A purely infinite AH-algebra and an application to AF-embeddability

We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs the Cuntz algebra O_\infty tensorially. Thus one can reach an O_\infty-absorbing C*-algebra as an inductive limit of the finite and elementary C*-algebras C_0([0,1),M_k). As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exact C*-algebra is AF-embeddable, and we exhibit a concrete AF-algebra into which this class of C*-algebras can be embedded.Comment: 20 pages, revised January 2004, to appear in Israel J. Mat

Structure of the core of magnetic vortices in d-wave superconductors with a subdominant triplet pairing mechanism

The quasiparticle states found in the vortex core of a high-T$_{\rm{c}}$ cuprate superconductor may be probed by scanning tunneling spectroscopy. Results of such experiments have revealed typical spectra that are quite different from what is seen in conventional low-Tc superconductors. In particular the Caroli-deGennes-Matricon state at $E\sim 0$ in the core center is not seen. Instead, in a high-T$_{\rm{c}}$ vortex core, quasiparticle states are found at energies that are at a sizable fraction of the gap energy. One explanation for this could be that a finite amplitude of a competing orderparameter stabilizes in the vortex-core center. Here I will explore the possibility of nucleating a vortex-core state that locally breaks inversion symmetry. The vortex-core orderparameter is of mixed parity, a $[d + i p]$-wave, and the quasiparticle spectra in the core center lacks the E=0 states.Comment: 6 pages, 5 figures, accepted for publication as a regular article in Physical Review

The stable and the real rank of Z-absorbing C*-algebras

Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is weakly unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then A is isomorphic to A tensor O_infty if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterise when A is of real rank zero.Comment: 24 pages. Minor revisions August 2004. To appear in International J. Mat

Shenfun -- automating the spectral Galerkin method

With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is made towards automating the implementation of the spectral Galerkin method for simple tensor product domains, consisting of (currently) one non-periodic and any number of periodic directions. The user interface to shenfun is intentionally made very similar to FEniCS (fenicsproject.org). Partial Differential Equations are represented through weak variational forms and solved using efficient direct solvers where available. MPI decomposition is achieved through the {mpi4py-fft} module (bitbucket.org/mpi4py/mpi4py-fft), and all developed solver may, with no additional effort, be run on supercomputers using thousands of processors. Complete solvers are shown for the linear Poisson and biharmonic problems, as well as the nonlinear and time-dependent Ginzburg-Landau equation.Comment: Presented at MekIT'17, the 9th National Conference on Computational Mechanic

On the Aharonov-Casher formula for different self-adjoint extensions of the Pauli operator with singular magnetic field

Two different self-adjoint Pauli extensions describing a spin-1/2 two-dimensional quantum system with singular magnetic field are studied. An Aharonov-Casher type formula is proved for the maximal Pauli extension and it is also checked that this extension can be approximated by operators corresponding to more regular magnetic fields