Postgraduate Thesis uoadl:1316482 648 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2016-06-24

2016

Μπατσής Αλέξανδρος

Αριστείδης Κατάβολος Καθηγητής

Άλγεβρες τελεστών και δυναμικά συστήματα

Greek

Operator algebras and dynamical systems

The relation between operator algebras and ergodic theory is known since the

time von Neumann began research on the subject. The theory of crossed products

was created which had applications to the classification of factors. The

crossed product is a method of constructing a new von Neumann algebra from a

given von Neumann algebra acted on by a locally compact group. There is also a

crossed product for C*-algebras. In this case the crossed product is a

C*-algebra that is constructed from a given C*-algebra acted on by a locally

compact group. The construction of the crossed product can be seen as a way to

code dynamical systems to operator algebras. An example of Hoare and Parry

showed that there is a pair of non-conjugate topological dynamical systems such

that the corresponding crossed products are not isomorphic. In 1967 a paper by

W. Arveson was published in which he constructed a nonselfadjoint operator

algebra for each ergodic automorphism of [0,1] and showed that two such

automorphisms are conjugate if and only if the corresponding algebras are

unitarily equivalent. In 1969 W. arveson and K. Josephson applied this work to

a specific class of topological dynamical systems that admit invariant

probability measures. For each such system a nonselfadjoint operator algebra

was constructed and it was proved that two systems of this type are conjugate

if and only if the corresponding algebras are isomorphic. From these ideas the

theory of semicrossed products arose and there were theorems proved which

significantly improved the results mentioned above. In this thesis we focus on

these first results of W. Arveson and K. Josephson.

time von Neumann began research on the subject. The theory of crossed products

was created which had applications to the classification of factors. The

crossed product is a method of constructing a new von Neumann algebra from a

given von Neumann algebra acted on by a locally compact group. There is also a

crossed product for C*-algebras. In this case the crossed product is a

C*-algebra that is constructed from a given C*-algebra acted on by a locally

compact group. The construction of the crossed product can be seen as a way to

code dynamical systems to operator algebras. An example of Hoare and Parry

showed that there is a pair of non-conjugate topological dynamical systems such

that the corresponding crossed products are not isomorphic. In 1967 a paper by

W. Arveson was published in which he constructed a nonselfadjoint operator

algebra for each ergodic automorphism of [0,1] and showed that two such

automorphisms are conjugate if and only if the corresponding algebras are

unitarily equivalent. In 1969 W. arveson and K. Josephson applied this work to

a specific class of topological dynamical systems that admit invariant

probability measures. For each such system a nonselfadjoint operator algebra

was constructed and it was proved that two systems of this type are conjugate

if and only if the corresponding algebras are isomorphic. From these ideas the

theory of semicrossed products arose and there were theorems proved which

significantly improved the results mentioned above. In this thesis we focus on

these first results of W. Arveson and K. Josephson.

Algebra, Operator, Dynamical, System, Analysis

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