Normed star rings and algebras #
A normed star group is a normed group with a compatible
star which is isometric.
A C⋆-ring is a normed star group that is also a ring and that verifies the stronger
‖x⋆ * x‖ = ‖x‖^2 for all
x. If a C⋆-ring is also a star algebra, then it is a
To get a C⋆-algebra
E over field
[NormedField 𝕜] [StarRing 𝕜] [NormedRing E] [StarRing E] [CstarRing E] [NormedAlgebra 𝕜 E] [StarModule 𝕜 E].
- Show that
‖x⋆ * x‖ = ‖x‖^2is equivalent to
‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖, which is used as the definition of C*-algebras in some sources (e.g. Wikipedia).