Wavelets and Graph C ∗-Algebras

Abstract

Here we give an overview on the connection between wavelet theory and representation theory for graph C * -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 [20], 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 [22] to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.2010 Mathematics Subject Classification: 46L05, 42C40

Wavelets and spectral triples for fractal representations of Cuntz algebras

“…, v k ], as was established in the proof of Theorem 6.5. Consequently, we can factor the term (µ One can also construct representations and wavelet spaces of O 2 associated to the weighted Bratteli diagram (∂B 2 , w r 2 ); see Theorem 3.8 of [8]. This is the analogue of Theorem 5.1 above for the uneven weight case.…”

“…Let µ r 2 be the Markov probability measure on the infinite path space Λ ∞ 2 corresponding to the weight assigning r to the vertex v 1 and 1 − r to the vertex v 2 . Then for the corresponding representation of O 2 on L 2 (Λ ∞ 2 , µ r 2 ) defined in Theorem 3.8 of [8], we haveW 0 = span η:|η|=0 {E r η }, where E r η are the eigenspaces of the Laplace-Beltrami operator defined in Proposition 6.9.…”

“…Proof. As in Theorem 3.8 of [8] and Section 5 above, we have an inner product on C 2 defined by (x j ), (y j ) = 2 j=1…”

Irreducible and permutative representations of ultragraph Leavitt path algebras

“…Permutative representations of combinatorial algebras such as the Cuntz-Krieger, graph and ultragraph algebras have connections with the theory of operator algebras, dynamical systems, and pure algebra (see [7,35,19,15,13]), and therefore are a subject of much interest. In this section we characterize the perfect, irreducible and permutative representations of an ultragraph Leavitt path algebra.…”

Representations and the reduction theorem for ultragraph Leavitt path algebras

“…of Kumjian-Pask in [8], of Steinberg algebras in [3,10]. Representations of various algebras, in connection with branching systems, were studied in [11,15,17,18,20,21,22,23,24,25,27]. To describe the connections of representations of ultragraph Leavitt path algebras with branching systems is the second goal of this paper.…”