Editing Operator theory (section)

===C*-algebras===
{{Main article|C*-algebra}}
A C*-algebra, ''A'', is a [[Banach algebra]] over the field of [[complex number]]s, together with a [[Map (mathematics)|map]] {{math|1=* : ''A'' → ''A''}}. One writes ''x*'' for the image of an element ''x'' of ''A''. The map * has the following properties:<ref>{{citation |first=William | last=Arveson|authorlink = William Arveson|title=An Invitation to C*-Algebra| publisher=Springer-Verlag | year=1976 |isbn=0-387-90176-0}}. An excellent introduction to the subject, accessible for those with a knowledge of basic [[functional analysis]].</ref>

* It is an [[Semigroup with involution|involution]], for every ''x'' in ''A'' <math display="block"> x^{**} = (x^*)^* =  x </math>
* For all ''x'', ''y'' in ''A'': <math display="block"> (x + y)^* = x^* + y^* </math> <math display="block"> (x y)^* = y^* x^*</math>
* For every λ in '''C''' and every ''x'' in ''A'': <math display="block"> (\lambda x)^* = \overline{\lambda} x^* .</math>
* For all ''x'' in ''A'': <math display="block"> \|x^* x \| = \left\|x\right\| \left\|x^*\right\|.</math>

'''Remark.''' The first three identities say that ''A'' is a [[*-algebra]]. The last identity is called the '''C* identity''' and is equivalent to:
<math display="block">\|xx^*\| = \|x\|^2,</math>

The C*-identity is a very strong requirement. For instance, together with the [[spectral radius|spectral radius formula]], it implies that the C*-norm is uniquely determined by the algebraic structure:
<math display="block"> \|x\|^2 = \|x^* x\| = \sup\{|\lambda| : x^* x - \lambda \,1 \text{ is not invertible} \}.</math>