An official website of the United States government
Motivated by the theory of Cuntz-Krieger algebras we define and study C ∗ C^\ast -algebras associated to directed quantum graphs. For classical graphs the C ∗ C^\ast -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to K K KK -equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C ∗ C^\ast -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of K K KK -theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
more »
« less
Monic representations of finite higher-rank graphs
In this paper, we define the notion of monic representation for the $$C^{\ast }$$ -algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $$C^{\ast }$$ -algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $$\unicode[STIX]{x1D6EC}$$ -semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs. J. Math. Anal. Appl. 434 (2015), 241–270) and also provide a universal representation model for non-negative monic representations.
more »
« less
- Award ID(s):
- 1800749
- PAR ID:
- 10189090
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 40
- Issue:
- 5
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1238 to 1267
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $$C^{\ast }-$$algebra, the $$C^{\ast }-$$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $$C^{\ast }-$$algebras; any *−representation of the Cuntz–Toeplitz $$C^{\ast }-$$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.more » « less
-
Bodlaender, Hans L (Ed.)In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in TC². Our approach builds on the framework of Köbler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. In order to control the depth of our circuit, we leverage the fact that any graph of rank-width k admits a rank decomposition of width ≤ 2k and height O(log n) (Courcelle & Kanté, WG 2007). This allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation. Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in TC¹. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the (6k+3)-dimensional Weisfeiler-Leman (WL) algorithm can identify graphs of rank-width k using only O(log n) rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by FO + C formulas with 6k+4 variables and quantifier depth O(log n). Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in NC.more » « less
-
Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure $$F: \Omega \to B(H)$$ has an integral representation of the form $$F(E) =\sum_{k=1}^{m} \int_{E}\, G_{k}(\omega)\otimes G_{k}(\omega) d\mu(\omega)$$ for some weakly measurable maps $$G_{k} \ (1\leq k\leq m) $$ from a measurable space $$\Omega$$ to a Hilbert space $$\mathcal{H}$$ and some positive measure $$\mu$$ on $$\Omega$$. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.more » « less
-
We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite rank groups, and we apply these descriptions to study polynomial representations and representation stability of parabolic subgroups of general linear groups.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2018.79
