[Paper Review] A characterization of weak (semi-)projectivity for commutative C*-algebras

Dominic Enders|arXiv (Cornell University)|Feb 16, 2011

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper completes the characterization of weak (semi-)projectivity for commutative C*-algebras by proving that the spectrum $X$ of a weakly (semi-)projective $C(X)$ must be at most one-dimensional. Using a lifting obstruction argument based on continuous maps from the 2-disk $\mathbb{D}^2$ into $X$, it shows that if $X$ is an AANR with $\dim(X) > 1$, a known unsolvable lifting problem can be constructed, leading to a contradiction. The key result is that $C(X)$ is weakly (semi-)projective if and only if $X$ is an AAR (AANR) and $\dim(X) \leq 1$. The work extends earlier results by Sørensen and Thiel to the weak case.

ABSTRACT

We show that the spectrum X of a weakly semiprojective, commutative C*-algebra C(X) is at most one dimensional. This completes the work of Sørensen and Thiel on the characterization of weak (semi-)projectivity for commutative C*-algebras.

Motivation & Objective

• To complete the classification of weak (semi-)projective commutative C*-algebras by determining the topological constraints on their spectra.
• To resolve the open problem of whether weak (semi-)projectivity implies finite-dimensional spectra, specifically dimension ≤1.
• To extend the framework of Sørensen and Thiel’s work on semiprojectivity to the weaker notion of weak (semi-)projectivity.
• To establish that the dimension bound is sharp by constructing a lifting obstruction when $\dim(X) > 1$.

Proposed method

• Constructing a continuous self-map $f: \mathbb{D}^2 \to \mathbb{D}^2$ with $|f(z) - z| < \sqrt{3}/2$ on $S^1$, which ensures that $f$ is close to the identity on the boundary.
• Defining a radial retraction $r: \mathbb{D}^2 \to \mathbb{D}^2$ based on the image of $f$ on $S^1$, using the minimum and maximum radial distances to define a region where $r$ is constant.
• Defining a map $g: \mathbb{D}^2 \to \mathbb{D}^2$ by $g(z) = r(z)z$, which satisfies $\|g \circ f - \text{id}_{\mathbb{D}^2}\|_\infty < 1$, showing that $g \circ f$ is homotopic to the identity in a weak sense.
• Using the existence of such maps from $\mathbb{D}^2$ to $X$ in the case of an AANR $X$ with $\dim(X) > 1$, to embed a known unsolvable lifting problem into $C(X)$.
• Applying a contradiction argument: if $C(X)$ were weakly semiprojective and $\dim(X) > 1$, such a lifting would exist, but it is known to be impossible.
• Using the fact that $\text{dist}(S^{\oplus k}, \{N+K\}) = 1$ for normal $N$ and compact $K$, to show that a lifting cannot exist, thus forcing $\dim(X) \leq 1$.

Experimental results

Research questions

• RQ1 What topological constraints must the spectrum $X$ of a weakly semiprojective commutative C*-algebra $C(X)$ satisfy?
• RQ2 Can the dimension of the spectrum $X$ exceed one if $C(X)$ is weakly (semi-)projective?
• RQ3 Is the existence of a continuous map $\mathbb{D}^2 \to X$ with a weak left inverse sufficient to construct a lifting obstruction in the weak (semi-)projective setting?
• RQ4 Does the characterization of weak (semi-)projectivity for commutative C*-algebras require the spectrum to be an AANR and of dimension at most one?
• RQ5 Can the results on semiprojectivity from Sørensen and Thiel be extended to the weak (semi-)projective case?

Key findings

• The spectrum $X$ of a weakly semiprojective commutative C*-algebra $C(X)$ must satisfy $\dim(X) \leq 1$, completing the classification.
• If $X$ is an AANR and $\dim(X) > 1$, then $C(X)$ cannot be weakly semiprojective, as it would allow a solution to an unsolvable lifting problem.
• The spectrum $X$ of a weakly projective $C(X)$ must be an AAR with $\dim(X) \leq 1$, which fully characterizes weak projectivity.
• The existence of a continuous map $f: \mathbb{D}^2 \to X$ with $|f(z) - z| < \sqrt{3}/2$ on $S^1$ implies the existence of a map $g: \mathbb{D}^2 \to \mathbb{D}^2$ such that $\|g \circ f - \text{id}\|_\infty < 1$, which is crucial for the obstruction argument.
• The lifting problem involving $C(\mathbb{D}^2) \to B/J_n \otimes M_k$ with target $\varphi^{\oplus k}$ is unsolvable when $\text{dist}(S^{\oplus k}, \{N+K\}) = 1$, which blocks weak (semi-)projectivity for higher-dimensional $X$.
• As a consequence, $C(X) \otimes M_k$ weakly (semi-)projective implies $X$ is an AANR and $\dim(X) \leq 1$, and hence $C(X)$ itself is weakly (semi-)projective.