Lie groups are important to describe symmetries, both in mathematics and in application (physics, chemistry, engineering, ...). The classical Lie groups are for example the orthogonal groups O(n), the unitary groups U(n), but mathematicians and physicists are also fascinated by more exotic examples such as the symmetry group of the octonions which is discussed a lot in modern mathematical physics. Many of these Lie groups can be represented as subgroups of Gl(k,R) for some sufficiently large k, but there are also Lie groups which cannot. Lie groups are manifolds G with a group structure on the underlying set such that the group multiplication is smooth. Lie groups and their representation is a mighty theory which allows effect calculations both for problems inside mathematics and also for applications outside (e.g. if in physics or chemistry the coupling of two electrons with various spin is described, this can be nicely described in terms of representaions of the Liegroup SU(2)=Spin(3)).
C*-algebras were first considered in quantum mechanics to model algebras of physical observables. In mathematics, they since then became ubiquitous tools in index theory, the theory of representations of locally compact groups and Alain Connes non-commutative geometry, just to name a few. They are also fundamental in coarse geometry, a subject that has been recently found to be relevant in the theory of topological phases, to circle back to physics.The lecture will start with a basic introduction to the C*-algebras, discussing in particular the result of Gelfand-Naimark that commutative C*-algebras are isomorphic to continuous functions on a locally compact Hausdorff space, while general C*-algebras can be realized as subalgebras of B(H) for some Hilbert space H.The lecture will continue with an introduction to K-theory, a tool that has revolutionized the study of C*-algebras in the last decades. Roughly speaking, the idea of K-theory is to understand an algebra by studying the category of modules over it. However, for C*-algebras, K-theory has this additional feature of Bott periodicity, which makes the theory particularly well-behaved.
Literature[B1] B. Blackadar. K-Theory for Operator Algebras. [B2] B. Blackadar. Operator Algebras. [RLL] M. Rørdam, F. Larsen, N.J. Laustsen. An Introduction to K-Theory for C*-Algebras. [WE] N.E. Wegge-Olsen. K-Theory and C*-Algebras. A Friendly Approach.
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