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In mathematics, an element x of a star-algebra is self-adjoint if the involution acts trivially upon it. In other words, <math>x^*=x<math>. When we work in an inner product space which is a star-algebra, being self-adjoint is the same as being hermitian.
A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if <math>x^*=y<math> then since <math>y^*=x^{**}=x<math> in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements.
In linear algebra, an operator T∈L(V) is called self-adjoint iff T = T*.
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