Idealizer
Encyclopedia
In mathematics, the idealizer R(A) of an additive subgroup A of a ring
R is the largest subring of R that contains A as a two-sided ideal. One has
Example: The multiplier algebra
M(A) of a C*-algebra A is isomorphic to the idealizer of π(A) where π is any faithful nondegenerate representation of A on a Hilbert space
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Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R is the largest subring of R that contains A as a two-sided ideal. One has
Example: The multiplier algebra
Multiplier algebra
In C*-algebras, the multiplier algebra, denoted by M, of a C*-algebra A is a unital C*-algebra which is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification...
M(A) of a C*-algebra A is isomorphic to the idealizer of π(A) where π is any faithful nondegenerate representation of A on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
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