A. Ülger, A Characterization of the Invertible Measures
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  • A. Ülger, A Characterization of the Invertible Measures

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A Characterization of the Invertible Measures - III




Ali Ülger


Koç University



                               Abstract. Let G be a locally compact abelian group and M(G) be its measure algebra. That is, this is the set of all the complex measures defined on the Borel sigma-algebra of G. The set M(G), under the convolution, is a commutative ring with a unit element.




The problem I am interested in is this: Characterize the invertible elements of M(G). This problem is open since the beginning of the theory (i.e. 1930’s).




Invertibility in any commutative unital ring involves the maximal ideals of that ring. The difficulty with the ring M(G) is that we know only a very few of its maximal ideals; and, the invertible elements of M(G) should be characterized in term of these maximal ideals.




In a series of talks, I shall start from the beginning of the subject and try to develop the necessary ingradients that will lead us to the solution of the above mentionned problem. 








A-    Banach Algebras: Definition and Examples.


B-     Commutative Semisimple Banach algebras: Gelfand spectrum, Gelfand Transform. Examples.


C-     The Multiplier Algebra M(A) of a Commutaive Semisimple Banach Algebra A. Definition, Examples and Gelfand Spectrum of M(A).


D-    Invertibility in a Commutative Unital Banach algebra.


E-     Locally Compact Abelian Groups:


a)      Potriyagin dual of a l.c.a group G, Examples.


b)      Haar Measure of G.


c)      The group algebra of G.


                         d) The Measure algebra M(G) of G.


                         e) M(G) as a multiplier algebra of the group algebra.


                         f) The idempotent elements of M(G).




            F-Glicksberg-Host-Parreau Theorem.


            G- A Characterization of the invertible measures.


            H- The General (Abstract) Problem of Invertible Multipliers and Results About this problem.








March 24, 2006, 15:30


Sabancı University, Karaköy İletişim Merkezi


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