# K-Theory of C*-algebras

Recall that two projections $p,q$ in a C*-algebra $A$ are equivalent, denoted $p\sim q,$ if there is a partial isometry $u$ such that $p=u^*u$ and $q=uu^*.$  Let $V_0(A)$ be the set of equivalence classes of projections and $V(A)=V_0(M_\infty(A)).$  We can define addition on $V(A)$ by

$[p]+[q]=[p'+q']$

where $p'\in [p]$, $q'\in [q]$, and $p'\perp q'$ (see [1] for why this is possible).  This gives an abelian monoid structure on $V(A)$ with identity $[0]$.  If $A$ is unital, then we define $K_0(A)=G(V(A))$ where $G(V(A))$ is the Grothendieck group of $V(A).$

To proceed with the nonunital case, first consider that $V:C^*algebras\to Mon$ is a covariant functor sending $A\mapsto V(A),$ since if $\varphi:A\to B$ is a *-homomorphism, then we can define $V(\varphi):V(A)\to V(B)$ by

$V(\varphi)([p])=V(\varphi\circ V^{-1}[p]).$

Recall the unitization of $A$ where $A^\dagger=A\oplus\mathbb{C}$ with funky structure.  We have a *-homomorphism $\mu:A^\dagger\to\mathbb{C}$ such that $\ker\mu=0.$  Since $V$ is a covariant functor, and by composition induces a covariant functor $K_0:C^*algebras\to Grp,$ we have a map

$\mu_*=K_0(\mu):K_0(A^\dagger)\to K_0(\mathbb{C})=\mathbb{Z}.$

In the nonunital case we then define $K_0(A)=\ker\mu_*.$  Now define

$U_n(A)=\{x\in U(M_n(A^\dagger)):x=1_n\mbox{~mod~}M_n(A)\}.$

If $A$ is unital, then the condition is always satisfied, so we have $U_n(A)=U(M_n(A)).$  Define a norm on $M_n(A)$ by $\|x\|=\max_{ij}\|x_{ij}\|$ and let $U_n(A)_0$ be the connected component (in the sense of the norm topology) containing $1_n.$  Then we define

$K_1(A)=\lim U_n(A)/U_n(A)_0.$

Definition 1.  Let $A$ be a C*-algebra.  We define its suspension, denoted $SA,$ as $C_0((0,1),A)=C_0(\mathbb{R},A).$

Hence these are continuous $A$-valued functions that vanish at infinity (so the the suspension is similar to the topological notion).

Theorem 2.  There is an isomorphism $\varphi:K_1(A)\to K_0(SA)$ defined by $\varphi([v])=[p]-[q_n]$ where $[p]=[p_t:(0,1)\to A]$ defined by $p_t=w_tq_nw_t^*$ and $w_t$ is a path in $U_{2n}(A)$ between $1_{2n}$ and $diag(v,v^*).$

$[q_n]$ in the image refers to the constant map $q_n:(0,1)\to\{q_n\}.$  We can then define by induction the higher K-groups: $K_{n+1}(A)=K_n(SA).$  All of these are clearly covariant functors since the definition can be reduced to $K_0$ of the $(n+1)$-th suspension of $A.$

Corollary 3.  Let

$0\longrightarrow J\stackrel{i}{\longrightarrow} A\stackrel{\pi}{\longrightarrow}A/J\to 0$

be a short exact sequence.  Then the induced sequence

$K_1(J)\stackrel{i_*}{\longrightarrow} K_1(A)\stackrel{\pi_*}{\longrightarrow}K_1(A/J)$

is exact in the middle.

Let us define a map $\partial_1:K_1(A/J)\to K_0(J)$ by

$\partial_1([u])=[wq_nw^*]-[q_n]$

for some $q_n\in J$ where $u\in U_n(A/J)$ and $w\in U_{2n}(A)$ be a lift of $diag(u,u^{-1}).$  Then $\partial_1([u])\in K_0(J)$ since $diag(u,u^{-1})$ commutes with $q_n$ since $u\notin J.$

Proposition 4$\partial_1$ makes the slightly longer sequence exact at $K_1(A/J)$ and $K_0(J).$

Corollary 5.  By induction we obtain maps $\partial=\{\partial_n\}$ and a long exact sequence

$\cdots K_n(J)\stackrel{i_*}{\to}K_n(A)\stackrel{\pi_*}{\to}K_n(A/J)\stackrel{\partial}{\to}K_{n-1}(J)\stackrel{i_*}{\to}\cdots\stackrel{\pi_*}{\to}K_0(A/J)\to 0.$

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.

# Covariant Systems

Recall a topological group $G$ is a group and a topological space such that the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are continuous.  Let $\Sigma$ be the $\sigma$-algebra generated by the compact subsets of $G$.  A measure $\mu$ on $\Sigma$ is left-invariant if $\mu(tU)=\mu(U)$ for all $t\in G$ and $U\in\Sigma$.  A left Haar measure on $G$ is a left-invariant Radon measure on $\Sigma$.

Theorem 1.  Let $G$ be a locally compact group.  Then there exists a unique left and unique right Haar measure on $G$ (up to multiplication by a constant).

See [2] for its construction.  Let $\mu$ be a left Haar measure on $G$.  We can define right translate Haar measures by $\mu_t(U)=\mu(Ut)$.  Clearly these are left invariant as they simply changes the input of the original left invariant Haar measure.

Proposition 2.  If $\mu$ is a left Haar measure, then there is a unique function $\Delta_G:G\to\mathbb{R},$ called the modular function of $G$, such that $\mu_t(U)=\Delta_G(t)\mu(U)$.

Let $\rho:G\to U(L(H))$ be a strongly continuous unitary representation of $G$ on a Hilbert space $H$.  That is, the group homomorphism is continuous with respect to the norm topology of $U(L(H))$, and $U(L(H))$ is the subset of $L(H)$ consisting of unitary linear operators on $H$.  Let $\mu$ be a left Haar measure on $G$ and $f\in L^1(G)$, then the operator

$\displaystyle\rho^*(f)=\int_G f(t)\rho(t)\,d\mu(t)$

in $L(H)$ is bounded and in fact defines a nondegenerate representation $\rho^*:L^1(G)\to L(H)$ as a Banach *-homomorphism with $f\mapsto\rho^*(f)$.  The product on $L^1(G)$ is convolution:

$\displaystyle (f*g)(t)=\int_G f(s)g(s^{-1}t)\,d\mu(s)$

and the involution is defined by

$f^*(t)=\Delta_G(t^{-1})\bar{f}(t^{-1}).$

Recall in the GNS construction we started with a positive functional $f$ on a C*-algebra $A$ and induced a representation $\rho_f:A\to L(H_f)$.  Here we start with one representation $\rho$ on $G$ and induce another, $\rho^*$, on $L^1(G)$.  We call $\rho^*$ the integrated form of $\rho$.  We can impose another norm (other than the default sup norm $\|f\|=\sup_{t\in G}|f(t)|$) on $L^1(G)$ defined by

$\|f\|=\sup_{\rho}\|\rho(f)\|$

where $\rho$ is a representation of $L^1(G)$.  The completion of $L^1(G)$ with respect to this norm is a C*-algebra called the group C*-algebra of $G$, which we denote by $C^*(G)$.  So every strongly continuous unitary representation of $G$ induces a nondegenerate representation of $L^1(G)$, and in fact, the converse is also true (that every nondegenerate representation of $L^1(G)$ is induced by a scu representation of $G$).  Moreover there is a bijection between the irreducible ones in each case.

Definition 3.  A covariant system is a triple $(G,A,\alpha)$ where $G$ is a locally compact group, $A$ is a C*-algebra, and $\alpha:G\to Aut(A)$ is a continuous representation (where $Aut(A)$ possesses the point-norm topology).  A covariant representation on a covariant system $(G,A,\alpha)$ is a pair $(\rho,\psi)$ of representations of $G$ and $A$ respectively on a Hilbert space $H$ such that $\rho$ is strongly continuous unitary, $\psi$ is nondegenerate, and

$\rho(t)\psi(a)\rho(t)^*=\psi(\alpha(t)(a))$

for all $t\in G$ and $a\in A$.

Definition 4.  We define the covariance algebra $L^1(G,A)$ of the covariant system $(G,A,\alpha)$ as the completion of $C_C(G,A)$ under the norm

$\displaystyle\|f\|_1=\int_G \|f(t)\|\,d\mu(t)$

where we define

$\displaystyle (f*g)(t)=\int_G f(s)\cdot\alpha(s)(g(s^{-1}t))\,d\mu(s)$

$\displaystyle f^*(t)=\Delta_G(t^{-1})\cdot\alpha(t)(f(t^{-1})^*).$

It is a Banach *-algebra.

Definition 5.  If $(\rho,\psi)$ is a covariant representation of $(G,A,\alpha)$, then there is a nondegenerate representation $\rho\times\psi:L^1(G,A)\to L(H)$ defined by

$\displaystyle(\rho\times\psi)(f)=\int_G \rho(f(t))\psi(t)\,d\mu(t)$

called the integrated form of $(\rho,\psi)$$L^1(G,A)$ together with a new norm

$\displaystyle\|f\|=\sup_{(\rho,\psi)}\|(\rho\times\psi)(f)\|$

is a C*-algebra called the crossed product of $(G,A,\alpha)$, which is denoted $C^*(G,A,\alpha)$.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.

[2]  Folland, Gerald.  Real Analysis: Modern Techniques and Their Applications.  2nd Edition.  John Wiley and Sons.  1999.

# Tensor Products of C*-algebras

Let $A$ and $B$ be C*-algebras.  We can define their *-algebra tensor product as the standard tensor product of algebras $A\otimes B$ with product $(a\otimes b)(a'\otimes b')=aa'\otimes bb'$ and involution $(a\otimes b)^*=a^*\otimes b^*$.  There are a variety of norms one can impose on this tensor product to make $A\otimes B$ a Banach *-algebra.  For example we may define

$\displaystyle\left\|\sum a_i\otimes b_i\right\|_\wedge=\sum \|a_i\|\|b_i\|$.

This seminorm becomes a norm on $A\otimes B$ modulo the appropriate subspace, and its completion is denoted $A\hat{\otimes}B$ and is called the projective tensor product of $A$ and $B$.  We also have

$\left\|\left(\sum a_i\otimes b_i\right)^*\right\|_\wedge=\left\|\sum a_i^*\otimes b_i^*\right\|_\wedge=\sum \|a_i^*\|\|b_i^*\|=\sum \|a_i\|\|b_i\|=\left\|\sum a_i\otimes b_i\right\|,$

so $A\hat{\otimes}B$ is a Banach *-algebra.  But it fails to satisfy the C*-axiom ($\|x^*x\|=\|x\|^2$):

$\begin{array}{lcl}\left\|\left(\sum a_i\otimes b_i\right)^*\left(\sum a_i\otimes b_i\right)\right\|&=&\left\|\left(\sum a_i^*\otimes b_i^*\right)\left(\sum a_i\otimes b_i\right)\right\|\\&=&\left\|\sum a_i^*a_j\otimes b_i^*b_j\right\|\\&=&\sum\|a_i^*a_j\|\|b_i^*b_j\|\\&\leq&\sum \|a_i\|\|a_j\|\|b_i\|\|b_j\|\\&=&\left(\sum \|a_i\|\|b_i\|\right)^2\\&=&\left\|\sum a_i\otimes b_i\right\|^2\end{array}.$

It turns out that representations on $A$ and $B$ allow us to define norms on $A\otimes B$ that make it a C*-algebra.

Definition 1.  Let $\rho_A:A\to L(H_1)$ and $\rho_B:B\to L(H_2)$ be representations on $A$ and $B$.  We define the product representation $\rho=\rho_A\otimes\rho_B$ on $H_1\otimes H_2$ as

$\rho(a\otimes b)=\rho_A(a)\otimes\rho_B(b)\in L(H_1)\otimes L(H_2)$.

Since we always have the trivial representations, the set of representations of $A$ on $H_1$ and $B$ on $H_2$ are never empty.  Let us define the minimal C*-norm on $A\otimes B$ by

$\begin{array}{lcl}\displaystyle\left\|\sum a_i\otimes b_i\right\|_{\mbox{min}}&=&\displaystyle\sup_{\rho_A,\rho_B}\left\|\rho\left(\sum a_i\otimes b_i\right)\right\|\\&=&\displaystyle\sup_{\rho_A,\rho_B}\left\|\sum \rho_A(a_i)\otimes\rho_B(b_i)\right\|\end{array}$

where the two norms on the right are operator norms.  This is clearly finite (hence a norm) and satisfies the C*-axiom.  The completion of $A\otimes B$ with this norm is a C*-algebra called the minimal (or spatial) tensor product of $A$ and $B$ with respect to $\rho_A$ and $\rho_B$, and is denoted $A\underline{\circledast} B$.

Definition 2.  Let $\rho_A:A\to L(H)$ be a representation and $N\leq H$ be the largest subspace of $H$ such that $\rho(a)(x)=0$ for all $a\in A$ and $x\in N$.  Then $N^\perp$ is called the essential subspace of $H$, and we will denote it $E(H)$.  If $E(H)=H$, then $\rho_A$ is said to be nondegenerate. $\rho_A$ is degenerate if it is not nondegenerate.

In other words, $\rho_A$ is nondegenerate if $N=0$.

Proposition 3.  If $\rho:A\otimes B\to L(H)$ is a nondegenerate representation, then there are unique nondegenerate representations $\rho_A:A\to L(H)$ and $\rho_B:B\to L(H)$ such that $\rho(a\otimes b)=\rho_A(a)\rho_B(b)=\rho_B(b)\rho_A(a)$.

But arbitrary representations of the tensor product of algebras cannot be broken into pieces.  This gives us the following.

Definition 4.  Let $H$ be a Hilbert space and $A,B$ be C*-algebras.  We define the maximal C*-norm on $A\otimes B$ as

$\displaystyle\left\|\sum a_i\otimes b_i\right\|_{\mbox{max}}=\sup_{\rho}\left\|\rho\left(\sum a_i\otimes b_i\right)\right\|$

where $\rho:A\otimes B\to L(H)$.  This is also a C*-norm, and the completion of $A\otimes B$ under this norm is a C*-algebra called the maximal tensor product of $A$ and $B$ and is denoted $A\overline{\circledast}B$.

We also have that $\|\cdot\|_{\mbox{min}}\leq\|\cdot\|_*\leq\|\cdot\|_{\mbox{max}}\leq\|\cdot\|_\wedge$ where $\|\cdot\|_*$ is any C*-norm.  It follows that $\|(a\otimes b)\|_*=\|a\|\|b\|$.

Definition 5.  A functional on $A\otimes B$ is positive if $f(x^*x)\geq 0$ for all $x\in A\otimes B$.  A state on $A\otimes B$ where $A$ and $B$ are unital is a positive linear functional $f$ on $A\otimes B$ such that $f(1\otimes 1)=1$.  We denote the set of states by $S(A\otimes B)$.

As in the previous post, there is a GNS construction that gives a representation $\rho_f:A\overline{\circledast}B\to L(H_f)$ for a positive linear functional $f$, although one must show the left action on $H_f$ is by bounded operators.

Definition 6.  A C*-algebra $A$ is nuclear if for every C*-algebra $B$, there is a unique C*-norm on $A\otimes B$.

Hence in such a case, we would have $A\underline{\circledast} B=A\overline{\circledast} B$, and thus denote the product C*-algebra by $A\circledast B$.  The class of nuclear C*-algebras includes all of the commutative ones, finite ones, and is itself closed under inductive products and quotients.  Non nuclear ones are exotic; $C^*(\mathbb{F}_2),$ the group C*-algebra of $\mathbb{F}_2$ (see next post), is an example.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag. 2006.

# States and Representations of C*-algebras

Definition 1.  Let $A$ be a C*-algebra and $x\in A$$x$ is positive if it is self-adjoint and $\sigma_A(x)\subseteq [0,\infty)$.  We denote the set of positive elements of $A$ by $A_+$ and write $x\geq 0$ for all $x\in A_+$.  We also write $x\leq y$ if $y-x\geq 0$.

It turns out that if $B$ is a C*-subalgebra of $A$, then $\sigma_B(x)=\sigma_A(x)$.  Hence $B_+=B\cap A_+$.

Definition 2.  A functional $f:A\to\mathbb{C}$ is positive if $x\geq 0\Rightarrow f(x)\geq 0$ (hence $f(x)\in\mathbb{R}$).  A state is a positive linear functional such that $\|f\|=\sup_{x\in A}|f(x)|=1$.  The set of all states is called the state space and is denoted $S(A)$.

If $f$ is a positive linear functional, it defines a pre inner product on $A$:

$\langle x,y\rangle_f=f(y^*x)$.

If $H$ is a Hilbert space, we can endow a *-algebra structure on $L(H)$, the space of linear operators on $H$ where multiplication is composition, and if $X\in L(H)$, then $X^*$ is defined as the adjoint of $X$.

Definition 3.  Let $A$ be a C*-algebra and $H$ be a Hilbert space.  A representation is a *-homomorphism $\rho:A\to L(H)$.  A subrepresentation is a representation $\rho':A\to L(H')$ where $H'\leq H$ is a closed subspace of $H$ which is invariant under action from $A$.  A representation is irreducible if it has no nontrivial subrepresentations.  A representation $\rho$ is faithful if $\ker\rho=0$.

We now present an important connection discovered by Gelfand, Naimark, and Segal.

Let $A$ be a C*-algebra and $f$ be a positive linear functional on $A$.  Define

$N_f=\{x\in A:f(x^*x)=0\}$.

$N_f$ is a closed left ideal in $A$, and $(A/N_f,\langle\cdot\rangle_f)$ is an inner product space.  Let $H_f=L^2(A,f)$ be its Hilbert completion.  Now define $\rho_f:A\to L(H_f)$ by action of $x$: $\rho_f(x)(a+N_f)=xa+N_f$$\rho_f$ is clearly a representation, and is called the GNS representation of $A$ associated with $f$.

If $A$ is unital, let $1_f$ denote the image of $1$ in the completion/quotient composition $A\to A/N_f\to H_f$.  Then $f$ induces a positive linear functional on $L(H_f)$ defined by

$\phi_f(X)=\langle X(1_f),1_f\rangle_f$

for $X\in L(H_f)$.

Theorem 4.  Let $f,g$ be positive linear functionals on a C*-algebra $A$ such that $g\leq f$ (meaning $g(x)\leq f(x)$ for all $x\in A_+$).  Then there is a unique operator $X\in\rho_f(A)\subseteq L(H_f)$ such that $0\leq X\leq 1$ and $g(x)=\phi_f(X\rho_f(x))=\langle (X\rho_f(x)(1_f),1_f\rangle_f$ for all $x\in A$.

This is a generalization of the Radon-Nikodym theorem (which is special case for $A=L^\infty(X,\mu)$ where $(X,\mu)$ is a finite measure space and $\phi(f)=\int f\,d\mu$).

Definition 5.  We define an extreme point in topological vector space $V$ as a point that does not belong to any open line segment in $V$.  A pure state is an extreme point in $S(A)$.  The set of pure states of $A$ is denoted $P(A)$.

It follows that $S(A)=hull(P(A))$.

Proposition 6.  Let $f$ be a state on a C*-algebra $A$.  Then $\rho_f$ is irreducible if and only if $f$ is pure.

[1] Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.

# C*-Algebras

Definition 1.  A Banach algebra is a complex algebra which is a Banach space and satisfies subnormality:

$\|xy\|\leq\|x\|\|y\|$.

A *-algebra is a complex algebra with a unary operation * that satisfies $(x+y)^*=x^*+y^*$, $(xy)^*=y^*x^*$, $x^{**}=x$, and $(\lambda x)^*=\overline{\lambda}x^*$.  A Banach *-algebra is a Banach algebra and a *-algebra that satisfies $\|x^*\|=\|x\|$.  A C*-algebra is a Banach *-algebra that satisfies $\|x^*x\|=\|x\|^2$.  A map between two *-algebras is a *-homomorphism if $f(x^*)=f(x)^*$.

Proposition 2.  Any nonunital Banach algebra $A$ has a unitization $A^\dagger$.

Proof.  Let $A^\dagger=A\oplus\mathbb{C}$.  Let $a\in A$, and send $a\mapsto (a,0)$.  Define $(a,\lambda)+(b,\mu)=(a+b,\lambda+\mu)$, $(a,\lambda)(b,\mu)=(ab+\lambda b+\mu a,\lambda\mu)$, and $\|(a,\lambda)\|=\|a\|+|\lambda|$.  Clearly $A^\dagger$ is a Banach space.  We also have

$\begin{array}{lcl}\|(a,\lambda)(b,\mu)\|&=&\|(ab+\lambda b+\mu a,\lambda\mu)\|\\&=&\|ab+\lambda b+\mu a\|+|\lambda\mu|\\&\leq&\|ab\|+\|\lambda b\|+\|\mu a\|+|\lambda||\mu|\\&\leq&\|a\|\|b\|+|\mu|\|a\|+|\lambda|\|b\|+|\lambda||\mu|\\&=&\|(a,\lambda)\|\|(b,\mu)\|.\end{array}$

The unit is $(0,1)$: $(a,\lambda)(0,1)=(a,\lambda)=(0,1)(a,\lambda)$.

In fact, if $A$ is a nonunital Banach *-algebra, then there is a unitization of $A$ which is also a Banach *-algebra.  We simply use $A^\dagger$ and define $(a,\lambda)^*=(a^*,\overline{\lambda})$.  And if $A$ is a C*-algebra, then we modify the norm on $A^\dagger$ to $\|(a,\lambda)\|=\sup\{\|ab+\lambda b\|:\|b\|=1\}$.

Definition 3.  Let $A$ be a Banach algebra and $x\in A$.  The spectrum of $x$ is the set

$\sigma_A(x)=\{\lambda:x-\lambda\cdot 1\mbox{~is not invertible in }A^\dagger\}$.

Clearly for nonunital $A$, $0\in\sigma_A(x)$ for all $x\in A$.

Lemma 4.  Let $A$ be a unital Banach algebra.  If $\|1-x\|<1$, then $x^{-1}$ exists and is defined by

$\displaystyle x^{-1}=1+\sum_{n=1}^\infty(1-x)^n$.

Proof.  Denote $x$ by $1-(1-x)$.  Then

$\displaystyle\begin{array}{lcl}xx^{-1}&=&\left(1-(1-x)\right)\left(1+\sum_{n=1}^\infty(1-x)^n\right)\\&=&1+(1-x)+(1-x)^2+\cdots-(1-x)-(1-x)^2-\cdots.\end{array}$

Since $\|1-x\|<1$, the series is absolutely convergent and hence convergent.  So the two sums cancel–leaving $1$.

Corollary 5.  If $|\lambda|>\|x\|$, then $\lambda\cdot 1-x$ is a unit.

Proposition 6.  Let $A$ be a unital Banach algebra and $x,y\in A$.

1. $\sigma_A(x)$ is a nonempty compact subset of $\mathbb{C}$.
2. $\rho(x)=\max_{\lambda\in\sigma_A(x)}|\lambda|=\lim_{n\to\infty}\|x^n\|^{1/n}=\inf_{n\in\mathbb{N}} \|x^n\|^{1/n}$.

Proof.  (1) By Heine-Borel, it suffices to show $\sigma_A(x)$ is closed and bounded for all $x\in A$.  Let $U$ denote the set of units of $A$ and a scalar $\lambda$ be called a regular point of $x$ if $\lambda\notin\sigma_A(x)$.  Let $\lambda_0$ be a regular point.  Then

$x_{\lambda_0}=x-\lambda_0\cdot 1\in U$.

Since $U$ is open, there is a neighborhood $B_\varepsilon(x_{\lambda_0})\subset U$.  Moreover, since the function $f:\mathbb{C}\to A$ defined by $f(\lambda)=x-\lambda\cdot 1$ is continuous, $f^{-1}(\lambda_0)$ is open in $\mathbb{C}$.  Thus the set of regular points is open–implying that the spectrum is closed.

Now if $|\lambda|>\|x\|$, then $\lambda$ is a regular point by the previous claim.  Hence $diam(\sigma_A(x))\leq 2\|x\|$.

Nonemptiness and (2) are omitted.

Corollary 7 (Gelfand-Mazur). If $A$ is a Banach division algebra, then $A=\mathbb{C}$.

Proposition 8.  Let $A$ be a C*-algebra and $x$ be self-adjoint ($x=x^*$).  Then $\rho(x)=\|x\|$.

Proof.  From the C*-axiom we have $\|x^*x\|=\|x^2\|=\|x\|^2$.  Iterating this yields $\|x^{2^n}\|=\|x\|^{2^n}$.  Hence by (2) of Proposition 6 and the C*-axiom,

$\displaystyle \rho(x)=\lim_{n\to\infty}\|x^{2^n}\|^{2^{-n}}=\|x\|$.

Since $x^*x$ is self-adjoint for all $x\in A$, this says that the norm of a C*-algebra is uniquely determined:  $\|x\|=\sqrt{\rho(x^*x)}$.

Theorem 9 (Gelfand-Naimark).  Let $A$ be a commutative unital C*-algebra and $\hat{A}$ be a closed subset of the unit ball of the dual space $A^*$.  Then the map $\varphi:A\to C_0(\hat{A})$ defined by $\varphi(x)(f)=f(x)$, called the Gelfand transform, is an isometric *-isomorphism.

[1] Bachman, George and Lawrence Narici.  Functional Analysis.  Dover Publications. 2000.

[2] Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.