A reference for the Tannaka-Krein theorem I am looking for a reference for the Tannaka-Krein theorem on compact groups. By the Tannaka-Krein theorem which is also called (classic) Tannaka duality (because of the quantum theory), I mean the theory which is to reconstruct a compact (Lie) group from its representations. 

I found it myself. It is in 
Hewitt, Edwin; Ross, Kenneth A. Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups.  Die Grundlehren der mathematischen Wissenschaften, Band 152 Springer-Verlag, New York-Berlin 1970 ix+771 pp.
Section 30. It is written with all the details and preliminaries that one may need.
