arXiv subjects

Sooran Kang

Publications and source records attributed to Sooran Kang.

At least 19 recordsLinked to original sources

arXiv2022

An Infinite Dimensional Analysis of Kernel Principal Components

We study non-linear data-dimension reduction. We are motivated by the classical linear framework of Principal Component Analysis. In nonlinear case, we introduce instead a new kernel-Principal Component Analysis, manifold and feature space transforms. Our results extend earlier work for probabilistic Karhunen-Loève transforms on compression of wavelet images. Our object is algorithms for optimization, selection of efficient bases, or components, which serve to minimize entropy and error; and hence to improve digital representation of images, and hence of optimal storage, and transmission. We prove several new theorems for data-dimension reduction. Moreover, with the use of frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identify when a choice of Gaussian kernel is optimal.

math.FA↗

arXiv2022

Yang-Mills connections and sigma-models on quantum Heisenberg manifolds

We construct a spectral triple on a quantum Heisenberg manifold, which generalizes the results of Chakraborty and Shinha, and associate to it an energy functional on the set of projections, following the approach of Mathai-Rosenberg to non-linear sigma models. The spectral triples that we construct extend the We derive a lower bound for this energy functional that is linked on the topological charge of the projection which depends on the curvature of a compatible connection. A detailed study of this lower bound is given for the Kang projection in quantum Heisenberg manifolds. These results display an intriguing interplay between non-linear sigma models and Yang-Mills theory on quantum Heisenberg manifolds, unlike in the well-studied case of noncommutative tori.

math.OA↗

arXiv2020

Computing the fundamental group of a higher-rank graph

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about $k$-graphs when we show that the abelianisation of fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

math.DS↗

arXiv2019

Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs

The aim of this paper is to study the heat kernel and jump kernel of the Dirichlet form associated to ultrametric Cantor sets $\partial\BB_Λ$ that is the infinite path space of the stationary $k$-Bratteli diagram $\BB_Λ$, where $Λ$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\PB_Λ), π_ϕ, \mathcal{H}, D, Γ)$ and is given by \[ Q_s(f,g)=\frac{1}{2} \int_Ξ \operatorname{Tr}\big(\vert D\vert^{-s} [D,π_ϕ(f)]^{\ast} [D,π_ϕ(g)] \big) \, dν(ϕ), \] where $Ξ$ is the space of choice functions on $\partial \BB_Λ\times \partial \BB_Λ$. There are two ultrametrics, $d^{(s)}$ and $d_{w_δ}$, on $\partial \BB_Λ$ which make the infinite path space $\PB_Λ$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of Laplace-Beltrami operator $Δ_s$ associated to $Q_s$, and the latter $d_{w_δ}$ is associated to a weight function $w_δ$ on $\BB_Λ$, where $δ\in (0,1)$. We show that the Perron-Frobenius measure $μ$ on $\partial \BB_Λ$ has the volume doubling property with respect to both $d^{(s)}$ and $d_{w_δ}$ and we study the asymptotic behaviors of the heat kernel associated to $Q_s$. Moreover, we show that the Dirichlet form $Q_s$ coincides with a Dirichlet form $\mathcal{Q}_{J_s, μ}$ which is associated to a jump kernel $J_s$ and the measure $μ$ on $\partial \BB_Λ$, and we investigate the asymptotic behavior and moments of displacements of the process.

math.PR↗

arXiv2019

Spectral triples and wavelets for higher-rank graphs

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Λ$, via the infinite path space $Λ^\infty$ of $Λ$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Λ^\infty$ 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 $Λ$. 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 $ζ$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $μ$, and show that $μ$ is a rescaled version of the measure $M$ on $Λ^\infty$ 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 $L^2(Λ^\infty, M)$ which was constructed by Farsi et al.

math.OA↗

arXiv2019

Yang-Mills connections on quantum Heisenberg manifolds

We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds $D^c_{μν}$, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over the QHMs. A compatible linear connection which is both a critical point and minimizer of YM is called a Yang-Mills connection. In this paper, we investigate Yang-Mills connections with constant curvature. We are interested in Yang-Mills connections on the following classes of modules over the QHMs: (i) Abadie's module $Ξ$ of trace $2μ$ and its submodules; (ii) modules $Ξ^\prime$ of trace $2ν$; (iii) tensor product modules of the form $P E^c_{μν}\otimes Ξ$, where $E^c_{μν}$ is Morita equivalent to $D^c_{μν}$ and $P$ is a projection in $E^c_{μν}$. We present a characterization of critical points and minimizers of YM, and provide a class of new Yang-Mills connections with constant curvature via concrete examples. In particular, we show that every Yang-Mills connection $\nabla$ on $Ξ$ over $D^c_{μν}$ with constant curvature should have a certain form of the curvature such that $Θ_\nabla(X,Y)=Θ_\nabla(X,Z)=0$ and $Θ(Y,Z)=\frac{πi}μ Id_E$. Also we show that these Yang-Mills connections with constant curvature do not provide global minima but only local minima, and give two other examples which show that the critical points and minimizers of YM depend crucially on the geometric structure of the QHMs and of the projective modules over them. Furthermore, we construct the Grassmannian connection on the projective modules $Ξ^\prime$ with trace $2ν$ over the QHMs and compute its corresponding curvature, and we construct a tensor product connections on $P E^c_{μν}\otimes Ξ$ whose coupling constant is $2ν$ and characterize the critical points of YM for this projective module.

math.OA↗

arXiv2018

Purely atomic representations of higher-rank graph C*-algebras

We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.

math.OA↗

arXiv2018

Spectral triples for higher-rank graph $C^*$-algebras

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Λ)$ of a strongly connected finite higher-rank graph $Λ$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Λ)$, and we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Λ$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of this spectral triple.

math.OA↗

arXiv2018

Monic representations of finite higher-rank graphs

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 $Λ$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

math.OA↗

arXiv2018

Representations of higher-rank graph $C^*$-algebras associated to $Λ$-semibranching function systems

In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $Λ$. We begin by giving an alternative characterization of the $Λ$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.

math.OA↗

arXiv2017

Separable representations of higher-rank graphs

In this monograph we undertake a comprehensive study of separable representations (as well as their unitary equivalence classes) of $C^*$-algebras associated to strongly connected finite $k$-graphs $Λ$. We begin with the representations associated to the $Λ$-semibranching function systems introduced by Farsi, Gillaspy, Kang, and Packer in \cite{FGKP}, by giving an alternative characterization of these systems which is more easily verified in examples. We present a variety of such examples, one of which we use to construct a new faithful separable representation of any row-finite source-free $k$-graph. Next, we analyze the monic representations of $C^*$-algebras of finite $k$-graphs. We completely characterize these representations, generalizing results of Dutkay and Jorgensen \cite{dutkay-jorgensen-monic} and Bezuglyi and Jorgensen \cite{bezuglyi-jorgensen} for Cuntz and Cuntz-Krieger algebras respectively. We also describe a universal representation for non-negative monic representations of finite, strongly connected $k$-graphs. To conclude, we characterize the purely atomic and permutative representations of $k$-graph $C^*$-algebras, and discuss the relationship between these representations and the classes of representations introduced earlier.

math.OA↗

arXiv2017

Wavelets and spectral triples for higher-rank graphs

In this paper, we present two new ways to associate a spectral triple to a higher-rank graph $Λ$. Moreover, we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Λ$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, to associate a family of ultrametric Cantor sets to a finite, strongly connected higher-rank graph $Λ$. Then we show that under mild hypotheses, the Pearson-Bellissard spectral triples of such Cantor sets have a regular $ζ$-function, whose abscissa of convergence agrees with the Hausdorff dimension of the Cantor set, and that the measure $μ$ induced by the associated Dixmier trace agrees with the measure $M$ on the infinite path space $Λ^\infty$ of $Λ$ which was introduced by an Huef, Laca, Raeburn, and Sims. Furthermore, we prove that $μ= M$ is a rescaled version of the Hausdorff measure of the ultrametric Cantor set. From work of Julien and Savinien, we know that for $ζ$-regular Pearson-Bellissard spectral triples, the eigenspaces of the associated Laplace-Beltrami operator constitute an orthogonal decomposition of $L^2(Λ^\infty, μ)$; we show that this orthogonal decomposition refines the wavelet decomposition of Farsi et al. In addition, we generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras, and prove that the wavelet decomposition of Farsi et al.~describes the eigenspaces of its Dirac operator.

math.OA↗

arXiv2016

KMS states on the operator algebras of reducible higher-rank graphs

We study the equilibrium or KMS states of the Toeplitz C*-algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of the real numbers in the torus. Here we use an embedding which leads to a dynamics which has previously been identified as "preferred", and we scale the dynamics so that 1 is a critical inverse temperature. As with 1-graphs, we study the strongly connected components of the vertices of the graph. The behaviour of the KMS states depends on both the graphical relationships between the components and the relative size of the spectral radii of the vertex matrices of the components. We test our theorems on graphs with two connected components. We find that our techniques give a complete analysis of the KMS states with inverse temperatures down to a second critical temperature beta_c<1.

math.OA↗

arXiv2016

Wavelets and spectral triples for fractal representations of Cuntz algebras

In this article we provide an identification between the wavelet decompositions of certain fractal representations of $C^*-$algebras of directed graphs of M. Marcolli and A. Paolucci, and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of $\mathcal{O}_D$. In particular, in this setting we use results of J. Pearson and J. Bellissard, and A. Julien and J. Savinien, to construct first the spectral triple and then the Laplace Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace Beltrami operators for the equal weight case for representations of Cuntz algebras, and in the uneven weight case for certain representations of $\mathcal{O}_2$, and show how the eigenspaces and wavelet subspaces at different levels are related.

math.OA↗

arXiv2016

Wavelets and graph $C^*$-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.

math.OA↗

arXiv2015

Separable representations, KMS states, and wavelets for higher-rank graphs

Let $Λ$ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of $C^*(Λ)$ on certain separable Hilbert spaces of the form $L^2(X,μ)$, by introducing the notion of a $Λ$-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, when $Λ$ is aperiodic, we obtain a faithful representation of $C^*(Λ)$ on $L^2(Λ^\infty, M)$, where $M$ is the Perron-Frobenius probability measure on the infinite path space $Λ^\infty$ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a $Λ$-semibranching function system gives rise to KMS states for $C^*(Λ)$. For the higher-rank graphs of Robertson and Steger, we also obtain a representation of $C^*(Λ)$ on $L^2(X, μ)$, where $X$ is a fractal subspace of $[0,1]$ by embedding $Λ^{\infty}$ into $[0,1]$ as a fractal subset $X$ of $[0,1]$. In this latter case we additionally show that there exists a KMS state for $C^*(Λ)$ whose inverse temperature is equal to the Hausdorff dimension of $X$. Finally, we construct a wavelet system for $L^2(Λ^\infty, M)$ by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

math.OA↗

arXiv2014

Spatial realisations of KMS states on the C*-algebras of higher-rank graphs

Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by intregrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

math.OA↗

arXiv2014

Quantum Heisenberg Manifolds as Twisted Groupoid $C^*$-Algebras

The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.

math.OA↗