WOAT 2016

International Workshop on Operator Theory and Operator Algebras

Instituto Superior Técnico, Lisbon, Portugal. July 5-8, 2016.

WOAT 2016

International Workshop on Operator Theory and Operator Algebras

Instituto Superior Técnico, Lisbon, Portugal. July 5-8, 2016.

- Carlos André
*(Faculdade de Ciências de Lisboa, Portugal)* -
**Extreme supercharacters of the infinite unitriangular group.**We define extreme supercharacters of the infinite unitriangular group, introduce the ramification scheme associated with the classical supercharacter theories of the finite unitriangular groups and describe how extreme supercharacters of the infinite unitriangular group appear as weak limits of supercharacters of those.

In order to prove that the set of extreme supercharacters is closed (with respect to the topology of weak convergence), we will deform the ramification sheme using appropriate operations of induction and restriction and show that the resulting scheme is multiplicative and determines a convenient Riesz ring.

- Pere Ara
*(Universitat Autònoma de Barcelona, Spain)* -
**Amenability and paradoxical decompositions: algebraic aspects.** *Amenability of groups was introduced by von Neumann in connection with the Banach-Tarski paradox, concerning paradoxical decompositions for subsets of $\mathbb{R}^3$. A very useful characterization of amenability in terms of almost invariant sets was obtained by Følner in 1955 [4]. Since then, the concept has been extended to various diferent contexts. I will introduce the concept of amenability for metric spaces with bounded geometry, and I will recall its characterization using invariant means and using paradoxical decompositions, which is due to Ceccherini-Silberstein, Grigorchuck and de la Harpe [2]. I will then introduce the notion of amenability of general algebras over a field, which is due to Gromov [5]. In a joint paper with Kang Li, Fernando Lledó and Jianchao Wu [1], we have obtained a characterization of amenability for general algebras, which parallels the corresponding characterization for metric spaces. This extends work of Gábor Elek [3], who considered the case of countably dimensional algebras with no zero-divisors. Both notions are unified by the concept of the translation algebra of a discrete metric space, which will be considered in the last part of the talk.*- References
- 1. P. Ara, K. Li, F. Lledó and J. Wu, Amenability in coarse geometry, algebra and Roe C*-algebras, Preprint.
- 2. T. Ceccherini-Silberstein, R. Grigorchuk and P. de la Harpe, Amenability and paradoxical decomposi- tion for pseudogroups and for discrete metric spaces, Proc. Steklov Inst. Math. 224 (1999) 57-97.
- 3. G. Elek, The amenability of affine algebras, J. Algebra 264 (2003) 469-478.
- 4. E. Følner, On groups with full Banach mean value, Math. Scand. 3 (1955) 243-254.

5. M. Gromov, Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I, Math.

Phys. Anal. Geom. 2 (1999) 323-415.

- Natália Bebiano
*(Universidade de Coimbra, Portugal)* -
**Non-Hermitian quantum mechanics of bosonic systems.** *In conventional formulations of non-relativistic quantum mechanics, observables are described by self-adjoint operators whose eigenvalues are the possible results of the respective measurements. In particular, the Hamiltonian operator is self-adjoint. However, certain relativistic extensions of quantum mechanics, such as the Klein-Gordon theory, lead to non self-adjoint Hamiltonian operators. We investigate spectral aspects of non self-adjoint bosonic operators. We show that the so-called equation of motion method, which is well known from the cintext of self-adjoint bosonic operators, is also useful to obtain the explicit form of the eigenvectors and eigenvalues of non self-adjoint bosonic Hamiltonians with real spectrum. We also demonstrate that these operators can be diagonalized when they are expressed in terms of the so called pseudo-bosons, which do not behave as true bosons under the adjoint transformation, but share with them some properties. In certain cases, these operators still obey the Weyl-Heisenberg commutation relations.*

- Ronald Douglas
*(Texas A&M University, USA)* -
**An analytic Grothendieck Riemann Roch theorem and the Arveson conjecture.** *Xiang Tang, Guoliang Yu, and the author, in joint work, extend the Boutet de Monvel Toeplitz index theorem to complex manifolds with isolated singularities following the relative $K$-homology theory of Baum, Taylor, and the author for manifolds with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $\mathbb{B}^m$ be the unit ball in $\mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $\mathbb{C}[z_1, \cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $\mathbb{S}^{2m-1}$ transversely, the representations of $\mathbb{C}[z_1, \cdots, z_m]$ on the closure of $I$ in $L^2_a(\mathbb{B}^m )$ and also the corresponding quotient space $Q_I$ are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on $Q_I$ by showing that the representation of $\mathbb{C}[z_1, \cdots, z_m]$ on the quotient space $Q_I$ gives the fundamental class of the boundary $Z_I\cap \mathbb{S}^{2m-1}$. Moreover, we prove with Kai Wang that if $f\in L^2_a(\mathbb{B}^m)$ vanishes on $Z_I\cap \mathbb{B}^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(\mathbb{B}^m)$. This establishes the Arveson-Douglas conjecture for these ideals.*

- Ruy Exel
*(Universidade Federal de Santa Catarina, Brasil)* -
**Partial actions and subshifts.** -
Given a finite alphabet $\Lambda $, and a not necessarily finite type subshift $X\subseteq \Lambda ^\infty $, we introduce a partial action of the free group $F (\Lambda )$ on a certain compactification $\Omega _X $ of $X$, which we call the spectral partial action. The space $\Omega _X $ has already appeared in many papers in the subject, arising as the spectrum of a commutative C*-algebra usually denoted by ${\cal D}_X $. Since the descriptions given of $\Omega _X$ in the literature are often somewhat terse and obscure, one of our main goals is to present a sensible model for it which allows for a detailed study of its structure, as well as of the spectral partial action, from various points of view, including topological freeness and minimality.

We then apply our results to study certain C*-algebras associated to $X$, introduced by Matsumoto and Carlsen. Most of the results we prove are already well known, but our proofs are hoped to be more natural and more in line with mainstream techniques used to treatsimilar C*-algebras. The clearer understanding of $\Omega _X $ provided by our model in turn allows for a fine tuning of some of these results, including a new necessary and sufficient condition for the simplicity of the Carlsen-Matsumoto C*-algebra, generalizing a similar result of Thomsen. This work is the result of a collaboration with M. Dokuchaev from São Paulo.

- Susana Furtado
*(Universidade do Porto, Portugal)* -
**Sign characteristic of Hermitian linearizations of Hermitian matrix polynomials.** *Let*

\begin{equation} P(\lambda)=A_{k}\lambda^{k}+\cdots+A_{1}\lambda+A_{0}, \label{pol} \end{equation} with $A_{i}\in\mathbb{C}^{n\times n}$, $i=1,\ldots,k$, be a matrix polynomial of degree $k$.

The most widely used approach to solve the polynomial eigenvalue problem $P(\lambda)x=0$ is to consider a linearization of the matrix polynomial $P(\lambda)$ and solve the corresponding linear eigenvalue problem. In case the matrix polynomial has some structure, it is important to consider linearizations with that structure (in case they exist) in order to preserve the properties of the eigenvalues imposed by it.

A matrix polynomial $P(\lambda)$ as in (\ref{pol}) is said to be Hermitian if $A_{i}^{\ast}=A_{i}$, for $i=1,\ldots,k.$ Hermitian matrix polynomials are one of the most important structured matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic of a Hermitian matrix polynomial $P(\lambda)$ with nonsingular leading coefficient is a set of signs attached to the elementary divisors of $P(\lambda)$ associated with the real eigenvalues which is crucial to understand the behavior of the eigenvalues under structured perturbations of $P(\lambda)$. It is known that the sign characteristic of a Hermitian linearization of a Hermitian matrix polynomial $P(\lambda)$ may be different from the sign characteristic of $P(\lambda)$.

In this talk we give a general characterization of the Hermitian linearizations of a Hermitian matrix polynomial $P(\lambda)$ that preserve the sign characteristic of $P(\lambda)$ and present several classes of such linearizations. In particular, we identify the Hermitian linearizations in the vector space $\mathbb{DL}(P)$, introduced by [Mackey, Mackey, Mehl, Mehrmann;2006], that preserve the sign characteristic of $P(\lambda)$ and give infinitely many new Hermitian linearizations with the same property. This talk is based on a joint work with Maria I. Bueno and Froilán Dopico.

- Charles Johnson
*(College of William & Mary, USA)* -
**The Nonnegative Inverse Eigenvalue Problem.** *The nonnegative inverse eigenvalue problem (NIEP) asks which collections of n complex numbers $a_1, \ldots, a_n$ occur as the eigenvalues of n-by-n entry-wise nonnegative matrices. This is a very difficult problem that is a very big goal for many matrix analysts. There are several variants: the R-NIEP (only real eigenvalues); the S-NIEP (requires symmetric realization); the DS-NIEP (requires a doubly stochastic realization), the diagonalizable NIEP, etc. However, none of these seems any easier. We survey selected historical work on this problem, as well as some recent work by the speaker and others. Among this will be the single eigenvalue problem, necessary conditions, low dimensions, adding 0's, consideration of eigenvectors, etc. The speaker would like to thank collaborators Pietro Papparella, Miriam Pisonero, and Carlos Marijuan for getting him re-interested in the problem.*

- Yuri Karlovich
*(Universidad Autónoma del Estado de Morelos, Mexico)* -
**$C^*$-algebra of convolution type operators with affine mappings and piecewise slowly oscillating data.** *The $C^*$-subalgebra $\mathfrak{B}$ of all bounded linear operators on the space $L^2(\mathbb{R})$, which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the solvable group of all affine mappings on $\mathbb{R}$, is studied. A faithful representation of the quotient $C^*$-algebra $\mathfrak{B}^\pi=\mathfrak{B}/\mathcal{K}$ in a Hilbert space, where $\mathcal{K}$ is the ideal of compact operators on $L^2(\mathbb{R})$, is constructed by applying a local-trajectory method, appropriate spectral measures and a lifting theorem. This gives a Fredholm symbol calculus for the $C^*$-algebra $\mathfrak{B}$ and a Fredholm criterion for the operators $B\in\mathfrak{B}$.*

The talk is based on a joint work with Iván Loreto-Hernández.

- Bartosz Kwasniewski
*(University of Southern Denmark, Denmark)* -
**Crossed products by endomorphisms of $C_0(X)$-algebras.** *We will review a notion of crossed product $A\rtimes_\alpha \mathbb{N}$ of a $C^*$-algebra $A$ by an endomorphism $\alpha:A\to A$, and discuss some general results. Next we consider a situation where $A$ is a $C_0(X)$-algebra and $\alpha$ is such that $\alpha(f a)=\Phi(f)\alpha(a)$, $a\in A$, $f\in C_0(X)$ where $\Phi$ is an endomorphism of $C_0(X)$. Pictorially speaking, $\alpha$ is a mixture of a topological dynamical system $(X,\varphi)$ dual to $(C_0(X),\Phi)$ and a continuous field of homomorphisms $\alpha_x$ between the fibers $A(x)$, $x\in X$, of the corresponding $C^*$-bundle.*

For such systems, we establish efficient conditions for the uniqueness property, ideal lattice description, and pure infiniteness of $A\rtimes_\alpha \mathbb{N}$. We apply these results to the case when $X=\text{Prim}(A)$ is a Hausdorff space. In particular, if the associated $C^*$-bundle is trivial, we obtain formulas for $K$-groups of all ideals in $A\rtimes_\alpha \mathbb{N}$. In this way, we constitute a large class of crossed products whose ideal structure and $K$-theory is completely described in terms of $(X,\varphi,\{\alpha_{x}\}_{x\in X})$.

- Fernando Lledó
*(Universidad Carlos III de Madrid, Spain)* -
**Amenability and paradoxical decompositions: analytic aspects.** *This talk explores amenability aspects in the context of operators and operator algebras and is a continuation of Pere Ara's talk in this conference, where metric space aspects and purely algebraic aspects were considered.*

We will introduce the notion and first properties of Følner sequences in the context of Operator Theory and Operator Algebras. Let ${\cal T}\subset\mathfrak{B}({\cal H})$ be a set of bounded linear operator acting on a complex separable Hilbert space ${\cal H}$. A sequence of non-zero finite rank orthogonal projections $\{P_n\}_{n\in \mathbb{N}}$ is called a Følner sequence for ${\cal T}$ , if

$$ \lim_n\frac{||TP_n-P_n T||}{||P_n||_2}=0\,,\quad T\in{\cal T}$$ where $||.||_2$ is the Hilbert-Schmidt norm. Følner sequences generalize the notion of quasi-diagonality for operators and can also be applied to spectral approximation problems.

We will give an intrinsic characterization for unital $C^*$-algebras having faithful representations with Følner sequences in terms of unital completely positive maps into matrices. Finally, we will introduce uniform Roe $C^*$-algebras ${\cal R}(X)$ associated to discrete metric spaces $ (X; d)$ with bounded geometry. We will show how, in this class of examples, the different notions of amenability for metric spaces, algebras and operator algebras unify in a natural way.- References
- 1. P. Ara, K. Li, F. Lledó and J. Wu, Amenability in coarse geometry, algebra and Roe $C^*$-algebras, Preprint.

2. P. Ara and F. Lledó, Amenable traces and Følner C*-algebras, Expo. Math. 32 (2014) 161-177.

3. P. Ara, F. Lledó and D. Yakubovich, Følner sequences in Operator Theory and Operator Algebras,Oper. Theory Adv. Appl. 242 (2014) 1-24.

4. N.P. Brown, Invariant means and nite representation theory of C*-algebras, Mem. Am. Math. Soc.184 (2006) no. 865, 1-105.

- Victor Nistor
*(Universite de Lorraine, France)* -
**Groupoid algebras and Fredholm conditions for geometric operators.** *I will introduce Lie groupoids and some of their associated operator algebras. Then I will discuss how these operator algebras can be used to model and study various operators appearing in geometric applications and in otherapplications. In particular, I will show how the concept of 'invertibility sufficient families' of representations, introduced by Rauch, can be used to obtain characterizations of the Fredholm operators using groupoids. The results of this talk are based on joint works with Ammann, Prudhon, Weinstein, and Xu.*

- Florin Radulescu
*(Inst Mat Romanian Academy/Universita Roma "Tor Vergata")* -
**Quantum Dynamics and Operator Algebras with Applications In Number Theory.** *We introduce an operator algebra repreesntation of Hecke operators. In this quantized representation the Hecke operators are completely postive maps, and we construct a Stinespring-Arveson Dilation. This is related to a canonical cohomology obstruction corresponding the Out group of the von Neumann algebra associated to the modular group.*

- Steffen Roch
*(Technische Universität Darmstadt, Germany)* -
**Extension-restriction theorems for algebras of approximation sequences.** *The $C^*$-algebra ${\mathcal S}({\sf T}(C))$ of the finite sections discretization for Toeplitz operators with continuous generating function is fairly well understood. Since its description by Böttcher and Silbermann in 1983, this algebra had served both as a source of inspiration and as an archetypal example of an algebra generated by an discretization procedure. The latter is no accident: it turns out that, after suitable extension by compact sequences and suitable fractal restriction,*every*separable $C^*$-algebra of approximation sequences has the same structure as ${\mathcal S}({\sf T}(C))$. We explain what this statement means and sketch a proof.*

- Peter Šemrl
*(University of Ljubljana, Slovenia)* -
**Automorphisms of Hilbert space effect algebras.** *The starting point of my talk will be the classical theorem of Ludwig describing the general form of ortho-order automorphisms of Hilbert space effect algebras. I will continue with some recent results of Molnar characterizing automorphisms of effect algebras.Then I will mention my results in this direction, present some proof techniques, and pose some open problems.*

- Eugene Shargorodsky
*(King's College London, UK)* -
**When does the norm of a Fourier multiplier dominate its $L^\infty$ norm?** *It is well known that the $L^\infty$ norm of a Fourier multiplier on $L^p(\mathbb{R}^n)$, $1 \le p \le \infty$ is less than or equal to its norm. The standard proof of this fact extends with almost no change to weighted $L^p$ spaces provided the weight $w$ is such that $w(x) = w(-x)$ for all $x \in \mathbb{R}^n$. It is natural to ask whether the norm of a Fourier multiplier on a weighted $L^p$ space still dominates its $L^\infty$ norm if the weight does not satisfy the above condition.*

If $w$ satisfies the Muckenhoupt $A_p$ condition, then the $L^\infty$ norm of a Fourier multiplier on $L^p(\mathbb{R}, w)$, $1 < p < \infty$ is less than or equal to its norm times a constant that depends only on $p$ and $w$. This result first appeared in 1998 in a paper by E. Berkson and T.A. Gillespie where it was attributed to J. Bourgain. It was extended to more general function spaces over $\mathbb{R}$ by A. Karlovich (2015). We prove that the above estimate holds with the constant equal to 1 for function spaces over $\mathbb{R}^n$ under considerably weaker restrictions. We also show that our result is in a sense optimal and that there exist weighted $L^p$ spaces with many unbounded Fourier multipliers.The talk is based on a joint work with Alexei Karlovich (Lisbon).

- Bernd Silbermann
*(Technische Universität Chemnitz, Germany)* -
**Remarks on the spectrum of the Hilbert matrix.** *For each $\lambda \in \mathbb{C}, \, \lambda \not= 0,-1,-2,\dots$, the (generalized) Hilbert matrix ${\cal H}_\lambda$ is given by*

$$ {\cal H}_\lambda : = \left(\frac{1}{n+m+\lambda}\right)_{n,m \ge 0}\,. $$ If $\lambda = 1$ then ${\cal H}_1$ is the classical Hilbert matrix. This talk is concerned with spectral properties of ${\cal H}_\lambda$ under some conditions, for instance of ${\cal H}_\lambda$ considered as a linear and bounded operator acting on the classical Hardy spaces $H^p, \, 1 < p < \infty$.

In particular, the essential spectrum of ${\cal H}_\lambda$ is identified. This will be done using the structure of some Banach algebras which was studied previously.

- Frank-Olme Speck
*(Instituto Superior Técnico, Portugal)* -
**Convolution type operators with symmetry in Bessel potential spaces.** *Convolution type operators with symmetry appear naturally in boundary value problems for elliptic PDEs in symmetric or symmetrizable domains. They are defined as truncations of translation invariant operators in a scale of Sobolev-like spaces that are convolutionally similar to subspaces of even or odd functionals. The present class, as a basic example, is closely related to the Helmholtz equation in a quadrant, where a possible solution is "symmetrically" extended to a half-plane. Explicit factorization methods allow the representation of resolvent operators in closed analytic form for a large class of boundary conditions including the two-impedance and the oblique derivative problems. Moreover they allow new results on the regularity and asymptotic behavior of the solutions. The talk is based upon joint work with L.P. Castro.*

- Ilya Spitkovsky
*(College of William & Mary, USA and New York University in Abu Dhabi, UAE)* -
**On the numerical range, normalized or not.** *The notion of the normalized numerical range $F_N(A)$ of an operator $A$ acting on a Hilbert space $\mathcal H$ was introduced by W. Auzinger in 2003 as the set of values $\{ (Ax,x)/||Ax||\colon ||x||=1, Ax\neq 0\}$, prompted by some applications to growth estimates of the resolvent of sectorial operators. The structure of $|F_N(A)$ is much more complicated than that of the classical numerical range $F(A)$; in particular, $F_N(A)$ is not necessarily convex (or even star-shaped).*

We will discuss some basic properties of $F_N(A)$, such as its simply connectedness, criteria for being open, closed, having empty interior, etc. An exact shape of $F_N(A)$ will be described for several classes of operators, e.g., weighted shifts and essentially Hermitian.

The talk is based on the joint work with Andrei-Florian Stoica.

- Nikolai Vasileski
*(CINVESTAV, Mexico)* -
**Algebras of Toeplitz operators on the unit ball.** *One of the common strategies in the study of Toeplitz operators consists in selecting of various special symbol classes $S \subset L_{\infty}$ so that the properties of both the individual Toeplitz operators $T_a$, with $a \in S$, and of the algebra generating by such Toeplitz operators can be characterized.*to lower the problem dimension

A motivation to study an algebra generating by Toeplitz operators (rather then just Toeplitz operators themselves) lies in a possibility to apply more tools, in particular those coming from the algebraic toolbox, and furthermore the results obtained are applicable not only for generating Toeplitz operators but also for a whole variety of elements of the algebra in question.

To make our approach more transparent we restrict the presentation to the case of the two-dimensional unit ball $\mathbb{B}^2$. We consider various sets $S$ of symbols that are invariant under a certain subgroup of biholomorphisms of $\mathbb{B}^2$ ($\{1\}\times \mathbb{T}$ in the talk). Such an invariance permits us*and to give a recipe, supplied by various concrete examples, on how the known results for the unit disk $\mathbb{D}$ can be applied to the study of various algebras (both commutative and non-commutative) that are generated by Toeplitz operators on the two-dimensional ball $\mathbb{B}^2$.*fixed

Although we consider the operators acting on the weighted Bergman space on $\mathbb{B}^2$ with a*weight parameter, the Berezin quantization effects (caused by a*growing*weight parameter of the corresponding weighted Bergman spaces on the unit disk $\mathbb{D}$) have to be taken into account.*

- João Xavier
*(ISR, Instituto Superior Técnico, Portugal)* -
**Collaborative processing in multi-agent networks.** *We describe some matrix problems that arise in the context of distributed processing for multi-agent networks.*

The main theme is how does the product of matrices, drawn randomly from given structured sets, behave asymptotically. Does it converge? In what sense? To what? At what rate? The matrices represent the connectivity between agents across time, and the answers to such questions are key to understand the fundamental limits of performance of many networked systems. We survey some partial results and point out several open questions.

- Ion Zaballa
*(Universidad del Pais Vasco, Spain)* -
**On the Inverse Symmetric Quadratic Eigenvalue Problem.** -

1. Lancaster P, and Zaballa I,A review of canonical forms for selfadjoint matrix polynomials, Operator Theory: Advances and Applications, 218, 2012, 425--443.*The detailed spectral structure of real symmetric quadratic matrix polynomials, including an orthogonality conditions on the eigenvectors, has been developed recently,[2]. Their canonical forms,[1], are shown to be useful to provide a detailed analysis of inverse problems of the form: construct the coefficients of a real symmetric quadratic matrix polynomial from the spectral data including the classical eigenvalue/eigenvector data and sign characteristics for the real eigenvalues. The aforementioned orthogonality condition dependent on the sign characteristic plays a vital role in this construction.*

Special attention is paid to the case when all coefficients are prescribed to be positive definite. A precise knowledge of the admissible sign characteristics of such matrix polynomials is revealed to be important.

The talk is based on the joint work with Peter Lancaster.

2. Lancaster P, Prells U, and Zaballa I, An orthogonality property for real symmetric matrix polynomials with application to the inverse problem. Operators and Matrices, 7 (2), 2013, 357--380.