MATH 546 C1 --  Hilbert Spaces
 
 
 

Time:  MWF 2:00-2:50pm

Location:  Altgeld Hall  141

Instructor: Marius Junge      Course email   Office hours: Tuesday 5-6 pm 147 Altgeld
 

Course description
:

Part I: Basic Hilbert spaces theory
      1) Banach space and linear maps
      2) sesquilinear forms and Cauchy Schwarz,
      2)
Gram Schmidt, ONB and best approximation
      4)
Riesz representation
      5)
geometric properties
      6) Compact and Fredholm operators
 Comments 


     

Part II: Basic Banach and C*-algebra theory and spectral
     1) Banach algebras with and without unit
      2) Spectrum of commutative Banach algebras
      3) Spectrum of commutative C*-algebras
      4) Aplication: Spectrum of compact normal operators
      5) Positive elements and positive functional
      6) approximate units and quotients
      7) Aplications



Part III: Spectral theory for normal operators

     1) Motivation: Volterra operator, Shift operator
      2) Riesz Representation  theorem
      3) Projection valued measures 
      4) Spectral decomposition for normal operators
      5) Multiplicity theory

Part V: Hilbert C*-modules and completely positive maps
     1) Definition of Hilbert C*-modules
     2) Projective modules, W*-modules
     3) Completely positive maps and the  GNS representation
     4)  Kasparov's representation  theorem
    
5) 
Further applications


Books:
John B. Conway:  A course in Functional Analysis, Springer, Graduate Text in Mathematics 1990 (second edition)

John B. Conway: A Course in Operator Theory, Graduate Studies in Mathematics, Vol. 21, 1999

E. Lance:
Hilbert C*-modules,   London Mathematical Society Lecture Note Series, 210., Cambridge University Press, Cambridge, 1995



Grading:
Homework   (50%)

Final exam or  presentation  (50%)


Comments: I expect everybody to give a presentation, please look at the books and reprot on your topic by mid October


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