Abstract
In this paper, we consider certain quantities that arise in the images of the so-called graph-tree indexes of graph groupoids. In text, the graph groupoids are induced by connected finite-directed graphs with more than one vertex. If a graph groupoid GG contains at least one loop-reduced finite path, then the order of G is infinity; hence, the canonical groupoid index G:K of the inclusion K⊆G is either ∞ or 1 (under the definition and a natural axiomatization) for the graph groupoids K of all “parts” K of G. A loop-reduced finite path generates a semicircular element in graph groupoid algebra. Thus, the existence of semicircular systems acting on the free-probabilistic structure of a given graph G is guaranteed by the existence of loop-reduced finite paths in G. The non-semicircularity induced by graphs yields a new index-like notion called the graph-tree index Γ of G. We study the connections between our graph-tree index and non-semicircular cases. Hence, non-semicircularity also yields the classification of our graphs in terms of a certain type of trees. As an application, we construct towers of graph-groupoid-inclusions which preserve the graph-tree index. We further show that such classification applies to monoidal operads.
Highlights
A directed graph G = (V(G), E(G)) is a combinatorial object consisting of the vertex set V(G) of all vertices and the edge set E(G) of all directed edges
Semicircular elements whose free distributions obey the semicircular law play key roles, as the semicircular law is the noncommutativeanalytic counterpart of the Gaussian distribution of classical functional analysis by the central limit theorem(s) (e.g., [16,17,18,19])
Based on our semicircularity characterization in terms of the loop-ness, we introduce the so-called graph-tree index of Axioms 2022, 11, 47 graph groupoids and show that these index quantities give information of elements of G, which are “not” semicircular in (MG, τ)
Summary
We assume that all given graphs are connected, finite, and have more than one vertex. Such a directed graph is depicted in a diagrammatic form as a set of dots (for vertices) jointed by arrowed curves (for directed edges), where the arrows of the curves indicate the direction on the graph (e.g., [1,2,3,4]). Free probability is one of the main areas of operator algebra theory studying “noncommutative” measure-theoretic and corresponding statistical analysis on operator-theoretic structures (e.g., [15,16,17,18]). The main results of this paper include (i) characterizing the semicircularity on (MG, τ) by the loop-ness on G; (ii) considering a certain measure on G called the non-loop index of G, providing the information of groupoidal elements in G that are not loop-reduced finite paths; (iii) showing how our measuring tool of (ii) implies the non-semicircularity on (MG, τ) with respect to (i); and (iv) constructing and studying a tower of C∗-probability spaces that are free homomorphic from the base to the top, preserving our non-loop index
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