I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask:
Question: What was the motivation and historical context for works of major contributors to the "Tannaka-Krein theory" (in a broad sense)? Just to name a few names: Tannaka, Krein, Saavedra, Deligne, Milne, Lurie, (it seems Grothendieck should also be in this list(?)).
Let me explain some points in the history which seems to me puzzling:
Mark Krein was a famous Soviet mathematician, but he was an expert in analysis,
it seems it is the only paper by him devoted to algebra(See discussion below). How did he come to it? Why did he not continue?
Similar question about Tadao Tannaka. "His interest in mathematics lied mainly in algebraic number theory", And it seems similar to Krein, it is the only work by devoted to group theory. (See his publication list).
P. Deligne seems to have devoted quite much efforts on "Tannakian formalism" and more generally on tensor categories. What was his motivation? He is a leading algebraic geometer. So probably the subject should be quite important in algebraic geometry? What is its importance?
Wikipedia article starts with a sentence: "...natural extension to the non-Abelian case is the Grothendieck duality theory." What is the role of Grothendieck in this history ? And what is "Grothendieck duality theory" - wikipedia links to something not related.
Important work was done by Saavedra. It seems not so much is known about him, his motivation, his other works.
J. Lurie seems to develop the theory further (see e.g. MO question Tannakian formalism). What is the motivation?
List of references (it seems original articles by Tannaka and Krein are not available electronically)
Tadao Tannaka, Über den Dualitätssatz der nichtkommutativen topologischen Gruppen, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!)
M.G. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady AN SSSR 69 (1949), 725–728. in Russian: М. Г. Крейн, Принцип двойственности для бикомпактной группы и квадратной блок-алгебры, Докл. АНСССР, 69:6 (1949), 725–728.
N. Saavedra Rivano, Cat´egories tannakienns, Lecture Notes in Math., vol. 265, Springer-Verlag, Berlin–New York, 1972.
Deligne, P., and Milne, J.S., Tannakian Categories, in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, 1982, pp. 101-228". ( http://www.jmilne.org/math/xnotes/tc.html )
Some remarks about Mark Krein. Part of his publication list is here, strangely enough the paper on "Tanaka-Krein duality" is not contained in this list.
I have found an article devoted to overview of his works related to group theory: L. I. Vainerman. On M. G. Krein's works in the theory of representations and harmonic analysis on topological groups Ukrainian Mathematical Journal 46 (1994), no. 3, 204-218.
It seems he had several papers dating from 1940-1949 which were related to "Tannaka-Krein theory".
He started as student of Nikolai Chebotaryov, who is famous for Chebotarev density theorem, but actually was also working on Lie groups: famous results Ado theorem and Jacobson-Morozov theorem were obtained by his students Igor Ado and Morozov, who worked in Kazan city Russia. But it is not clear whether Krein was influenced by Chebotarev in this respect, since they meet around 1924 in Odessa city, and the paper was written in 1949, when Chebotarev already passed and long before he moved from Odessa to Kazan city, while Krein stayed in Odessa.
Anatoly Vershik in his paper devoted to 100-anniversary of M. Krein suggests that it might be that "success of Gelfand's theory of commutative normed rings" influenced Krein.