The Cuntz splice does not preserve ∗*-isomorphism of Leavitt path algebras over Z\mathbb{Z}

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in $L_{\mathbb{Z}}(E)$ for $E$ a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in $L_{2,\mathbb{Z}}$ given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just $\mathbb{Z}$.Comment: 17 pages. Since version 2 we extended the arguments from Z to more general ring

Leavitt RR-algebras over countable graphs embed into L2,RL_{2,R}

For a commutative ring $R$ with unit we show that the Leavitt path algebra $L_R(E)$ of a graph $E$ embeds into $L_{2,R}$ precisely when $E$ is countable. Before proving this result we prove a generalised Cuntz-Krieger Uniqueness Theorem for Leavitt path algebras over $R$.Comment: 17 pages. At the request of a referee the previous version of this paper has been split into two papers. This version is the first of these papers. The second will also be uploaded to the arXi

L2,Z⊗L2,ZL_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}} does not embed in L2,ZL_{2,\mathbb{Z}}

For a commutative ring $R$ with unit we investigate the embedding of tensor product algebras into the Leavitt algebra $L_{2,R}$. We show that the tensor product $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$ (as a unital $*$-algebra). We also prove a partial non-embedding result for the more general $L_{2,R} \otimes L_{2,R}$. Our techniques rely on realising Thompson's group $V$ as a subgroup of the unitary group of $L_{2,R}$.Comment: 16 pages. At the request of a referee the paper arXiv:1503.08705v2 was split into two papers. This is the second of those paper

Almost Commuting Orthogonal Matrices

We show that almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all pairs of almost commuting unitaries are close to commuting pairs. Our techniques also yield results about almost normal matrices over the reals and the quaternions.Comment: 13 pages, 3 figure

Three-dimensional theory of stimulated Raman scattering

We present a three-dimensional theory of stimulated Raman scattering (SRS) or superradiance. In particular we address how the spatial and temporal properties of the generated SRS beam, or Stokes beam, of radiation depends on the spatial properties of the gain medium. Maxwell equations for the Stokes field operators and of the atomic operators are solved analytically and a correlation function for the Stokes field is derived. In the analysis we identify a superradiating part of the Stokes radiation that exhibit beam characteristics. We show how the intensity in this beam builds up in time and at some point largely dominates the total Stokes radiation of the gain medium. We show how the SRS depends on geometric factors such as the Fresnel number and the optical depth, and that in fact these two factors are the only factors describing the coherent radiation.Comment: 21 pages 14 figure

Estimates of Effective Hubbard Model Parameters for C20 isomers

We report on an effective Hubbard Hamiltonian approach for the study of electronic correlations in C$_{20}$ isomers, cage, bowl and ring, with quantum Monte Carlo and exact diagonalization methods. The tight-binding hopping parameter, $t$, in the effective Hamiltonian is determined by a fit to density functional theory calculations, and the on-site Coulomb interaction, $U/t$, is determined by calculating the isomers' affinity energies, which are compared to experimental values. For the C$_{20}$ fullerene cage we estimate $t_{\rm cage}\simeq 0.68-1.36$ eV and $(U/t)_{\rm cage} \simeq 7.1-12.2$. The resulting effective Hamiltonian is then used to study the shift of spectral peaks in the density of states of neutral and one-electron-doped C$_{20}$ isomers. Energy gaps are also extracted for possible future comparison with experiments.Comment: 6 pages, 5 figure

Invariance of the Cuntz splice

We show that the Cuntz splice induces stably isomorphic graph $C^*$-algebras.Comment: Our arguments to prove invariance of the Cuntz splice for unital graph C*-algebras in arXiv:1505.06773 applied with only minor changes in the general case. Since most of the results of that preprint have since been superseded by other forthcoming work, we do not intend to publish it, whereas this work is intended for publication. arXiv admin note: substantial text overlap with arXiv:1505.0677