Description:  Let G be a Polish group. We say that G is an algebraically determined Polish group if given any Polish group L and any algebraic isomorphism from L to G, then the algebraic isomorphism is a topological isomorphism. We will prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This will smooth the way for the proofs for some special groups. For example, let H be a separable Hilbert space and let G be a subset of the unitary group U(H) acting transitively on the unit sphere. Assume that I in G and G is a Polish topological group in some topology such that H x G to H, (x,U) to U(x) is continuous, then H x G is a Polish topological group. Hence H x G is an algebraically determined Polish group. In addition, we apply the above the above result on the unitary group U(A) of a separable irreducible C*algebra A with identity acting transitively on the unit sphere in a separable Hilbert space H and proved that the natural semidirect product H x U(A) is an algebraically determined Polish group. A similar theorem is true for the natural semidirect product R^{n} x G(n), where G(n) = GL(n,R), or GL^{+}(n,R), or SL(n,R), or SL(n,R)={A in GL(n,R) : det(A)=1}. On the other hand, it is known that the Heisenberg group H_{3}(R) , (R, +), (R{0}, x), and GL^{+}(n,R) are not algebraically determined Polish groups. 

Creator(s):  Jasim, We'am Muhammad 
Creation Date:  May 2011 
Partner(s): 
UNT Libraries

Collection(s): 
UNT Theses and Dissertations

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Publisher Name: University of North Texas
Place of Publication: Denton, Texas


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Description:  Let G be a Polish group. We say that G is an algebraically determined Polish group if given any Polish group L and any algebraic isomorphism from L to G, then the algebraic isomorphism is a topological isomorphism. We will prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This will smooth the way for the proofs for some special groups. For example, let H be a separable Hilbert space and let G be a subset of the unitary group U(H) acting transitively on the unit sphere. Assume that I in G and G is a Polish topological group in some topology such that H x G to H, (x,U) to U(x) is continuous, then H x G is a Polish topological group. Hence H x G is an algebraically determined Polish group. In addition, we apply the above the above result on the unitary group U(A) of a separable irreducible C*algebra A with identity acting transitively on the unit sphere in a separable Hilbert space H and proved that the natural semidirect product H x U(A) is an algebraically determined Polish group. A similar theorem is true for the natural semidirect product R^{n} x G(n), where G(n) = GL(n,R), or GL^{+}(n,R), or SL(n,R), or SL(n,R)={A in GL(n,R) : det(A)=1}. On the other hand, it is known that the Heisenberg group H_{3}(R) , (R, +), (R{0}, x), and GL^{+}(n,R) are not algebraically determined Polish groups. 

Degree: 
Name:
Doctor of Philosophy
Level:
Doctoral
Discipline:
Mathematics
Department:
Department of Mathematics
Grantor:
University of North Texas


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Keyword(s):  Polish groups  descriptive set theory  semidirect product  
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Partner: 
UNT Libraries


Collection: 
UNT Theses and Dissertations


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Resource Type:  Thesis or Dissertation  
Format:  Text  
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Holder:
Jasim, We'am Muhammad
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Copyright is held by the author, unless otherwise noted. All rights reserved.
