MATH 546 B1 --  Hilbert Spaces
 
 
 

Time:  MWF 9:00-9:50

Location:  Altgeld Hall  341

Instructor: Marius Junge      Course email   Office hours:  tba
 

Course description :

Part I: Hilbert Spaces
      1) sesquilinear forms,
      2) pre-Hilbert spaces
      3) ONB
      4) Riesz representation
      5) geometric properties
      6) least square

Part II: Spectral theory for normal compact operators
     1) Linear and compact operators on Banach spaces
      2) Weak topologies and duality for compact operators
      3) Spectral theorem for normal compact operators

Part III:  Introduction to (commutative) C*-algebras
     1) Elementary properties
      3) Holomorphic spectral calculus in Banach algebras
      3) Gelfand transform for commutative Banach algebras
      4) Positive elements and positive functionals
 
Part IV: 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)

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



Grading:
Homework (weekly) and projects  (50%)

Final exam or  presentation  (50%)


Comments: Some of the material will be prepared by individual students or groups of students  (projects)-That is good for  all of us.


hw1

hw2

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