March 7, 2017

C*-algebras and their automorphism groups by Gert Kjaergard Pedersen

By Gert Kjaergard Pedersen

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Dl g, dl (g −1 ) = ∆G (g)dl g. 130, we propose the proof as an exercise based on the application of Haar’s Theorem. 68/9, but the modular function there is the inverse to our function, which here is defined following Kirillov. 2: If G is compact, then the Haar measure is finite and biinvariant. Proof: The image of a compact group under the continuous homomorphism ∆G is again a compact group. The multiplicative group R>0 has only {1} as a compact subgroup. 1, we have a biinvariant measure which can be normalized to G dg = 1.

1, we recommend to do this and to determine the characters of all the representations π1 , π2 , π3 , and π0 introduced there. 1. 53]). 1 to the central fact that the characters χ1 , . . , χh belonging to a complete set of equivalence classes of irreducible representations of G form an orthonormal basis of the space of class functions H0 . We recall that a function f defined on G is a class function if one has f (g) = f (tgt−1 ) for all g, t ∈ G. 6: Let f be a class function on G and (π, V ) a representation of G.

Continuous Representations • An operator T is said to be bounded iff there exists a constant C such that one has T v ≤ C v for all v ∈ D(T ). • The norm T of a bounded operator T is defined as the smallest C such that Tv ≤ C v for all v ≤1 . 5: A bounded linear operator is uniformly continuous. g. v0 = 0), then T is bounded. 2: Prove this. • Let A be a linear operator in H with dense domain D(A) ⊂ H. Then one can prove that there are v, v ∈ H such that < Au, v > = < u, v > holds for all u ∈ H. One sets v =: A∗ v, verifies that A∗ is a linear operator, and calls A∗ the adjoint of A.

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