Browsing by Subject "C*algebras"
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Ro, Min (University of Oregon, August 18, 2015)[more][less]Lin, Huaxin Ro, Min 20150818T22:52:07Z 20150818T22:52:07Z 20150818 http://hdl.handle.net/1794/19199 In this dissertation, we explore the approximate diagonalization of unital homomorphisms between C*algebras. In particular, we prove that unital homomorphisms from commutative C*algebras into simple separable unital C*algebras with tracial rank at most one are approximately diagonalizable. This is equivalent to the approximate diagonalization of commuting sets of normal matrices. We also prove limited generalizations of this theorem. Namely, certain injective unital homomorphisms from commutative C*algebras into simple separable unital C*algebras with rational tracial rank at most one are shown to be approximately diagonalizable. Also unital injective homomorphisms from AHalgebras with unique tracial state into separable simple unital C*algebras of tracial rank at most one are proved to be approximately diagonalizable. Counterexamples are provided showing that these results cannot be extended in general. Finally, we prove that for unital homomorphisms between AFalgebras, approximate diagonalization is equivalent to a combinatorial problem involving sections of lattice points in cones. en_US University of Oregon All Rights Reserved. approximate diagonalization C*algebras Elliott classification Approximate Diagonalization of Homomorphisms Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Gardella, Eusebio (University of Oregon, August 18, 2015)[more][less]Phillips, N. Christopher Gardella, Eusebio 20150818T23:13:52Z 20150818T23:13:52Z 20150818 http://hdl.handle.net/1794/19345 This dissertation is concerned with representations of locally compact groups on different classes of Banach spaces. The first part of this work considers representations of compact groups by automorphisms of C*algebras, also known as group actions on C*algebras. The actions we study enjoy a freenesstype of property, namely finite Rokhlin dimension. We investigate the structure of their crossed products, mainly in relation to their classifiability, and compare the notion of finite Rokhlin dimension with other existing notions of noncommutative freeness. In the case of Rokhlin dimension zero, also known as the Rokhlin property, we prove a number of classification theorems for these actions. Also, in this case, much more can be said about the structure of the crossed products. In the last chapter of this part, we explore the extent to which actions with Rokhlin dimension one can be classified. Our results show that even for Z_2actions on O_2, their classification is not Borel, and hence it is intractable. The second part of the present dissertation focuses on isometric representations of groups on Lpspaces. For p=2, these are the unitary representations on Hilbert spaces. We study the Lpanalogs of the full and reduced group \ca s, particularly in connection to their rigidity. One of the main results of this work asserts that for p different from 2, the isometric isomorphism type of the reduced group Lpoperator algebra recovers the group. Our study of group algebras acting on Lpspaces has also led us to answer a 20yearold question of Le Merdy and Junge: for p different from 2, the class of Banach algebras that can be represented on an Lpspace is not closed under quotients. We moreover study representations of groupoids, which are a generalization of groups where multiplication is not always defined. The algebras associated to these objects provide new examples of Lpoperator algebras and recover some previously existing ones. Groupoid Lpoperator algebras are particularly tractable objects. For instance, while groupoid Lpoperator algebras can be classified by their K_0group (an ordered, countable abelian group), we show that UHFLpoperator algebras not arising from groupoids cannot be classified by countable structures. This dissertation includes unpublished coauthored material. en_US University of Oregon All Rights Reserved. C*algebras Classification Cossed product Group action Lpspace ppseudofunctions Compact Group Actions on C*algebras: Classification, NonClassifiability and Crossed Products and Rigidity Results for Lpoperator Algebras Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Archey, Dawn Elizabeth, 1979 (University of Oregon, June , 2008)[more][less]Archey, Dawn Elizabeth, 1979 20081220T02:10:58Z 20081220T02:10:58Z 200806 http://hdl.handle.net/1794/8155 viii, 107 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. This dissertation consists of two related parts. In the first portion we use the tracial Rokhlin property for actions of a finite group G on stably finite simple unital C *algebras containing enough projections. The main results of this part of the dissertation are as follows. Let A be a stably finite simple unital C *algebra and suppose a is an action of a finite group G with the tracial Rokhlin property. Suppose A has real rank zero, stable rank one, and suppose the order on projections over A is determined by traces. Then the crossed product algebra C * ( G, A, Ã Ã Â±) also has these three properties. In the second portion of the dissertation we introduce an analogue of the tracial Rokhlin property for C *algebras which may not have any nontrivial projections called the projection free tracial Rokhlin property . Using this we show that under certain conditions if A is an infinite dimensional simple unital C *algebra with stable rank one and Ã Ã Â± is an action of a finite group G with the projection free tracial Rokhlin property, then C * ( G, A, Ã Ã Â±) also has stable rank one. Adviser: Phillips, N. Christopher en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph.D., 2008; Mathematics Cuntz subequivalence Stable rank one Tracial Rokhlin property Finite group actions Crossed product C*algebras Crossed product C*algebras by finite group actions with a generalized tracial Rokhlin property Thesis

Crossed product C*algebras of certain nonsimple C*algebras and the tracial quasiRokhlin propertyBuck, Julian Michael, 1982 (University of Oregon, June , 2010)[more][less]Buck, Julian Michael, 1982 20101130T23:48:55Z 20101130T23:48:55Z 201006 http://hdl.handle.net/1794/10849 viii, 113 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. This dissertation consists of four principal parts. In the first, we introduce the tracial quasiRokhlin property for an automorphism α of a C *algebra A (which is not assumed to be simple or to contain any projections). We then prove that under suitable assumptions on the algebra A , the associated crossed product C *algebra C *([Special characters omitted.] , A , α) is simple, and the restriction map between the tracial states of C *([Special characters omitted.] , A , α) and the αinvariant tracial states on A is bijective. In the second part, we introduce a comparison property for minimal dynamical systems (the dynamic comparison property) and demonstrate sufficient conditions on the dynamical system which ensure that it holds. The third part ties these concepts together by demonstrating that given a minimal dynamical system ( X, h ) and a suitable simple C *algebra A , a large class of automorphisms β of the algebra C ( X, A ) have the tracial quasiRokhlin property, with the dynamic comparison property playing a key role. Finally, we study the structure of the crossed product C *algebra B = C *([Special characters omitted.] , C ( X , A ), β) by introducing a subalgebra B { y } of B , which is shown to be large in a sense that allows properties B { y } of to pass to B . Several conjectures about the deeper structural properties of B { y } and B are stated and discussed. Committee in charge: Christopher Phillips, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Huaxin Lin, Member, Mathematics; Marcin Bownik, Member, Mathematics; Van Kolpin, Outside Member, Economics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Dynamical systems Minimal homeomorphisms Crossed product algebras Tracial property Automorphisms C*algebras QuasiRokhlin property Mathematics Theoretical mathematics Crossed product C*algebras of certain nonsimple C*algebras and the tracial quasiRokhlin property Thesis

Sun, Wei, 1979 (University of Oregon, June , 2010)[more][less]Sun, Wei, 1979 20101222T01:32:21Z 20101222T01:32:21Z 201006 http://hdl.handle.net/1794/10912 vii, 124 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. This dissertation is a study of the relationship between minimal dynamical systems on the product of the Cantor set ( X ) and torus ([Special characters omitted]) and their corresponding crossed product C *algebras. For the case when the cocyles are rotations, we studied the structure of the crossed product C *algebra A by looking at a large subalgebra A x . It is proved that, as long as the cocyles are rotations, the tracial rank of the crossed product C *algebra is always no more than one, which then indicates that it falls into the category of classifiable C *algebras. In order to determine whether the corresponding crossed product C *algebras of two such minimal dynamical systems are isomorphic or not, we just need to look at the Elliott invariants of these C *algebras. If a certain rigidity condition is satisfied, it is shown that the crossed product C *algebra has tracial rank zero. Under this assumption, it is proved that for two such dynamical systems, if A and B are the corresponding crossed product C *algebras, and we have an isomorphism between K i ( A ) and K i ( B ) which maps K i (C(X ×[Special characters omitted])) to K i (C( X ×[Special characters omitted])), then these two dynamical systems are approximately K conjugate. The proof also indicates that C *strongly flip conjugacy implies approximate K conjugacy in this case. We also studied the case when the cocyles are Furstenberg transformations, and some results on weakly approximate conjugacy and the K theory of corresponding crossed product C *algebras are obtained. Committee in charge: Huaxin Lin, Chairperson, Mathematics Daniel Dugger, Member, Mathematics; Christopher Phillips, Member, Mathematics; Arkady Vaintrob, Member, Mathematics; LiShan Chou, Outside Member, Human Physiology en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Tracial rank Approximate conjugacy C*algebras Minimal dynamical systems Cantor set Torus Mathematics Theoretical mathematics Crossed product C*algebras of minimal dynamical systems on the product of the Cantor set and the torus Thesis

Sun, Michael (University of Oregon, September 29, 2014)[more][less]Lin, Huaxin Sun, Michael 20140929T17:46:18Z 20140929T17:46:18Z 20140929 http://hdl.handle.net/1794/18368 In this dissertation we explore the question of existence of a property of group actions on C*algebras known as the tracial Rokhlin property. We prove existence of the property in a very general setting as well as specialise the question to specific situations of interest. For every countable discrete elementary amenable group G, we show that there always exists a Gaction ω with the tracial Rokhlin property on any unital simple nuclear tracially approximately divisible C*algebra A. For the ω we construct, we show that if A is unital simple and Zstable with rational tracial rank at most one and G belongs to the class of countable discrete groups generated by finite and abelian groups under increasing unions and subgroups, then the crossed product A ω G is also unital simple and Zstable with rational tracial rank at most one. We also specialise the question to UHF algebras. We show that for any countable discrete maximally almost periodic group G and any UHF algebra A, there exists a strongly outer product type action α of G on A. We also show the existence of countable discrete almost abelian group actions with the "pointwise" Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear C*algebras with tracial rank zero and a unique tracial state appearing as crossed products. en_US University of Oregon All Rights Reserved. C*algebras classification crossed product existence group actions tracial Rokhlin property The Tracial Rokhlin Property for Countable Discrete Amenable Group Actions on Nuclear Tracially Approximately Divisible C*Algebras Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon
Now showing items 16 of 6