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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

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Finite Blaschke products and the construction of rational $\Gamma$-inner functions

Agler Jim, Lykova Zinaida A., Young N. J.
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.

Algebraic and geometric aspects of rational $\Gamma$-inner functions

Agler Jim, Lykova Zinaida A., Young Nicholas J.
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 $\Gamma$.

3-extremal holomorphic maps and the symmetrised bidisc

Agler Jim, Lykova Zinaida A., Young N. J.
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 $\mathcal{G}$.

Extremal holomorphic maps and the symmetrised bidisc

Agler Jim, Lykova Zinaida A., Young N. J.
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.

Projectivity of Banach and $C^*$-algebras of continuous fields

Cushing David, Lykova Zinaida A.
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.

Pseudo-Taylor expansions and the Carath\'{e}odory-Fej\'{e}r problem

Agler Jim, Lykova Zinaida A., Young N. J.
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.

The boundary Carath\'{e}odory-Fej\'{e}r interpolation problem

Agler Jim, Lykova Zinaida A., Young N. J.
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.

The higher-dimensional amenability of tensor products of Banach algebras

Lykova Zinaida A.
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.

The K\"unneth formula for nuclear $DF$-spaces and Hochschild cohomology

Lykova Zinaida A.
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.

Cyclic cohomology of certain nuclear Fr\'echet and DF algebras

Lykova Zinaida A.
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 $G$.