A characterization of weak (semi-)projectivity for commutative C*-algebras

⇔ X is an AR and dim(X) ≤ 1 C(X) is semiprojective in S 1 ⇔ X is an ANR and dim(X) ≤ 1

Here we denote by S 1 the category of all unital, separable C*-algebras with unital -homomorphisms. The first equivalence stated above is due to Chigogidze and Dranishnikov, see [CD10], while the second one was recently proved by Sørensen and Thiel in [ST11]. In this article we show that furthermore: C(X) is weakly projective in S 1 ⇔ X is an AAR and dim(X) ≤ 1 C(X) is weakly semiprojective in S 1 ⇔ X is an AANR and dim(X) ≤ 1 Note that the " ⇐ “-implications have already been proved in [ST11,Corollary 6.16]. Hence the only (non-trivial) part left to show is the dimension estimate for weakly (semi-)projective C-algebras C(X). This will be the content of this paper. The idea of proof is the same as the one in [ST11, Proposition 3.1]: We show that if C(X) was weakly semiprojective and X an AANR of dimension > 1, we could solve a lifting problem which is known to be unsolvable. We would like to point out that, as noted in [ST11, Remark 3.3], the existence of an inclusion D 2 ֒→ X would be a sufficient, but not necessary condition to construct such a lifting problem. Note also that, since the closed two-dimensional disc D 2 is an absolute retract, an embedding D 2 ֒→ X would admit a leftinverse X → D 2 . We will show that for our purpose it is enough to have (possibly non-injective) maps D 2 → X which are leftinvertible in an extremely weak sense. The existence of such maps in the case of an AANR X with dimension > 1 is the crucial point in our argumentation and its proof makes up the greatest part of this paper. As an application of our main result, we illustrate by an example how one can generalize the results of [ST11, Section 6] to the setting of weakly (semi-)projective, commutative C*-algebras.

We refer the reader to section 2 of [ST11] for definitions of weakly (semi-)projective C*-algebras, of AA(N)Rs, for further terminology, notations and everything else necessary.

Let us start off by a technical lemma concerning continuous self-maps of the closed, two-dimensional disc

2 for all z ∈ S 1 ⊂ D 2 . Then there exists a continuous map g :

Proof. A simple compactness argument shows that there is a 1 -

Notice that r max and r min are well defined, continuous and satisfy

by the assumption on f and the choice of the parameter t. Now let r : D 2 → [0, 1] be the function

• which equals 0 on {se iϕ : s ≤ r min (e iϕ ) -t} ∪ {se iϕ : s ≥ r max (e iϕ ) + t},

• which equals t on {se iϕ : r min (e iϕ ) ≤ s ≤ r max (e iϕ )} and

This map will be continuous. Finally, let g : D 2 → D 2 be given by

Then g is continuous with g ∞ = t and we claim that

Now consider a fixed z with 1-t ≤ |z| < 1 and write z = |z|e iϕ0 . The parameter t was chosen in such a way that f

Using compactness of X and continuity of g • f we get the desired uniform estimate.

The following is a refined version of an argument used in the proof of [ST11, Proposition 3.1].

Proposition 2.2. Let X be a compact AANR with dim(X) > 1. Then there exists a point x 0 ∈ X such that every neighbourhood of x 0 admits a topological embedding of S 1 .

Proof. As shown in [Gor99, Theorem 3.4] there exists an ANR Y such that X is (homeomorphic to) an approximative retract of Y. This means we have X ⊆ Y and for all ǫ > 0 there exists an ǫ-retract r ǫ : Y → X (which again means that d(x, r ǫ (x)) < ǫ for every x ∈ X). We claim that for some ǫ > 0 we have dim(r ǫ (Y )) = dim(X). This follows from [HW48, Corollary to Theorem V.9], since (following the terminology from [HW48]) for every ǫ-mapping p from r ǫ (Y ) to some space Z, the composition

will be a (3ǫ)-mapping on X. Now fix some ǫ > 0 such that dim(r ǫ (Y )) = dim(X) > 1. By compactness we have locdim(r ǫ (Y )) = dim(r ǫ (Y )) > 1. This means there exists a point x 0 ∈ r ǫ (Y ) such that dim(U ) > 1 for every closed neighbourhood U of x 0 in r ǫ (Y ). Since Y is an ANR, it is a Peano continuum and so will be every continuous image of it. Hence r ǫ (Y ) is a Peano continuum of dimension > 1 and so will be every closure of an open, connected neighbourhood U of x 0 in r ǫ (Y ). Thus we may apply [CD10, Proposition 3.1] to obtain topological embeddings S 1 ֒→ U for every such U . Now if V is a closed neighbourhood of x 0 in X, V ∩ r ǫ (Y ) is a closed neighbourhood of x 0 in r ǫ (Y ) and hence S 1 ֒→ V ∩ r ǫ (Y ) ⊆ V as shown above.

Theorem 2.3. Let X be a compact AANR with dim(X) > 1. Then the following holds: There exist continuous maps f : D 2 → X and g : X → D 2 such that the diagram

commutes up to a constant strictly less then 1, i.e. g • f -id D 2 ∞ < 1.

Proof. By [Gor99, Theorem 3.4] there exists an ANR Y such that X is an approximative retract of Y. This means we have X ⊆ Y and for all ǫ > 0 there exists an ǫ-retract r ǫ : Y → X. Applying Proposition 2.2, we find some x 0 ∈ X such that every closed neighbourhood of x 0 in X admits a topolog

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