[Paper Review] On the K-theory of C*-algebras for substitution tilings (a pedestrian version)

Daniel Gonçalves, Maria Ramirez-Solano|arXiv (Cornell University)|Dec 27, 2017

Advanced Operator Algebra Research35 references4 citations

TL;DR

This paper presents a computationally efficient method to compute the K-theory of C*-algebras associated with substitution tilings using a single cochain complex with connecting maps. It establishes that the K-theory of stable (S) and unstable (U) C*-algebras can be derived from stable cohomology and stable-transpose homology, respectively, and proves that for 1D tilings, K-theory groups are torsion-free, while in 2D, only K₀(U) and K₁(S) may contain torsion. The method is implemented in Mathematica and offers a simpler alternative to existing approaches.

ABSTRACT

Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article we show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, we show that the $K$-theory groups for tilings of dimension 1 are always torsion free. For tilings of dimension 2, only $K_0(U)$ and $K_1(S)$ can contain torsion.

Motivation & Objective

• To develop a unified, computable method for computing the K-theory of C*-algebras associated with substitution tilings.
• To establish a direct relationship between the K-theory of stable and unstable C*-algebras via cohomological and homological constructions.
• To characterize the presence of torsion in K-theory groups for tilings of dimension 1 and 2.
• To provide explicit formulas and algorithms for computing K₀ and K₁ groups using a single cochain complex with connecting maps.
• To implement the method in Mathematica for practical computation and comparison with existing techniques.

Proposed method

• Construct three equivalence relations—stable (Rs), unstable (Ru), and asymptotic (Ra)—from a Smale space structure derived from substitution tilings.
• Define the corresponding C*-algebras S, U, and A via Renault's groupoid C*-algebra construction using inductive limit topologies and Haar systems.
• Introduce the concept of stable cohomology and stable-transpose homology as dual tools to compute K-theory of S and U, respectively.
• Use a single cochain complex with connecting maps to unify the computation of K-theory for both S and U.
• Apply direct limit constructions of a substitution matrix and its transpose to compute cohomology and homology in the absence of torsion.
• Implement the algorithm in Mathematica to enable practical computation of K-theory groups for specific tiling examples.

Experimental results

Research questions

• RQ1 Can the K-theory of the stable and unstable C*-algebras for substitution tilings be computed from a single cochain complex with connecting maps?
• RQ2 What is the role of stable cohomology and stable-transpose homology in relating the K-theory of S and U?
• RQ3 Under what conditions are the K-theory groups of 1D substitution tilings torsion-free?
• RQ4 Which K-theory groups in 2D tilings can contain torsion, and how can they be identified?
• RQ5 How does the proposed method compare in efficiency and simplicity to existing methods for computing K-theory of unstable C*-algebras?

Key findings

• For 1D substitution tilings, the K-theory groups of the stable and unstable C*-algebras are always torsion-free.
• In 2D tilings, only K₀(U) and K₁(S) can contain torsion, while K₀(S) and K₁(U) are torsion-free.
• The K-theory of S is computed via stable cohomology, and the K-theory of U is computed via stable-transpose homology, both derived from a single cochain complex with connecting maps.
• The method provides explicit formulas for computing K₀ and K₁ groups using direct limits of a substitution matrix and its transpose when torsion is absent.
• The proposed method is more computationally efficient than prior approaches and has been implemented in Mathematica for practical use.
• The construction establishes a direct relationship between the K-theory of S and U, confirming a conjecture by the first author regarding their duality.