• Volume 118, Issue 3

August 2008,   pages  321-494

• Sharp Bounds on the Ranks of Negativity of Certain Sums

If 𝑀 is a complex vector space and $\langle\cdot p,\cdot p\rangle$ a Hermitian sesquilinear form on 𝑀 with a finite rank of negativity 𝑘 (i.e., 𝑘 is the maximal dimension of any linear subspace 𝐸 of 𝑀 satisfying $\langle x,x\rangle &lt; 0$ for each nonzero 𝑥 in 𝐸), if 𝑛 is a positive integer, and if $a_1,\ldots,a_n$ are endomorphisms of 𝑀, then it is easy to see that the Hermitian sesquilinear form

\begin{equation*}(x, y)\mapsto\sum\limits_{v=1}^n\langle a_v x, a_v y\rangle\end{equation*}

on 𝑀 has rank of negativity at most $nk$. It is also fairly easy to see that the bound $nk$ cannot be improved in general. Less trivial is the fact that it cannot be improved by making the following assumption:

(a) the space 𝑀 is the *-algebra $A:=C[[\omega_1,\omega_2]]$ of polynomials in two self-adjoint non-commuting indeterminates; there is a (necessarily Hermitian) linear form 𝜑 on 𝐴 such that $\langle x, y\rangle =\varphi(y^\ast x)(x, y, \in A)$; and each $a_v$ is just left multiplication by some element of 𝐴 (which we may denote by a_v' at no great risk of confusion).

Now suppose that, with $M,\langle \cdot p ,\cdot p\rangle, k,n$, and $a_1,\ldots,a_n$ as initially, the following two conditions are satisfied:

(i) each $a_v$ has a formal adjoint $a^\ast_v$, being an endomorphism of 𝑀 such that

$$\langle a_v x, y\rangle = \langle x,a^\ast_v y\rangle\quad (x, y \in M)$$;

(ii) the mappings $a_1,\ldots,a_n, a^\ast_1,\ldots,a^\ast_n$ commute pairwise.

Then the bound $nk$ can be replaced by 𝑘 (regardless of how large 𝑛 may be). This result cannot be improved in general since it may happen that each $a_v$ is a scalar multiple of the identical mapping of 𝑀 into itself (not all $a_v$ equal to 0), in which case the form (1) is a positive multiple of $\langle \cdot p ,\cdot p\rangle itself.$

There are ties with the subjects of positive semidefinite submodules' (positive semidefinite left ideals') and definitisation'.

• On Split Lie Algebras with Symmetric Root Systems

We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such algebras 𝐿 is of the form $L=\mathcal{U}+\sum_j I_j$ with $\mathcal{U}$ a subspace of the abelian Lie algebra 𝐻 and any $I_j$ a well described ideal of 𝐿, satisfying $[I_j,I_k]=0$ if $j\neq k$. Under certain conditions, the simplicity of 𝐿 is characterized and it is shown that 𝐿 is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.

• Decomposition and Removability Properties of John Domains

In this paper we characterize John domains in terms of John domain decomposition property. In addition, we also show that a domain 𝐷 in $\mathbb{R}^n$ is a John domain if and only if $D\backslash P$ is a John domain, where 𝑃 is a subset of 𝐷 containing finitely many points of 𝐷. The best possibility and an application of the second result are also discussed.

• The Structure of some Classes of 𝐾-Contact Manifolds

We study projective curvature tensor in 𝐾-contact and Sasakian manifolds. We prove that (1) if a 𝐾-contact manifold is quasi projectively flat then it is Einstein and (2) a 𝐾-contact manifold is 𝜉-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a 𝐾-contact manifold to be quasi projectively flat and 𝜑-projectively flat are obtained. We also prove that for a $(2n+1)$-dimensional Sasakian manifold the conditions of being quasi projectively flat, 𝜑-projectively flat and locally isometric to the unit sphere $S^{2n+1}(1)$ are equivalent. Finally, we prove that a compact 𝜑-projectively flat 𝐾-contact manifold with regular contact vector field is a principal $S^1$-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.

• Holomorphic Two-Spheres in Complex Grassmann Manifold $G(2, 4)$

In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold $G(2,4)$. We show that if the Gaussian curvature 𝐾 (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies $K\leq 2$ (or $K\geq 2$), then 𝐾 must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature $K\leq 1$ (or $K\geq 1$), then $K=1$. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in $G(2,4)$ must be $U(4)$-equivalent.

• The Motive of the Moduli Stack of 𝐺-Bundles over the Universal Curve

We define relative motives in the sense of André. After associating a complex in the derived category of motives to an algebraic stack we study this complex in the case of the moduli of 𝐺-bundles varying over the moduli of curves.

• The $L_p$-Curvature Images of Convex Bodies and $L_p$-Projection Bodies

Associated with the $L_p$-curvature image defined by Lutwak, some inequalities for extended mixed 𝑝-affine surface areas of convex bodies and the support functions of $L_p$-projection bodies are established. As a natural extension of a result due to Lutwak, an $L_p$-type affine isoperimetric inequality, whose special cases are $L_p$-Busemann–Petty centroid inequality and $L_p$-affine projection inequality, respectively, is established. Some $L_p$-mixed volume inequalities involving $L_p$-projection bodies are also established.

• Limit Algebras of Differential Forms in Non-Commutative Geometry

Given a C∗-normed algebra A which is either a Banach ∗-algebra or a Frechet ∗-algebra, we study the algebras ∞A and A obtained by taking respectively the projective limit and the inductive limit of Banach ∗-algebras obtained by completing the universal graded differential algebra ∗A of abstract non-commutative differential forms over A. Various quantized integrals on ∞A induced by a K-cycle on A are considered. The GNS-representation of ∞A defined by a d-dimensional non-commutative volume integral on a d+-summable K-cycle on A is realized as the representation induced by the left action of A on ∗A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).

• Quantum Random Walks and their Convergence to Evans-Hudson Flows

Using coordinate-free basic operators on toy Fock spaces, quantum random walks are defined following the ideas of Attal and Pautrat. Extending the result for one dimensional noise, strong convergence of quantum random walks associated with bounded structure maps to Evans–Hudson flow is proved under suitable assumptions. Starting from the bounded generator of a given uniformly continuous quantum dynamical semigroup on a von Neumann algebra, we have constructed quantum random walks which converges strongly and the strong limit gives an Evans–Hudson dilation for the semigroup.

• Hartman-Mycielski Functor of Non-Metrizable Compacta

We investigate certain topological properties of the normal functor 𝐻, introduced by the first author, which is a certain functorial compactification of the Hartman–Mycielski construction $HM$. We prove that 𝐻 is always open and we also find the condition when $HX$ is an absolute retract, homeomorphic to the Tychonov cube.

• Growth of Preferential Attachment Random Graphs Via Continuous-Time Branching Processes

Some growth asymptotics of a version of `preferential attachment’ random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.