In mathematics, specifically in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the Sherman–Takeda theorem states that if ''A'' is a
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
then its double dual is a
W*-algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebr ...
, and is isomorphic to the
weak closure of ''A'' in the
universal representation of ''A''.
The theorem was announced by and proved by . The double dual of ''A'' is called the
universal enveloping W*-algebra
In operator algebras, the enveloping von Neumann algebra of a C*-algebra is a von Neumann algebra that contains all the operator-algebraic information about the given C*-algebra. This may also be called the ''universal'' enveloping von Neumann alg ...
of ''A''.
References
*
*
Banach algebras
C*-algebras
Theorems in functional analysis
Operator theory
Von Neumann algebras
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