By Ilwoo Cho
This e-book introduces the learn of algebra triggered 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 publication provides contemporary examine in operator algebra thought hooked up with discrete and combinatorial mathematical items. It additionally covers instruments and strategies from various mathematical parts, together with algebra, operator conception, and combinatorics, and gives various functions of fractal conception, entropy thought, K-theory, and index theory.
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Additional info for Algebras, Graphs and their Applications
Now, let Gk be given as above, for k = 1, 2, and let G be the unioned graph G2 ∪ G1 . Then, by the very definition, the unioned graph G is graphisomorphic to the unioned graph G = G1 ∪ G2 . This guarantees the following proposition. 1 Let X = G1 + G2 be the sum of graph groupoids G1 and G2 . Then X is groupoid-isomorphic to G2 + G1 . Proof. Let X = G1 + G2 be the sum of the graph groupoids Gk of graphs Gk . We know that the groupoid X is groupoid-isomorphic to the graph groupoid G of the unioned graph G = G1 ∪ G2 .
Define a groupoid action α of G acting on M in B(K ⊗ HG ) by a nonunital partial representation satisfying αw (m)Lw L∗w = L∗w mLw = Lw−1 mLw , for all m ∈ M and w ∈ G. We call the above relation of α, the G-representation of G. Here, the operators Lw ’s are understood as 1K ⊗ Lw ’s in B(K ⊗ HG ). Remark that, in the G-representation, the operator Lw L∗w is Lww−1 , and hence it is a projection on K ⊗ HG . If θ : M → B(K) is a ∗-homomorphism (or a representation of M ), then αw (m) and m in the G-representation are understood as π (αw (m)) and π(m) in π(M ), respectively, for all m ∈ M.
This action L is called the canonical (left) groupoid action of G (acting on HG ). 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).
Algebras, Graphs and their Applications by Ilwoo Cho