By Ilwoo Cho

This e-book introduces the learn of algebra triggered by way of combinatorial items referred to as directed graphs. those graphs are used as instruments within the research of graph-theoretic difficulties and within the characterization and resolution of analytic difficulties. The booklet provides fresh learn in operator algebra conception attached with discrete and combinatorial mathematical gadgets. It additionally covers instruments and techniques from quite a few mathematical components, together with algebra, operator concept, and combinatorics, and gives various purposes of fractal conception, entropy concept, *K*-theory, and index theory.

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**Example text**

Recall that, for each v ∈ V (G), αv (m) = mv = m, for all m ∈ M, by definition of α. So, we have that: (VII) mLv = mv Lv = mv Lv L∗v = Lv mv L∗v = Lv mv Lv = Lv mLv . , we have a groupoid W ∗ -dynamical system (M, G, α), where G has its canonical representation (HG , L). 3 Graph von Neumann Algebras 51 def MG = vNα (M, {Lw : w ∈ G}) , generated by M and {Lw : w ∈ G}, satisfying the G-representation determined by α. 6 Let (M, G, α) be given as above. Define the crossed product MG = M × α G of M and G via α, by the von Neumann algebra vNα (M, L(G)), generated by M and {Lw : w ∈ G} in B(K ⊗ HG ), satisfying the G-representation of α.

The above proposition shows that if G is the product graph G1 × G2 of the shadowed graphs Gk of connected graphs Gk , for k = 1, 2, then there always exists “a” graph G0 , whose shadowed graph G0 of G0 is graphisomorphic to G, by the self-shadowedness of G. Note that the choice of G0 is not unique. , if G10 and G20 satisfy the above proposition, then the both shadowed graphs Gk0 of Gk0 , for k = 1, 2, are graph-isomorphic to the product graph G. The important fact here is the existence of a graph G0 , having its shadowed graph G0 , graph-isomorphic to the product graph G1 × G2 of the shadowed graphs Gk of Gk , for k = 1, 2.

And the graph G1:2 with the identification rule (I R) is called the quotient graph of G1 by G2 . Sometimes, we denote G1:2 by G1 /G2 , to emphasize it is generated by the relation G2 ≤ G1 . Let G2 ≤ G1 be given connected graphs, and let G1:2 = G1 /G2 be the quotient graph of G1 by G2 . Then, it is an independent graph, and hence it has its own graph groupoid G1:2 . The following theorem shows that the quotient groupoid G1 /G2 is groupoid-isomorphic to the graph groupoid G1:2 of the quotient graph G1 /G2 , whenever the graph groupoids G1 and G2 satisfies (I) and (II).