Structure in operator algebras
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- Tez No: 400101
- Danışmanlar: DR. DAVİD BLECHER
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2011
- Dil: İngilizce
- Üniversite: University of Houston
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 123
Özet
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Özet (Çeviri)
In this dissertation, we define two new classes of operator algebras; matricialoperator algebras and scattered operator algebras. The C-algebras of compact operatorsplay an important role in C-algebra theory, and they are widely used inmathematical physics and quantum mechanics. We define 1-matricial algebras usinga sequence of invertible operators on a Hilbert space, and -matricial algebras arec0-sums of 1-matricial algebras. These operator algebras, in some sense, generalizethe class of C-algebras of compact operators to a non-selfadjoint setting. Theypossess many properties similar to the properties of the C-algebras of compact operators.We present a `Wedderburn type' structure theorem that characterizes the-matricial algebras. We define scattered operator algebras using a composition serieswhere each consecutive quotient is a 1-matricial algebra. Note that a C-algebrais scattered if and only if it has a composition series where each consecutive quotientis a C-algebra of compact operators. Hence, our definition of scattered operatoralgebras is quite natural. We present many results on the structure of the scatteredoperator algebras and show that they have some properties generalizing theproperties of scattered C-algebras. For example, the dual of a scattered operatoralgebra has the Radon-Nikodym property and scattered operator algebras are Asplundspaces. Working with a composition series requires us to develop some toolsfor general operator algebras, and in particular, quotient operator algebras. For examplewe utilize frequently the isomorphism theorems and a correspondence theoremfor operator algebras; as well as the results about the structure of the diagonal of aquotient operator algebra.
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