Some examples of quantum graphs

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“…• a self-adjoint operator 𝐴 on 𝐿 2 (𝐵) satisfying certain conditions corresponding to, in the classical case, being {0, 1}-valued, being undirected, and having a loop at every vertex. See [27,34] with generalisations to non-tracial states in [8,9,33]; see Section 2. • when 𝐵 ⊆ ℬ(𝐻) for some Hilbert space 𝐻, a unital, self-adjoint subspace 𝑆 of ℬ(𝐻) which is a bimodule over 𝐵 ′ , the commutant of 𝐵, meaning that for 𝑎, 𝑏 ∈ 𝐵 ′ , 𝑥 ∈ 𝑆 also 𝑎𝑥𝑏 ∈ 𝑆.…”

“…For example, [27] takes this as the principal definition. We also extend the "completely positive" approach suggested in [14] to arbitrary (finite-dimensional) 𝐶 *algebras.…”

“…However, we find some of the literature here a little hard to follow. For example, [27,Section 4] is readable, but only looks at the Hopf * -algebra case, while we work with the 𝐶 * -algebra picture of compact quantum groups. We hope that this section will hence be helpful to the non-expert.…”

“…Again when 𝜓 is a 𝛿-form, [9, Definition 3.3] just considers axiom (1) (and without the requirement that 𝐴 be self-adjoint). A different starting point is chosen in [27,Definition 1.8] which corresponds to the projections 𝑒 ∈ 𝐵 ⊗ 𝐵 op which we consider in Section 5, again when 𝜓 is a tracial 𝛿-form, and again graphical methods are used to make links to adjacency matrices.…”

“…Also here links with Quantum Groups arose, which has been further explored in [8]. Recent papers exploring quantum graphs are [27,33], while generalisations to the non-tracial case have been given in [8,33], and a more minimal set of axioms explored in [9,10,14]. Papers making use of quantum graphs include [7,13,14,[23][24][25]41] to list but a few.…”