Operator algebras and dynamical systems

Postgraduate Thesis uoadl:1316482 648 Read counter

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Original Title:
Άλγεβρες τελεστών και δυναμικά συστήματα
Translated title:
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.
Algebra, Operator, Dynamical, System, Analysis
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