0 for all g ∈ C(X) ⊗ B implies that D1 contains f (C(X) ⊗ B)f ∼ = C(X) ⊗ B. Now if we let e11 be a minimal projection in K, then C(X) ⊗ O∞ ⊗ e11 → C(X) ⊗ B. It follows that D1 contains C(X) ⊗ O∞ . Then if we consider the extension τ associated with p, then τ (a) = π(ι(a)) for a ∈ D1 where the map ι : D1 → M(C(X) ⊗ B) is an embedding into M(C(X) ⊗ B). Note that the intersection D1 ∩ (C(X) ⊗ B) is trivial by the assumption. By Proposition 3.5, it follows that ι ⊕ idD1 ≈u idD1 . This implies that τ is trivial, so we are done. 2 However, for other stable Kirchberg algebras B the above proof breaks down since the automatic triviality of a C(X)-algebra all of whose fibers are isomorphic to B is no longer true. A conditional trivialization is provided due to a recent progress of Classification program, particularly on continuous fields of self-absorbing C ∗ -algebras. Since stable Kirchberg algebras are stable self-absorbing C ∗-algebras, we show that additional assumptions on the preimage of p and an underlying space X induces the Fell condition of Dadarlat and Pennig in our situation, thus we obtain the global triviality. Corollary 3.11. Let p ∈ Q(C(X) ⊗ B) be a projection where B is O∞ ⊗ K and X an even dimensional sphere S 2k for k > 0. (Note that K1 (B) = K1 (C(X)) = 0 in this case.) If the preimage f of p satisfies that the C ∗ -algebra generated by f in M(C(X) ⊗ B) has no nonzero intersection with C(X) ⊗ B, and f [f, g]f = 0 for all g ∈ C(X) ⊗ B, and that evx (f ) is properly infinite and full for all x ∈ X, then p is liftable. Proof. Since B is a Kirchberg algebra with K1 (B) = 0, then M(B) has real rank zero [11]. Thus using Lemma 3.2 we can regard f as a locally projection valued function. In other words, for every point x in X, we can find a (compact) neighborhood U such that the restriction of f on U is a projection valued function. Then on such a neighborhood f is properly infinite and full as an element of M(C(U ) ⊗ B). Therefore, there is an element w in M(C(U ) ⊗ B) such that w∗ w = 1 and ww∗ = f . Then w(1 ⊗ e)w∗ is Murray–von Neumann equivalent to 1 ⊗ e where e is a minimal projection in K. Hence the C ∗ -algebra f (C(X) ⊗ B)f satisfies the Fell condition in the sense of Dadarlat and Pennig [7, Lemma 2.14]. Therefore it is locally trivial. There are topological obstructions to the global triviality of locally trivial C(X)-algebra with all fibers isomorphic to O∞ ⊗ K in terms of the cohomology of underlying space X as in [7, Corollary 4.7]. In case X is even-dimensional spheres S 2k with k > 0, H 2m+1 (S 2k , Z) = 0 for m ≥ 0 implies that such a C(X)-algebra is trivial. Therefore f (C(X) ⊗ B)f is actually isomorphic to C(X) ⊗ O∞ ⊗ K. Hence there is a natural C(X)-embedding of C(X) ⊗ O∞ into f (C(X) ⊗ B)f . Then we can follow the same argument as in Proposition 3.10. 2

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Corollary 3.12. Let p ∈ Q(C(X) ⊗ B) be a projection where B is MQ ⊗ O∞ ⊗ K and X an even dimensional sphere S 2k for k > 0 where MQ is the universal UHF-algebra. (Note that K1 (B) = K1 (C(X)) = 0 in this case.) If the preimage f of p satisfies that the C ∗ -algebra generated by f in M(C(X) ⊗ B) has no nonzero intersection with C(X) ⊗ B, and f [f, g]f = 0 for all g ∈ C(X) ⊗ B, and that evx (f ) is properly infinite and full for all x ∈ X, then p is liftable. Proof. We note that MQ ⊗O∞ is strongly self-absorbing and therefore simple. It is also purely infinite, since it is separable, nuclear, simple and O∞ -absorbing. Using Kuenneth theorem it follows that K0 (MQ ⊗O∞ ) = Q(rational numbers) and K1 (MQ ⊗ O∞ ) = 0. Thus MQ ⊗ O∞ ⊗ K is also a stable Kirchberg algebra whose multiplier algebra has real rank zero. It follows that the C ∗ -algebra f (C(X) ⊗ B)f also satisfies the Fell condition, and thereby locally trivial. Again for the global triviality of a C(X)-algebra with all fibers are isomorphic to a MQ ⊗ O∞ ⊗ K there are topological obstructions in terms of H 1 (X, Q× ) and H 2m+1 (X, Q) for m ≥ 1. If X is simply connected, then H 1 (X, Q× ) vanishes, and further if X = S 2k even dimensional spheres, then H 2m+1 (X, Q) also vanishes for all m > 1. 2 Acknowledgment This work was supported by 2014 Research Fund of University of Ulsan (2014-0044). References [1] Charles A. Akemann, Gert K. Pedersen, Jun Tomiyama, Multipliers of C ∗ -algebras, J. Funct. Anal. 13 (1973) 277–301, MR0470685 (57 #10431). [2] Etienne Blanchard, Subtriviality of continuous fields of nuclear C ∗ -algebras, J. Reine Angew. Math. 489 (1997) 133–149, http://dx.doi.org/10.1515/crll.1997.489.133, MR1461207 (98d:46072). [3] Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of C ∗ -algebras, Pacific J. Math. 71 (1977) 335–348, MR454645 (56#:12894). [4] Lawrence G. Brown, Hyun Ho. Lee, Homotopy classification of projections in the corona algebra of a non-simple C ∗ -algebra, Canad. J. Math. 64 (4) (2012) 755–777, MR2957229. [5] Lawrence G. Brown, Gert K. Pedersen, C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1) (1991) 131–149, http://dx.doi. org/10.1016/0022-1236(91)90056-B, MR1120918 (92m:46086). [6] Marius Dadarlat, Continuous fields of C ∗ -algebras over finite dimensional spaces, Adv. Math. 222 (5) (2009) 1850–1881, MR2555914 (2010j:46102). [7] Marius Dadarlat, Ulrich Pennig, A Dixmier–Douady theory for strongly self-absorbing C ∗ -algebras, J. Reine Angew. Math. (2016), in press, arXiv:1302.4468v1. [8] Hyun Ho Lee, Proper asymptotic unitary equivalence in KK-theory and projection lifting from the corona algebra, J. Funct. Anal. 260 (1) (2011) 135–145, http://dx.doi.org/10.1016/j.jfa.2010.09.003, MR2733573 (2011k:46079). [9] M. Pimsner, S. Popa, D. Voiculescu, Homogeneous C ∗ -extensions of C(X) ⊗ K(H). I, J. Operator Theory 1 (1) (1979) 55–108, MR526291 (82e:46093a). [10] J. Rosenberg, C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55 (2) (1987) 431–474, MR0894590 (88i:46091). [11] Shuang Zhang, Certain C ∗ -algebras with real rank zero and their corona and multiplier algebras. Part I, Pacific J. Math. 155 (1) (1992) 169–197.