## PUBLICATIONS

## PAPERS

1. (with K.-H. Neeb, J. Schober) Reflection positivity and Hankel operators -- the multiplicity free case

Submitted, arXiv:2105.08522

ABSTRACT: We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples (G,S,τ), where G is a group, τ an involutive automorphism of G and S⊆G a subsemigroup with τ(S)=S^{−1}. For the triples (Z,N,−id_Z), corresponding to reflection positive operators, and (R,R_+,−id_R), corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method consists in using the measure μ_H on R_+ defined by a positive Hankel operator H on H^2(C_+) to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for H.

2. (with M. Fragoulopoulou) Tensor products of normed and Banach quasi *-algebras

Journal of Mathematical Analysis and Applications (2020), arXiv:2002.07930

ABSTRACT: Quasi *-algebras form an essential class of partial *-algebras, which are algebras of unbounded operators. In this work, we aim to construct tensor products of normed, respectively Banach quasi *-algebras, and study their capacity to preserve some important properties of their tensor factors, like for instance, *-semisimplicity and full representability.

3. (with C. Trapani) Unbounded derivations and *-automorphisms groups of Banach quasi *-algebras

Annali di Matematica Pura ed Applicata (1923 -), (2019), arXiv:1807.11525

ABSTRACT: This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one-parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense, i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.

4. (with C. Trapani) Representable and continuous functionals on a Banach quasi *-algebra

Mediterranean Journal of Mathematics (2017) 14: 157, arXiv:1703.02862

ABSTRACT: In the study of locally convex quasi *-algebras an important role is played by representable linear functionals, i.e., functionals which allow a GNS-construction. This paper is mainly devoted to the study of the continuity of representable functionals in Banach and Hilbert quasi *-algebras. Some other concepts related to representable functionals (full-representability, *-semisimplicity, etc) are revisited in these special cases. In particular, in the case of Hilbert quasi *-algebras, which are shown to be fully representable, the existence of a 1-1 correspondence between positive, bounded elements (defined in an appropriate way) and continuous representable functionals is proved.

5. (with C. Vetro) Fixed point and homotopy results for mixed multi-valued mappings in 0-complete partial metric spaces

Nonlinear Analysis: Modelling and Control 20 vol.2 (2015),159-174.

ABSTRACT: We give sufficient conditions for the existence of common fixed points for a pair of mixed multi-valued mappings in the setting of 0-complete partial metric spaces. An example is given to demonstrate the usefulness of our results over the existing results in metric spaces. Finally, we prove a homotopy theorem via fixed point results.

## PROCEEDINGS

1. About tensor products of Hilbert quasi *-algebras and their representability properties, arXiv:2002.07926

Published in the Proceedings of IWOTA 2019, Operator Theory: Advances and Applications.

ABSTRACT: This note aims to investigate the tensor product of two given Hilbert quasi *-algebras and its properties. The construction proposed in this note turns out to be again a Hilbert quasi *-algebra, thus interesting representability properties studed in \cite{AT} are maintained. Furthermore, if two functionals are representable and continuous respectively on the two Hilbert quasi *-algebras, then so is their tensor product.

2. On some applications of representable and continuous functionals of Banach quasi *-algebras, arXiv:2002.08775

Published in the Proceedings of OT 27.

ABSTRACT: This survey aims to highlight some of the consequences that representable (and continuous) functionals have in the framework of Banach quasi *-algebras. In particular, we look at the link between the notions of *-semisimplicity and full representability in which representable functionals are involved. Then, we emphasize their essential role in studying *-derivations and representability properties for the tensor product of Hilbert quasi *-algebras, a special class of Banach quasi *-algebras.

3. The interplay between representable functionals and derivations on Banach quasi *-algebras, arXiv:1809.00470

Abel, Mart (Ed.). Proceedings of the ICTAA 2018 (48-59). Tartu: Eesti Matemaatika Selts. (Mathematics Studies; 7)

ABSTRACT: This note aims to highlight the link between representable functionals and derivations on a Banach quasi *-algebra, i.e. a mathematical structure that can be seen as the completion of a normed *-algebra in the case the multiplication is only separately continuous. Representable functionals and derivations have been investigated in previous papers for their importance concerning the study of the structure properties of a Banach quasi *-algebra and applications to quantum models.

## PH.D. THESIS

Representable functionals and derivations on Banach quasi *-algebras, University of Catania (Italy) 2019.