23,683 research outputs found
Quantum Ultrametrics on AF Algebras and The Gromov-Hausdorff Propinquity
We construct quantum metric structures on unital AF algebras with a faithful
tracial state, and prove that for such metrics, AF algebras are limits of their
defining inductive sequences of finite dimensional C*-algebras for the quantum
propinquity. We then study the geometry, for the quantum propinquity, of three
natural classes of AF algebras equipped with our quantum metrics: the UHF
algebras, the Effros-Shen AF algebras associated with continued fraction
expansions of irrationals, and the Cantor space, on which our construction
recovers traditional ultrametrics. We also exhibit several compact classes of
AF algebras for the quantum propinquity and show continuity of our family of
Lip-norms on a fixed AF algebra. Our work thus brings AF algebras into the
realm of noncommutative metric geometry.Comment: 45 pages. v2: minor typos corrected; accepted in Studia Mathematic
Jones index theory for Hilbert C*-bimodules and its equivalence with conjugation theory
We introduce the notion of finite right (respectively left) numerical index
on a bimodule over C*-algebras A and B with a bi-Hilbertian structure. This
notion is based on a Pimsner-Popa type inequality. The right (respectively
left) index element of X can be constructed in the centre of the enveloping von
Neumann algebra of A (respectively B). X is called of finite right index if the
right index element lies in the multiplier algebra of A. In this case we can
perform the Jones basic construction. Furthermore the C*--algebra of bimodule
mappings with a right adjoint is a continuous field of finite dimensional
C*-algebras over the spectrum of Z(M(A)), whose fiber dimensions are bounded
above by the index. We show that if A is unital, the right index element
belongs to A if and only if X is finitely generated as a right module.
We show that bi-Hilbertian, finite (right and left) index C*-bimodules are
precisely those objects of the tensor 2-C*-category of right Hilbertian
C*-bimodules with a conjugate object, in the sense of Longo and Roberts, in the
same category.Comment: 59 pages, amste
Sufficient Dimension Reduction and Modeling Responses Conditioned on Covariates: An Integrated Approach via Convex Optimization
Given observations of a collection of covariates and responses , sufficient dimension reduction (SDR)
techniques aim to identify a mapping
with such that is independent of . The image
summarizes the relevant information in a potentially large number of covariates
that influence the responses . In many contemporary settings, the number
of responses is also quite large, in addition to a large number of
covariates. This leads to the challenge of fitting a succinctly parameterized
statistical model to , which is a problem that is usually not addressed
in a traditional SDR framework. In this paper, we present a computationally
tractable convex relaxation based estimator for simultaneously (a) identifying
a linear dimension reduction of the covariates that is sufficient with
respect to the responses, and (b) fitting several types of structured
low-dimensional models -- factor models, graphical models, latent-variable
graphical models -- to the conditional distribution of . We analyze the
consistency properties of our estimator in a high-dimensional scaling regime.
We also illustrate the performance of our approach on a newsgroup dataset and
on a dataset consisting of financial asset prices.Comment: 34 pages, 1 figur
Nullspaces and frames
In this paper we give new characterizations of Riesz and conditional Riesz
frames in terms of the properties of the nullspace of their synthesis
operators. On the other hand, we also study the oblique dual frames whose
coefficients in the reconstruction formula minimize different weighted norms.Comment: 16 page
- …