The history of C*-algebras starts with two groundbreaking articles by the Russian mathematicians Gelfand and Naimark in 1941 and 1943. Meanwhile C*-algebras  have become indispensable in modern mathematics. They have found numerous and deep applications in Analysis, Index Theory, Noncommutative Geometry, Topology and Quantum Physics. 

Formally, a C*-algebra is a (complex) Banach algebra A with an involution *, in which the identity  ||a*a||=||a||^2$ holds for all a in A.
 
A simple example is given by the algebra C(X) of all continuous complex-valued functions on a compact Hausdorff space X  with complex conjugation as involution or, more generally, the algebra C₀(X) of all continuous functions on a locally compact Hausdorff space X with the following property: For every f in C₀(X) and every ε>0 there exists a compact subset K_ε of X with |f(x)|< ε for x outside K_ε. 
Also, every closed subalgebra A of the algebra L(H) of all bounded linear operators on a Hilbert space H with the adjoint as involution is a C*-algebra.

These two examples are in fact central to the theory. Every C*-algebra is isomorphic to a closed subalgebra of  L(H) for a suitable Hilbert space H. Moreover, every commutative C*-algebra is isomorphic to one of the form C_0(X).

We will start this seminar by learning the basics of the theory and studying important examples. Depending on prior knowledge and interests of the participants, different applications can be covered in the sequel. 

The seminar may serve as the foundation of a Bachelor's Thesis. 

Literature

1. K. Davidson. C*-Algebras by Example. AMS, Providence, RI, 1996
2. J. Dixmier, C*-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
3.  G. Murphy. C*-Algebras and Operator Theory. Academic Press, San Diego 1990
4. N.E. Wegge-Olsen. K-Theory and C*-Algebras. Oxford University Press, Oxford, New York, Tokyo 1993

Questions: schrohe@math.uni-hannover.de.

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