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Alexandru Chirv\\u{a}situ L.
Publication date
2020
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English
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We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples $(X_1,cdots,X_d)$ of traceless self-adjoint operators in the $ntimes n$ matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: $2le dle n^2-3$. Moreover, the automorphism group is generically abelian in the larger parameter range $1le dle n^2-2$. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of $X_i$s (mimicking the ErdH{o}s-R{e}nyi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.
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In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $C^*$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $Lambda$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.
In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Lambda$, via the infinite path space $Lambda^infty$ of $Lambda$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Lambda^infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $Lambda$. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are $zeta$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $mu$, and show that $mu$ is a rescaled version of the measure $M$ on $Lambda^infty$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2(Lambda^infty, M)$ which was constructed by Farsi et al.
Using coordinate-free basic operators on toy Fock spaces cite{AP}, quantum random walks are defined following the ideas in cite{LP,AP}. Strong convergence of quantum random walks associated with bounded structure maps is proved under suitable assumptions, extendings the result obtained in cite{KBS} in case of one dimensional noise. To handle infinite dimensional noise we have used the coordinate-free language of quantum stochastic calculus developed in cite{GS1}.
We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.
Given a conditionally completely positive map $mathcal L$ on a unital $ast$-algebra $A$, we find an interesting connection between the second Hochschild cohomology of $A$ with coefficients in the bimodule $E_{mathcal L}=B^a(A oplus M)$ of adjointable maps, where $M$ is the GNS bimodule of $mathcal L$, and the possibility of constructing a quantum random walk (in the sense of cite{AP,LP,L,KBS}) corresponding to $mathcal L$.
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