Unit:
Τομέας Μαθηματικής ΑνάλυσηςLibrary of the School of Science
Author:
Κακαριάδης Ευγένιος
Dissertation committee:
Καθηγητής Μιχάλης Ανούσης, Καθηγητής Αριστείδης Κατάβολος (επιβλέπων), Καθηγητής Stephen Power
Original Title:
Xώροι τελεστών και άλγεβρες τελεστών: ημισταυρωτά γινόμενα αλγεβρών τελεστών
Summary:
The core of the Ph.D. thesis is the investigation of non-selfadjoint operator
algebras that arise from dynamical systems, i.e. the semicrossed products. The
first time the notion of semicrossed products appears in the literature is by
Arveson (1966) and were encoded by Peters (1984). There are considered the
non-selfadjoint analogue of the crossed products as they codify properties of
the dynamical system.
One of the problems in the theory of the crossed products was if isomorphic
crossed products give conjugacy of the actions. This particular problem cannot
be answered in the selfadjoint context since there are non-trivial
counterexamples. Nevertheless, recently Davidson and Katsoulis proved that two
actions are conjugate if and only if the respective semicrossed products are
isomorphic, for the case of commutative dynamical systems. Moreover, in
contrast to the crossed products the semicrossed products are more flexible
since they can be defined even when the action is not an automorphism.
The current thesis consists of four chapters and the appendix. In the first
chapter we give the main definitions of Operator Theory and important tools
that we use. In the second chapter we find the C*-envelope of dynamical systems
of C*-algebras. In the third chapter we consider dynamical systems of
non-selfadjoint operator algebras. An example shows that there is an essential
difference between these systems and those considered in the second chapter. In
the fourth chapter we examine properties of the semicrossed products in the
context of W*-theory such as the reflexive cover and the bicommutant property.
Moreover we give an introduction in W*-theory. In the appendix we give proofs
for some of the claims and constructions used.
Keywords:
operator algebras, C*-envelope, semicrossed product, crossed product, reflexive cover
Number of pages:
XXII, 184
