Searcharxiv⌕ Search

arXiv subjects

Nathanial P. Brown

Publications and source records attributed to Nathanial P. Brown.

28 records · Page 2Linked to original sources
arXiv2003

AF embeddings and the numerical computation of spectra in irrational rotation algebras

The spectral analysis of discretized one-dimensional Schrödinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is that of finding finite dimensional matrices whose eigenvalues approximate the spectrum of an infinite dimensional operator. In this note we observe that the seminal work of Pimsner-Voiculescu on AF embeddings of irrational rotation algebras provides a nice answer to the finite dimensional spectral approximation problem for a broad class of operators including the quasiperiodic case of the Schrödinger operators mentioned above. Indeed, the theory of continued fractions not only provides good matrix models for spectral computations (i.e. the Pimsner-Voiculescu construction) but also yields {\em sharp} rates of convergence for spectral approximations of operators in irrational rotation algebras.

math.NA↗
arXiv2003

Quasidiagonality and the finite section method

Quasidiagonal operators on a Hilbert space are a large and important class (containing all self-adjoint operators for instance). They are also perfectly suited for study via the finite section method (a particular Galerkin method). Indeed, the very definition of quasidiagonality yields finite sections with good convergence properties. Moreover, simple operator theory techniques yield estimates on certain rates of convergence. In the case of quasidiagonal band operators both the finite sections and rates of convergence are explicitly given.

math.NA↗
arXiv2003

Excision and a Theorem of Popa

We give an elementary C*-algebraic proof of a result of Sorin Popa which is of fundamental importance to Elliott's Classification Program.

math.OA↗
arXiv2003

Connes' Embedding Problem and Lance's WEP

A II_1 factor embeds into the ultraproduct of the hyperfinite II_1 factor if and only if it satisfies the W*-analogue of Lance's weak expectation property. This note gives a self contained proof of this fact.

math.OA↗
arXiv2002

Popa algebras in free group factors

For each 1<s<\infty, a Popa algebra A_s is constructed that embeds as a weakly dense C*-subalgebra of the interpolated free group factor L(F_s). Certain approximation properties for A_s are shown. It follows that L(F_s) has the weak expectation property of Lance with respect to A_s. In the course of the demonstration, it is proved that under certain conditions, full amalgamated free products of matrix algebras are residually finite dimensional.

math.OA↗
arXiv2002

Tracial invariants, classification and II_1 factor representations of Popa algebras

Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary self adjoint operators. (2) A McDuff factor embeds into the ultrapower of the hyperfinite II_1 factor if and only if it contains a weakly dense operator system which is injective. (3) There exists a simple, quasidiagonal, real rank zero C*-algebra with non-hyperfinite II_1 factor representations and which is not tracially AF. This answers negatively questions of Sorin Popa and, respectively, Huaxin Lin.

math.OA↗
arXiv2000

On Quasidiagonal C*-algebras

We give a detailed survey of the theory of quasidiagonal C*-algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and not known) about extensions, quotients, tensor products, ect. of quasidiagonal C*-algebras. We also point out how quasidiagonality is connected to some important open problems.

math.OA↗
arXiv1997

On the AF embeddability of crossed products of AF algebras by the integers

It is shown that if A is an AF algebra then a crossed product of A by the integers can be embedded into an AF algebra if and only if the crossed product is stably finite. This equivalence follows from a simple K-theoretic characterization of AF embeddability. It is then shown that if a crossed product of an AF algebra by the integers is AF embeddable then the AF embedding can be chosen in such a way as to induce a rationally injective map on K_0 of the crossed product.

funct-an↗