• Medientyp: Dissertation; Elektronische Hochschulschrift; E-Book
  • Titel: A new family of universal products and aspects of a non-positive quantum probability theory
  • Beteiligte: Lachs, Stephanie [Verfasser:in]
  • Erschienen: Publication Server of Greifswald University, 2015-06-01
  • Sprache: Englisch
  • Schlagwörter: Fock-Raum ; Monoidale Kategorie ; Kategorientheorie ; Hopf-Algebra ; Algebra ; Bialgebra ; Grenzwertsatz ; Lévy-Prozess ; Funktionalanalysis
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  • Beschreibung: This thesis revolves around a new concept of independence of algebras. The independence nicely fits into the framework of universal products, which have been introduced to classify independence relations in quantum probability theory; the associated product is called (r,s)-product and depends on two complex parameters r and s. Based on this product, we develop a theory which works without using involutive algebras or states. The following aspects are considered: 1. Classification: Universal products are defined on the free product of algebras (the coproduct in the category of algebras) and model notions of independence in quantum probability theory. We distinguish universal products according to their behaviour on elements of length two, calling them (r,s)-universal products with complex parameters r and s respectively. In case r and s equal 1, Muraki was able to show that there exist exactly five universal products (Muraki’s five). For r equals s nonzero we get five one parameter families (q-Muraki’s five). We prove that in the case r not equal to s the (r,s)-product, a two parameter deformation of the Boolean product, is the only universal product satisfying our set of axioms. The corresponding independence is called (r,s)-independence. 2. Dual pairs and GNS construction: By use of the GNS construction, one can associate a product of representations with every positive universal product. Since the (r,s)-product does not preserve positivity, we need a substitute for the usual GNS construction for states on involutive algebras. In joint work with M. Gerhold, the product of representations associated with the (r,s)-product was determined, whereby we considered representations on dual pairs instead of Hilbert spaces. This product of representations is - as we could show - essentially different from the Boolean product. 3. Reduction and quantum Lévy processes: U. Franz introduced a category theoretical concept which allows a reduction of the Boolean, monotone and antimonotone independence to the tensor ...
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