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By Ilwoo Cho

This ebook introduces the research of algebra brought on via combinatorial gadgets referred to as directed graphs. those graphs are used as instruments within the research of graph-theoretic difficulties and within the characterization and answer of analytic difficulties. The ebook provides contemporary study in operator algebra conception hooked up with discrete and combinatorial mathematical gadgets. It additionally covers instruments and techniques from numerous mathematical parts, together with algebra, operator concept, and combinatorics, and provides a number of purposes of fractal thought, entropy thought, K-theory, and index theory.

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Example text

Let G = G1 ∪ G2 be the unioned graph of G1 and G2 . Then, as a new directed graph, G has its own graph groupoid G. This groupoid G is groupoid-isomorphic to the reduced free product groupoid ∗G Ge , where e∈E(G) Ge are the subgroupoids of G, consisting of all reduced words only in {e, e−1 }, for e ∈ E(G). Thus, we obtain: G Groupoid = ∗G Ge e∈E(G) Groupoid = Groupoid = ∗G1 (G1 )e e∈E(G1 ) ∗G ∗G2 (G2 )x x∈E(G2 ) G1 ∗G G2 . , def V1:2 = V (G1 ) ∩ V (G2 ). Then the above formula can be re-written as follows: G Groupoid = G1 ∗G V1:2 G2 , 18 Algebra on Graphs where “∗G V1:2 ” means that every pair (w1 , w2 ) ∈ G1 × G2 of nonempty elements can generate a new nonempty element w1 w2 in G, if and only if the terminal vertex of w1 , denoted by r(w1 ), and the initial vertex of w2 , denoted by s(w2 ) are identical in V1:2 .

Proof. Suppose that w1 and w2 are not admissible in G. Then w1 w2 = ∅ and hence Lw1 w2 = L∅ = 0. Therefore, Lw1 Lw2 = 0 = Lw1 w2 , whenever w1 and w2 are not admissible in G. Assume now that w1 and w2 are admissible in G. By definition, Lw1 Lw2 ξ w = Lw1 ξ w2 w = ξ w1 w2 w = ξ w1 w2 ξ w = Lw1 w2 ξ w , for all w ∈ G. Thus Lw1 Lw2 = Lw1 w2 , whenever w1 and w2 are admissible. ✷ Also, we can obtain the following lemma by the very definition of multiplication operators on HG . 2 Let v ∈ V (G) be a vertex in G.

If G = G1 ∪ G2 is the unioned graph of G1 and G2 , then the graph groupoid G of G is groupoid-isomorphic to the sum G1 + G2 of the graph groupoids G1 and G2 . Proof. Let G = G1 ∪ G2 be the unioned graph of G1 and G2 . Then, as a new directed graph, G has its own graph groupoid G. This groupoid G is groupoid-isomorphic to the reduced free product groupoid ∗G Ge , where e∈E(G) Ge are the subgroupoids of G, consisting of all reduced words only in {e, e−1 }, for e ∈ E(G). Thus, we obtain: G Groupoid = ∗G Ge e∈E(G) Groupoid = Groupoid = ∗G1 (G1 )e e∈E(G1 ) ∗G ∗G2 (G2 )x x∈E(G2 ) G1 ∗G G2 .

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