{"id":{"repo_id":"catania","oai_identifier":"oai:www.iris.unict.it:20.500.11769/582731"},"canonical_url":"https://search.dev.ndltd.org/etd/catania/oai:www.iris.unict.it:20.500.11769/582731","repository":{"repo_id":"catania","name":"Università degli Studi di Catania","base_url":"https://www.iris.unict.it/oai/request"},"display":{"title":"Representable functionals and derivations on Banach quasi *-algebras","abstract":"Locally convex quasi *-algebras, in particular Banach quasi *-algebras, have been deeply investigated by many mathematicians in the last decades in order to describe quantum physical phenomena (see \\cite{ankar, ankar1, Ant1, Bag2, Bag6, Frag3,ino, ino1,kschm,Trap3,FragCt}). Banach quasi *-algebras constitute the framework of this thesis. They form a special family of locally convex quasi *-algebras, whose topology is generated by a single norm, instead of a separating family of seminorms (see, for instance, \\cite{Bag1,Bag4,Bag5,btt_meas}). The first part of the work concerns the study of representable functionals and their properties. The analysis is carried through the key notions of \\textit{fully representability} and \\textit{*-semisimplicity}, appeared in the literature in \\cite{Ant1,Bag1,Bag5,Frag2}. In the case of Banach quasi *-algebras, these notions are equivalent up to a certain \\textit{positivity condition}. This is shown in \\cite{AT}, by proving first that every sesquilinear form associated to a representable functional is everywhere defined and continuous. In particular, Hilbert quasi *-algebras are always fully representable. The aforementioned result about sesquilinear forms allows one to select {\\em well behaved} Banach quasi *-algebras where it makes sense to reconsider in a new framework classical problems that are relevant in applications (see \\cite{Bade,Brat1,HP,Kish,Sakai,trap,weigt,WZ1,WZ2}). One of them is certainly that of derivations and of the related automorphisms groups (for instance see \\cite{AT2,Alb,Ant4,Bag8,Brat2}). Definitions of course must be adapted to the new situation and for this reason we introduce weak *-derivations and weak automorphisms in \\cite{AT2}. We study conditions for a weak *-derivation to be the generator of such a group. An infinitesimal generator of a continuous one-parameter group of uniformly bounded weak *-automorphisms is shown to be closed and to have certain properties on its spectrum, whereas, to acquire such a group starting with a certain closed * derivation, extra regularity conditions on its domain are required. These results are then applied to a concrete example of weak *-derivations, like inner qu*-derivation occurring in physics. Another way to study representations of a Banach quasi *-algebra is to construct new objects starting from a finite number of them, like \\textit{tensor products} (see \\cite{ada,fiw,fiw1,hei,hel,lau,lp,sa}). In \\cite{AF} we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, like the aforementioned full representability and *-semisimplicity. It has been shown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors. About the viceversa, it is true if only the pre-completion is considered, i.e. if the factors are fully representable (resp. *-semisimple), then the tensor product pre-completion normed quasi *-algebra is fully representable (resp. *-semisimple). Several examples are investigated from the point of view of Banach quasi *-algebras.","abstract_html":"Locally convex quasi *-algebras, in particular Banach quasi *-algebras, have been deeply investigated by many mathematicians in the last decades in order to describe quantum physical phenomena (see \\cite{ankar, ankar1, Ant1, Bag2, Bag6, Frag3,ino, ino1,kschm,Trap3,FragCt}). Banach quasi *-algebras constitute the framework of this thesis. They form a special family of locally convex quasi *-algebras, whose topology is generated by a single norm, instead of a separating family of seminorms (see, for instance, \\cite{Bag1,Bag4,Bag5,btt_meas}). The first part of the work concerns the study of representable functionals and their properties. The analysis is carried through the key notions of \\textit{fully representability} and \\textit{*-semisimplicity}, appeared in the literature in \\cite{Ant1,Bag1,Bag5,Frag2}. In the case of Banach quasi *-algebras, these notions are equivalent up to a certain \\textit{positivity condition}. This is shown in \\cite{AT}, by proving first that every sesquilinear form associated to a representable functional is everywhere defined and continuous. In particular, Hilbert quasi *-algebras are always fully representable. The aforementioned result about sesquilinear forms allows one to select {\\em well behaved} Banach quasi *-algebras where it makes sense to reconsider in a new framework classical problems that are relevant in applications (see \\cite{Bade,Brat1,HP,Kish,Sakai,trap,weigt,WZ1,WZ2}). One of them is certainly that of derivations and of the related automorphisms groups (for instance see \\cite{AT2,Alb,Ant4,Bag8,Brat2}). Definitions of course must be adapted to the new situation and for this reason we introduce weak *-derivations and weak automorphisms in \\cite{AT2}. We study conditions for a weak *-derivation to be the generator of such a group. An infinitesimal generator of a continuous one-parameter group of uniformly bounded weak *-automorphisms is shown to be closed and to have certain properties on its spectrum, whereas, to acquire such a group starting with a certain closed * derivation, extra regularity conditions on its domain are required. These results are then applied to a concrete example of weak *-derivations, like inner qu*-derivation occurring in physics. Another way to study representations of a Banach quasi *-algebra is to construct new objects starting from a finite number of them, like \\textit{tensor products} (see \\cite{ada,fiw,fiw1,hei,hel,lau,lp,sa}). In \\cite{AF} we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, like the aforementioned full representability and *-semisimplicity. It has been shown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors. About the viceversa, it is true if only the pre-completion is considered, i.e. if the factors are fully representable (resp. *-semisimple), then the tensor product pre-completion normed quasi *-algebra is fully representable (resp. *-semisimple). Several examples are investigated from the point of view of Banach quasi *-algebras.","abstract_has_math":false,"creators":["ADAMO, MARIA STELLA"],"institution":"Università degli studi di Catania","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["RUSSO, Giovanni"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-11-30","date_published":"2018-11-30","updated_at":"2026-07-24T01:35:03Z","subjects":["Banach quasi *-algebras, representable functionals, problem of continuity for representable functionals, weak *-derivation, infinitesimal generator of weak *-automorphisms group, topological tensor products of Banach quasi *-algebras"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess","license:PUBBLICO - Pubblico con Copyright","license uri:iris.PUB02"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.11769/582731","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Adamo, MARIA STELLA","RUSSO, Giovanni"]},{"key":"dc:creator","label":"Author","values":["ADAMO, MARIA STELLA"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-11-30"]},{"key":"dc:publisher","label":"Institution","values":["Università degli studi di Catania","place:Catania"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Banach quasi *-algebras, representable functionals, problem of continuity for representable functionals, weak *-derivation, infinitesimal generator of weak *-automorphisms group, topological tensor products of Banach quasi *-algebras"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess","license:PUBBLICO - Pubblico con Copyright","license uri:iris.PUB02"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/20.500.11769/582731"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Locally convex quasi *-algebras, in particular Banach quasi *-algebras, have been deeply investigated by many mathematicians in the last decades in order to describe quantum physical phenomena (see \\cite{ankar, ankar1, Ant1, Bag2, Bag6, Frag3,ino, ino1,kschm,Trap3,FragCt}). Banach quasi *-algebras constitute the framework of this thesis. They form a special family of locally convex quasi *-algebras, whose topology is generated by a single norm, instead of a separating family of seminorms (see, for instance, \\cite{Bag1,Bag4,Bag5,btt_meas}). The first part of the work concerns the study of representable functionals and their properties. The analysis is carried through the key notions of \\textit{fully representability} and \\textit{*-semisimplicity}, appeared in the literature in \\cite{Ant1,Bag1,Bag5,Frag2}. In the case of Banach quasi *-algebras, these notions are equivalent up to a certain \\textit{positivity condition}. This is shown in \\cite{AT}, by proving first that every sesquilinear form associated to a representable functional is everywhere defined and continuous. In particular, Hilbert quasi *-algebras are always fully representable. The aforementioned result about sesquilinear forms allows one to select {\\em well behaved} Banach quasi *-algebras where it makes sense to reconsider in a new framework classical problems that are relevant in applications (see \\cite{Bade,Brat1,HP,Kish,Sakai,trap,weigt,WZ1,WZ2}). One of them is certainly that of derivations and of the related automorphisms groups (for instance see \\cite{AT2,Alb,Ant4,Bag8,Brat2}). Definitions of course must be adapted to the new situation and for this reason we introduce weak *-derivations and weak automorphisms in \\cite{AT2}. We study conditions for a weak *-derivation to be the generator of such a group. An infinitesimal generator of a continuous one-parameter group of uniformly bounded weak *-automorphisms is shown to be closed and to have certain properties on its spectrum, whereas, to acquire such a group starting with a certain closed * derivation, extra regularity conditions on its domain are required. These results are then applied to a concrete example of weak *-derivations, like inner qu*-derivation occurring in physics. Another way to study representations of a Banach quasi *-algebra is to construct new objects starting from a finite number of them, like \\textit{tensor products} (see \\cite{ada,fiw,fiw1,hei,hel,lau,lp,sa}). In \\cite{AF} we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, like the aforementioned full representability and *-semisimplicity. It has been shown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors. About the viceversa, it is true if only the pre-completion is considered, i.e. if the factors are fully representable (resp. *-semisimple), then the tensor product pre-completion normed quasi *-algebra is fully representable (resp. *-semisimple). Several examples are investigated from the point of view of Banach quasi *-algebras."]},{"key":"dc:title","label":"Title","values":["Representable functionals and derivations on Banach quasi *-algebras"]}]}],"canonical_facts":{"dc:contributor":["Adamo, MARIA STELLA","RUSSO, Giovanni"],"dc:creator":["ADAMO, MARIA STELLA"],"dc:date":["2018-11-30"],"dc:description":["Locally convex quasi *-algebras, in particular Banach quasi *-algebras, have been deeply investigated by many mathematicians in the last decades in order to describe quantum physical phenomena (see \\cite{ankar, ankar1, Ant1, Bag2, Bag6, Frag3,ino, ino1,kschm,Trap3,FragCt}). Banach quasi *-algebras constitute the framework of this thesis. They form a special family of locally convex quasi *-algebras, whose topology is generated by a single norm, instead of a separating family of seminorms (see, for instance, \\cite{Bag1,Bag4,Bag5,btt_meas}). The first part of the work concerns the study of representable functionals and their properties. The analysis is carried through the key notions of \\textit{fully representability} and \\textit{*-semisimplicity}, appeared in the literature in \\cite{Ant1,Bag1,Bag5,Frag2}. In the case of Banach quasi *-algebras, these notions are equivalent up to a certain \\textit{positivity condition}. This is shown in \\cite{AT}, by proving first that every sesquilinear form associated to a representable functional is everywhere defined and continuous. In particular, Hilbert quasi *-algebras are always fully representable. The aforementioned result about sesquilinear forms allows one to select {\\em well behaved} Banach quasi *-algebras where it makes sense to reconsider in a new framework classical problems that are relevant in applications (see \\cite{Bade,Brat1,HP,Kish,Sakai,trap,weigt,WZ1,WZ2}). One of them is certainly that of derivations and of the related automorphisms groups (for instance see \\cite{AT2,Alb,Ant4,Bag8,Brat2}). Definitions of course must be adapted to the new situation and for this reason we introduce weak *-derivations and weak automorphisms in \\cite{AT2}. We study conditions for a weak *-derivation to be the generator of such a group. An infinitesimal generator of a continuous one-parameter group of uniformly bounded weak *-automorphisms is shown to be closed and to have certain properties on its spectrum, whereas, to acquire such a group starting with a certain closed * derivation, extra regularity conditions on its domain are required. These results are then applied to a concrete example of weak *-derivations, like inner qu*-derivation occurring in physics. Another way to study representations of a Banach quasi *-algebra is to construct new objects starting from a finite number of them, like \\textit{tensor products} (see \\cite{ada,fiw,fiw1,hei,hel,lau,lp,sa}). In \\cite{AF} we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, like the aforementioned full representability and *-semisimplicity. It has been shown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors. About the viceversa, it is true if only the pre-completion is considered, i.e. if the factors are fully representable (resp. *-semisimple), then the tensor product pre-completion normed quasi *-algebra is fully representable (resp. *-semisimple). Several examples are investigated from the point of view of Banach quasi *-algebras."],"dc:identifier":["https://hdl.handle.net/20.500.11769/582731"],"dc:language":["eng"],"dc:publisher":["Università degli studi di Catania","place:Catania"],"dc:rights":["info:eu-repo/semantics/openAccess","license:PUBBLICO - Pubblico con Copyright","license uri:iris.PUB02"],"dc:subject":["Banach quasi *-algebras, representable functionals, problem of continuity for representable functionals, weak *-derivation, infinitesimal generator of weak *-automorphisms group, topological tensor products of Banach quasi *-algebras"],"dc:title":["Representable functionals and derivations on Banach quasi *-algebras"],"dc:type":["info:eu-repo/semantics/doctoralThesis"]},"updated_at":"2026-07-24T01:35:03Z"}