$C^*$- Colored graph algebras

TL;DR

This paper extends graph $C^*$-algebra theory to $C^*$-colored graphs by introducing a two-layer colored-graph framework and CK-family constructions. It develops both directed and undirected, color-labeled graphs, and uses Hamiltonian-path analysis and overlay/connect operations to realize CK $\Gamma$-families, yielding infinite- and finite-dimensional $C^*$-algebras. A key result is a concrete finite-dimensional example: the CK-family associated with a four-partite directed graph $\prescript{c}{}{\Sq}_{3}^{d}$ gives $C^*(\prescript{c}{}{\Sq}_{3}^{d}) \cong M_3( C(\mathbb T)) \oplus M_3( C(\mathbb T)) \oplus C(\mathbb T) \oplus C(\mathbb T) \cong \mathcal{M}_{3,3}( C(\mathbb T^2))$. The work broadens the scope of graph $C^*$-algebras to colored, layered structures and sketches pathways to connect these algebras with quantum/group-theoretic frameworks and potential non-commutative graph concepts.

Abstract

Following our previous works on $C^*$-graph algebras and the associated Cuntz-Krieger graph families, in this paper we will try to have a look at the colored version of these structures and to see what a $C^*$-colored graph algebra might mean by employing some constructive examples very close to the toy example used in our previous works, and we also will try to study their graph theoretical properties as possible.

Figures (8)

• Figure 1: Directed locally connected graph related to $\Pi_2$
• Figure 2: Two connected graph $\mathsf{Sq}_2$
• Figure 3: Four connected graph $\mathsf{Sq}_3$
• Figure 4: Six connected graph $\mathsf{Sq}_4$
• Figure 5: Five connected directed graph $\mathsf{Sq}_{3}^{d}$

Theorems & Definitions (27)

• Remark 2.3
• Remark 2.4
• Remark 2.7
• Remark 3.1
• Definition 3.2
• Remark 3.6
• Proposition 3.7
• proof
• Remark 3.8
• Definition 3.9