B*-Algebra

A Banach Algebra with an Antiautomorphic Involution $*$ which satisfies

$$x^{**} = x$$ (1)

$$x^*y^* = (yx)^*$$ (2)

$$x^*+y^* = (x+y)^*$$ (3)

$$(cx)^* = \bar cx^*$$ (4)

and whose Norm satisfies

$$\vert\vert xx^*\vert\vert=\vert\vert x\vert\vert^2.$$ (5)

A C*-Algebra is a special type of $B^*$-algebra.

See also Banach Algebra, C*-Algebra