We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and $K$-groups of our algebras.

Let $\phi :\,X\,\to \,M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension ≥ 2. Let $D$ be a reduced hypersurface in $M$, and $\gamma $ a general linear automorphism of $M$. We show that, under a certain differential-geometric condition on $\phi (X)$ and $D$, the fundamental group ${{\text{ }\!\!\pi\!\!\text{ }}_{1}}\left( {{\left( \gamma \,o\,\phi \right)}^{-1}}\,\left( M\,\backslash \,D \right) \right)$ is isomorphic to a central extension of ${{\pi }_{1}}\left( M\,\backslash \,D \right)\,\,\times \,{{\pi }_{1}}\left( X \right)$ by the cokernel of ${{\pi }_{2}}\left( \phi \right)\,:\,{{\pi }_{2}}\left( X \right)\,\to {{\pi }_{2}}\left( M \right)$.

Every classical inner function $\varphi$ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when $\varphi (z)\,=\,z$. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.

James gave an integral homotopy decomposition of $\sum \Omega \sum X$, Hilton-Milnor one for $\Omega (\sum X\,\vee \,\sum Y)$, and Cohen-Wu gave $p$-local decompositions of $\Omega \sum X$ if $X$ is a suspension. All are natural. Using idempotents and telescopes we show that the James and Hilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative $\text{co-}H$ spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative $\text{co-}H$ space.

Equivariant holomorphic maps of Hermitian symmetric domains into Siegel upper half spaces can be used to construct families of abelian varieties parametrized by locally symmetric spaces, which can be regarded as complex torus bundles over the parameter spaces. We extend the construction of such torus bundles using 2-cocycles of discrete subgroups of the semisimple Lie groups associated to the given symmetric domains and investigate some of their properties. In particular, we determine their cohomology along the fibers.

We describe the irrational pencils on surfaces of general type with ${{p}_{g}}=q=2$.

Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain.

Let ${{l}^{\Phi }}$ and ${{L}^{\Phi }}\left( \Omega \right)$ be the Orlicz sequence space and function space generated by $N$-function $\Phi (u)$ with Orlicz norm. We give equivalent expressions for the nonsquare constants ${{C}_{J}}\left( {{l}^{\Phi }} \right),\,{{C}_{J}}\left( {{L}^{\Phi }}\left( \Omega \right) \right)$ in sense of James and ${{C}_{S}}\left( {{l}^{\Phi }} \right),\,{{C}_{S}}\left( {{L}^{\Phi }}\left( \Omega \right) \right)$ in sense of Schäffer. We are devoted to get practical computational formulas giving estimates of these constants and to obtain their exact value in a class of spaces ${{l}^{\Phi }}$ and ${{L}^{\Phi}}\left(\Omega \right)$.

Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$ over ${{\mathbf{Q}}_{p}}$, $O$ the ring of integers in $K,\,B\,=\,\{{{\beta }_{1}},\ldots .{{\beta }_{r}}\}$ an integral basis of $K$ over ${{\mathbf{Q}}_{p}}$, $u$ a unit in $O$ and consider sets of the form $N\,=\,\{{{n}_{1}}{{\beta }_{1}}\,+\ldots +\,{{n}_{r}}{{\beta }_{r}}\,:\,1\,\le \,{{n}_{j}}\,\le \,{{N}_{j}},\,1\,\le \,j\,\le \,r\}$. We show under certain growth conditions that the pair correlation of $\{u{{z}^{2}}\,:\,z\,\in N\}$ becomes Poissonian.