The Cstar Algebra [Definition and Examples]

Definition

Let $\mathcal{A}$ be an associative $\mathbb{C}$-algebra, let $\|.\|$ be a norm on the vector $\mathbb{C}$-vector space $\mathcal{A}$, and let $*:\mathcal{A}\rightarrow\mathcal{A}$, $a\mapsto{a^*}$ be a $\mathbb{C}$-antilinear map. Then $(A,\|.\|,*)$ is called a $C^*$-algebra if $(A,\|.\|)$ is complete and we have for all $a,b\in{\mathcal{A}}$:

i. $a^{**}=a$ (* is an involution)

ii. $(ab)^*=b^*a^*$

iii. $\|ab\|\leq\|a\|\|b\|$ (submultiplicativity)

iv. $\|a^*\|=\|a\|$ ($*$ is an isometry)

v. $\|a^*a\|=\|a\|^2$ ($C*$ property)

A norm(not necessarily complete) satisfying(i)-(v) is called a $C^*$-norm.

Example

 Let $(\mathcal{H},\langle{.,.}\rangle)$ be a complete Hilbert space be a complex Hilbert space, let $A=L(H)$ be the algebra of bounded linear operators on $\mathcal{H}$. Let $\|.\|$ be the operator norm i.e
$\|a\|:\sup_{\|x\|=1}\|ax\|,\ x\in{\mathcal{H}}$ Let $a^*$ be the operator adjoint to $a$, i.e

$\langle{ax,y}\rangle=\langle{x,a^*y}\rangle$ for all

$\ xy\in\mathcal{H}$

Here axioms $i-iv$ are easily checked.

Using axioms $iii$ and $iv$ and the Cauchy-Schwarz inequality we see

$\|a\|^2=\sup\|ax\|^2=\sup_{\|x\|=1}\langle{ax,ax}\rangle$
$=\sup_{\|x\|=1}\langle{x,a^{*}ax}\rangle$

$\leq{\sup_{\|x\|=1}}\|x\|.\|a^{*}ax\|$

$=\|a^{*}a\|\leq{\|a^{*}\|}.\|a\|$

by axiom $iv$
$=\|a\|^2$

Example
Let $X$ be a locally compact Hausdorff space. Put
 $A:=C_0(X):=\{f:X\rightarrow\mathbb{C}$ continuous for all
$\epsilon>0$ there exists

$K\subset{X}$ is compact so that for all

$x\in{X\backslash{K}}:|f(x)|<\epsilon\}$

We $C_0(X)$ the algebra of continuous function varnishing at infinity.
If $X$ is compact then $A=C_0(X)=C(X)$.
All $f\in{C_0(X)}$ are bounded and we may define
 $\|f\|:=\sup_{x\in{X}}\|f(x)\|$
Moreover let
$f^*(x):=\overline{f(x)}$ Then $(C_0(X),\|.\|,*)$ is a cummutative $C^*$-algebra.