pp 321-350
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 $\langle x,x\rangle < 0$ 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 $A:=C[[𝜔_1,𝜔_2]]$ of polynomials in two self-adjoint non-commuting indeterminates; there is a (necessarily Hermitian) linear form 𝜑 on 𝐴 such that $\langle x, y\rangle =\varphi(y^* x)(x, y, \in A)$; 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 $a^*_v$, being an endomorphism of 𝑀 such that
$$\langle a_v x, y\rangle = \langle x,a^*_v y\rangle\quad (x, y \in M)$$;
(ii) the mappings $a_1,\ldots,a_n, a^*_1,\ldots,a^*_n$ 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'.
pp 351-356
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 $j≠ k$. 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.
pp 357-370
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.
pp 371-379
The Structure of some Classes of 𝐾-Contact Manifolds
Mukut Mani Tripathi Mohit Kumar Dwivedi
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.
pp 381-388
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≤ 2 (or 𝐾 ≥ 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 ≤ 1 (or 𝐾 ≥ 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.
pp 389-411
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.
pp 413-424
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.
pp 425-441
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]).
pp 443-465
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.
pp 467-472
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.
pp 473-494
Growth of Preferential Attachment Random Graphs Via Continuous-Time Branching Processes
Krishna B Athreya Arka P Ghosh Sunder Sethuraman
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.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.