Sherman–Takeda theorem

In mathematics, the Sherman–Takeda theorem states that if A is a C*-algebra then its double dual is a W*-algebra, and is isomorphic to the weak closure of A in the universal representation of A. The theorem was announced by (Sherman 1950) and proved by (Takeda 1954). The double dual of A is called the universal enveloping W*-algebra of A.

References

• Sherman, S. (1950), "The second adjoint of a C* algebra", Proceedings of the International Congress of Mathematicians 1950, 1, Providence, R.I.: American Mathematical Society, pp. 470, https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.1/ICM1950.1.ocr.pdf 
• Takeda, Zirô (1954), "Conjugate spaces of operator algebras", Proceedings of the Japan Academy 30 (2): 90–95, doi:10.3792/pja/1195526177, ISSN 0021-4280, https://www.jstage.jst.go.jp/article/pjab1945/30/2/30_2_90/_article/-char/en