Editing C*-algebra (section)

== Abstract characterization ==

We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gelfand and Naimark.

A C*-algebra, ''A'', is a [[Banach algebra]] over the field of [[complex number]]s, together with a [[Map (mathematics)|map]] <math display="inline"> x \mapsto x^* </math> for <math display="inline"> x\in A</math>     with the following properties:

* It is an [[Semigroup with involution|involution]], for every ''x'' in ''A'':
::<math> x^{**} = (x^*)^* =  x </math>

* For all ''x'', ''y'' in ''A'':
::<math> (x + y)^* = x^* + y^* </math>
::<math> (x y)^* = y^* x^*</math>

* For every complex number <math>\lambda\in\mathbb{C}</math> and every ''x'' in ''A'':
::<math> (\lambda x)^* = \overline{\lambda} x^* .</math>

* For all ''x'' in ''A'':
::<math>  \|x x^* \| = \|x\|\|x^*\|.</math>

'''Remark.''' The first four identities say that ''A'' is a [[*-algebra]]. The last identity is called the '''C* identity''' and is equivalent to:

<math>\|xx^*\| = \|x\|^2,</math>

which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the [[#Some_history:_B.2A-algebras_and_C.2A-algebras|history]] section below.

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> \|x\|^2 = \|x^* x\| = \sup\{|\lambda| : x^* x - \lambda \,1 \text{ is not invertible} \}.</math>

A [[bounded linear map]], ''π'' : ''A'' → ''B'', between C*-algebras ''A'' and ''B'' is called a '''*-homomorphism''' if

* For ''x'' and ''y'' in ''A''

::<math> \pi(x y) = \pi(x) \pi(y) \,</math>

* For ''x'' in ''A''

::<math> \pi(x^*) = \pi(x)^* \,</math>

In the case of C*-algebras, any *-homomorphism ''π'' between C*-algebras is [[Contraction (operator theory)|contractive]], i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C*-algebras is [[isometry|isometric]]. These are consequences of the C*-identity.

A bijective *-homomorphism ''π'' is called a '''C*-isomorphism''', in which case ''A'' and ''B'' are said to be '''isomorphic'''.