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Lykova Zinaida A. | Lykova Z. A. | Lykova Zinaida

A | A Gustavo Bruzual | A Dang Quang | A Krishna Chaitanya | A Lazarian | A Germina K | A M. | A Pranav | A Antony Franklin | A Azeef Muhammed P

10 May 2015
math.CV
arxiv.org/abs/1505.02415

Let [ \Gamma = {(z+w, zw): |z|\leq 1, |w|\leq 1} \subset \mathbb{C}^2. ]
A $\Gamma$-inner function is defined to be a holomorphic map $h$ from the unit
disc $\mathbb{D}$ to $\Gamma$ whose boundary values at almost all points of the
unit circle $\mathbb{T}$ belong to the distinguished boundary $b\Gamma$ of
$\Gamma$. A rational $\Gamma$-inner function $h$ induces a continuous map
$h|*\mathbb{T}$ from the unit circle to $b\Gamma$. The latter set is
topologically a M\"obius band and so has fundamental group $\mathbb{Z}$. The
{\em degree} of $h$ is defined to be the topological degree of $h|*\mathbb{T}$.
In a previous paper the authors showed that if $h=(s,p)$ is a rational
$\Gamma$-inner function of degree $n$ then $s^2-4p$ has exactly $n$ zeros in
the closed unit disc $\mathbb{D}^-$, counted with an appropriate notion of
multiplicity. In this paper, with the aid of a solution of an interpolation
problem for finite Blaschke products, we explicitly construct the rational
$\Gamma$-inner functions of degree $n$ with the $n$ zeros of $s^2-4p$ and the
corresponding values of $s$, prescribed.

14 Feb 2015
math.CV
arxiv.org/abs/1502.04216

The set [ \Gamma {\stackrel{\rm def}{=}} {(z+w,zw):|z|\leq 1,|w|\leq 1} \subset {\mathbb{C}}^2 ] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the M\"obius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $\Gamma$ that map the boundary of the disc to the distinguished boundary of $\Gamma$.

26 Jul 2013
math.CV
arxiv.org/abs/1307.7081

We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ {(z+w,zw): z,w\in\mathbb{D}}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be called {\em aligned} and {\em caddywhompus} functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are 3-extremal. We describe a method for the construction of aligned rational $\mathcal{G}$-inner functions; with the aid of this method we reduce the solution of a 3-point interpolation problem for aligned holomorphic maps from $\mathbb{D}$ to $\mathcal{G}$ to a collection of classical Nevanlinna-Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for $\mathcal{G}$.

29 Apr 2012
math.CV
arxiv.org/abs/1204.6517

We introduce the class of $n$-extremal holomorphic maps, a class that
generalises both finite Blaschke products and complex geodesics, and apply the
notion to the finite interpolation problem for analytic functions from the open
unit disc into the symmetrised bidisc $\Gamma$. We show that a well-known
necessary condition for the solvability of such an interpolation problem is not
sufficient whenever the number of interpolation nodes is 3 or greater. We
introduce a sequence $\mathcal{C}*\nu, \nu \geq 0,$ of necessary conditions for
solvability, prove that they are of strictly increasing strength and show that
$\mathcal{C}*{n-3}$ is insufficient for the solvability of an $n$-point problem
for $n\geq 3$. We propose the conjecture that condition $\mathcal{C}*{n-2}$ is
necessary and sufficient for the solvability of an $n$-point interpolation
problem for $\Gamma$ and we explore the implications of this conjecture.
We introduce a classification of rational $\Gamma$-inner functions, that is,
analytic functions from the disc into $\Gamma$ whose radial limits at almost
all points on the unit circle lie in the distinguished boundary of $\Gamma$.
The classes are related to $n$-extremality and the conditions
$\mathcal{C}*\nu$; we prove numerous strict inclusions between the classes.

26 Apr 2011
math.FA math.OA
arxiv.org/abs/1104.4935

We give necessary and sufficient conditions for the left projectivity and
biprojectivity of Banach algebras defined by locally trivial continuous fields
of Banach algebras. We identify projective $C^*$-algebras $\A$ defined by
locally trivial continuous fields $\mathcal{U} = {\Omega,(A_t)_{t \in
\Omega},\Theta}$ such that each $C^*$-algebra $ A_{t}$ has a strictly positive
element. For a commutative $C^*$-algebra $\D$ contained in ${\cal B}(H)$, where
$H$ is a separable Hilbert space, we show that the condition of left
projectivity of $\D$ is equivalent to the existence of a strictly positive
element in $\D$ and so to the spectrum of $\D$ being a Lindel$\ddot{\rm o}$f
space.

06 Jan 2011
math.CV
arxiv.org/abs/1101.1251

We give a new solvability criterion for the boundary Carath\'{e}odory-Fej\'{e}r problem: given a point $x \in \mathbb{R}$ and, a finite set of target values $a^0,a^1,...,a^n \in \mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \to x$ nontangentially in the upper half plane is $a^k$ for $k= 0,1,...,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.

05 Nov 2010
math.CV
arxiv.org/abs/1011.1399

We give an elementary proof of a solvability criterion for the {\em boundary Carath\'{e}odory-Fej\'{e}r problem}: given a point $x \in \R$ and, a finite set of target values, to construct a function $f$ in the Pick class such that the first few derivatives of $f$ take on the prescribed target values at $x$. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem. The proofs are based on a reduction method due to Julia and Nevanlinna.

29 Apr 2009
math.KT math.FA
arxiv.org/abs/0904.4548

We investigate the higher-dimensional amenability of tensor products $\A \ptp
\B$ of Banach algebras $\A$ and $\B$. We prove that the weak bidimension $db_w$
of the tensor product $\A \ptp \B$ of Banach algebras $\A$ and $\B$ with
bounded approximate identities satisfies [ db_w \A \ptp \B = db_w \A + db_w
\B. ] We show that it cannot be extended to arbitrary Banach algebras. For
example, for a biflat Banach algebra $\A$ which has a left or right, but not
two-sided, bounded approximate identity, we have $db_w \A \ptp \A \le 1$ and
$db_w \A + db_w \A =2.$ We describe explicitly the continuous Hochschild
cohomology $\H^n(\A \ptp \B, (X \ptp Y)^*)$ and the cyclic cohomology
$\H\C^n(\A \ptp \B)$ of certain tensor products $\A \ptp \B$ of Banach algebras
$\A$ and $\B$ with bounded approximate identities; here $(X \ptp Y)^*$ is the
dual bimodule of the tensor product of essential Banach bimodules $X$ and $Y$
over $\A$ and $\B$ respectively.

12 Sep 2007
math.KT math.FA
arxiv.org/abs/0709.1911

We consider complexes $(\X, d)$ of nuclear Fr\'echet spaces and continuous
boundary maps $d_n$ with closed ranges and prove that, up to topological
isomorphism, $ (H_{n}(\X, d))^*$ $\iso$ $H^{n}(\X^*,d^*),$ where
$(H_{n}(\X,d))^*$ is the strong dual space of the homology group of $(\X,d)$
and $ H^{n}(\X^*,d^*)$ is the cohomology group of the strong dual complex
$(\X^*,d^*)$. We use this result to establish the existence of topological
isomorphisms in the K\"{u}nneth formula for the cohomology of complete nuclear
$DF$-complexes and in the K\"{u}nneth formula for continuous Hochschild
cohomology of nuclear $\hat{\otimes}$-algebras which are Fr\'echet spaces or
$DF$-spaces for which all boundary maps of the standard homology complexes have
closed ranges. We describe explicitly continuous Hochschild and cyclic
cohomology groups of certain tensor products of $\hat{\otimes}$-algebras which
are Fr\'echet spaces or nuclear $DF$-spaces.

08 Apr 2007
math.KT math.FA
arxiv.org/abs/0704.1019

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $\phi: \X\to \Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism of cohomology groups $H^n(\phi): H^n(\X) \to H^n(\Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $\hat{\otimes}$-algebras: the tensor algebra $E \hat{\otimes} F$ generated by the duality $(E, F, < \cdot, \cdot >)$ for nuclear Fr\'echet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$; nuclear biprojective K\"{o}the algebras $\lambda(P)$ which are Fr\'echet spaces or $DF$-spaces; the algebra of distributions $\mathcal{E}^*(G)$ on a compact Lie group $G$.