By Ilwoo Cho
This publication introduces the examine of algebra prompted by way of combinatorial items known 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 e-book provides contemporary learn in operator algebra conception attached with discrete and combinatorial mathematical items. It additionally covers instruments and techniques from quite a few mathematical parts, together with algebra, operator thought, and combinatorics, and provides quite a few functions of fractal idea, entropy concept, K-theory, and index theory.
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Extra resources for Algebras, Graphs and their Applications
Let G0 be the graph whose shadowed graph G0 is graph-isomorphic to the product graph G = G1 × G2 , and assume that it has its graph groupoid G0 . Then the graph groupoid G0 of G is generated by E(G0 ), since all elements of G0 are the reduced words in E(G0 ). , the graph groupoid G0 is generated by the edge set E(G) of G, equivalently, G0 Groupoid = the groupoid generated by E(G). , all elements of the groupoid G are the reduced words in E. Then we have that E G1 × E G2 = E G1 × G2 = E(G). , def G = the groupoid generated by E Groupoid = the groupoid generated by E(G).
3 Let G be a graph with its graph groupoid G, and let HG be the graph Hilbert space of G. Also, let L be the canonical groupoid action of G. Then the pair (HG , L) is said to be the canonical representation of G. We have seen that, if there is a countable directed graph G, then the graph groupoid G of G is embedded in an operator algebra B(HG ); moreover, the elements of G become partial isometries and their initial or final projections on HG , under the canonical representation (HG , L). 2 Groupoid W ∗ -Dynamical Systems Let H be a Hilbert space and B(H), the operator algebra consisting of all (bounded linear) operators on H.
Now, take (v1 , v2 ) ∈ V (G). Then, by the connectedness of the graphs G1 and G2 , there exists at least one pair (e1 , e2 ) ∈ E(G), such that (v1 , v2 ) = −1 (e1 e−1 1 , e2 e 2 ), −1 e−1 e , e e 1 2 1 2 ), or or −1 (e1 e−1 1 , e2−1e2 ), or (e−1 e , e 1 1 2 e2 ), under the reduction (RR) on the graph groupoid G of G. This shows that (v1 , v2 ) satisfies (V). Thus, V (G) ⊆ V (G1 ) × V (G2 ). Remark that, in general, the equality does not hold (See the examples below). 4 The graph G, defined above from the connected graphs G1 and G2 , is called the edge-product graph of G1 and G2 .
Algebras, Graphs and their Applications by Ilwoo Cho