# Canonical Representation of Finitely Presented Coxeter Groups

Definition 1.  A Coxeter group is a group with presentation

$\langle r_1,...,r_n:(r_ir_j)^{m_{ij}}=1\rangle$

where $m_{ii}=1$ and $m_{ij}\geq 2$ for $i\neq j.$  Also $m_{ij}=\infty$ says that $(r_ir_j)^n\neq 1$ for any $n.$

To be clear, the above is technically finitely presented.  Although we could have Coxeter groups with an infinite number of generators.

Proposition 2.  Let $G$ be a Coxeter group with generators $r_1,...,r_n.$  Then we have

1. $r_i=r_i^{-1}$;
2. If $m_{ij}=2$ for all distinct $i,j,$ then $G$ is abelian;
3. $m_{ij}=m_{ji}$.

Proof.  (i)  Trivial since $m_{ii}=1.$  (ii)  We have $(r_ir_j)^2=1.$  We also of course have that $r_i^2=r_j^2=1.$  Thus

$r_ir_j=r_i(r_ir_j)(r_ir_j)r_j=r_i^2r_jr_ir_j^2=r_jr_i.$

(iii)  $(r_ir_j)^{m_{ij}}=1$ so

$(r_jr_i)^{m_{ij}}=(r_jr_i)^{m_{ij}}r_j^2=r_j(r_ir_j)^{m_{ij}}r_j=1.$

Thus $m_{ji}\leq m_{ij}.$  The other inequality is dually shown.

Let $M$ be the matrix defined by $M_{ij}=m_{ij}.$  This is called the Coxeter matrix of $G.$

Let $G$ be a Coxeter group with $n$ generators and define an inner product on $\mathbb{R}^n$

$\displaystyle\langle e_i,e_j\rangle_G=-\cos\left(\frac{\pi}{m_{ij}}\right).$

Let $a\in\mathbb{R}^n$  such that $\langle a,a\rangle_G\neq 0.$  Then define

$s_a(x)=x-2\langle a,x\rangle_Ga.$

We then define the representation $\rho:G\to L(\mathbb{R}^n)$ of the Coxeter group on $\mathbb{R}^n$ by

$\rho(r_i)=s_{e_i}.$

We also assume $1$ goes to $1.$  Let us now show that the relations carry over:  $(s_{e_i}s_{e_j})^{m_{ij}}=1.$  First suppose $i=j,$ then

$\begin{array}{lcl}(s_{e_i}s_{e_i})^{m_{ii}}(x)&=&s_{e_i}s_{e_i}(x)\\&=&s_{e_i}(x-2\langle e_i,x\rangle_Ge_i)\\&=&x-2\langle e_i,x\rangle_Ge_i-2\langle e_i,x-2\langle e_i,x\rangle_Ge_i\rangle_G e_i\\&=&x-4\langle e_i,x\rangle_Ge_i+4\langle e_i,x\rangle_G\langle e_i,e_i\rangle_Ge_i\\&=&x.\end{array}$

For $i\neq j,$ note that $s_{e_i}s_{e_j}$ is acting on the subspace $\mathbb{R}e_i\oplus\mathbb{R}e_j.$  One can show that $s_{e_i}s_{e_j}$ has order $m_{ij}$ on this subspace.

[1]  Abramenko, Peter and Kenneth Brown.  Buildings.  Graduate Texts in Mathematics.  Vol. 248.  Springer Science and Business Media.  2008.

# Direct Integral Decomposition

Let $\{H_x\}_{x\in X}$ be a collection of Hilbert spaces such that $(X,\Sigma,\mu)$ is a measure space.  Now define

$\displaystyle H=\left\{s\in\bigoplus_{x\in X} H_x:\int_X |s(x)|^2\,d\mu<\infty\right\}$

where $s(x)=s_x$ (the $x$th component of $s$).  Then

$\displaystyle\langle s,t\rangle=\int_X\langle s_x,t_x\rangle_x\,d\mu$

defines a pre inner product on $H$.  Now let $H_0=span\{s-t:s=t\,a.e.\}$.  Then $H/H_0$ is an inner product space, and its completion is a Hilbert space called the direct integral of $\{H_x\}_{x\in X}$.  We denote the direct integral by

$\displaystyle H_X^\oplus=\int_X^\oplus H_x\,d\mu.$

Now suppose $\{T_x\}_{x\in X}$ is a collection of linear operators where $T_x\in L(H_x)$ such that $\{\|T_x\|\}$ is uniformly bounded.  Then there is an operator $T\in L(H_X^\oplus)$ where $T(s)_x=T_x(s_x)$ and where we can define

$\displaystyle \|T\|={\mbox{ess}\sup}_{x\in X}\{\|T_x\|\}.$

This gives a representation $\rho:L^\infty(X,\mu)\to L(H_X^\oplus)$ defined by

$\left(\rho(f)(s)\right)_x=f(x)s_x,$

which is essentially just a component-wise left action of $f$ on $s$$\rho(L^\infty(X,\mu))$ is called the algebra of diagonalizable operators of $H_X^\oplus$, which we will denote $D(H_X^\oplus)$.

Theorem 1.  Let $\rho:A\to L(H)$ be a representation of von Neumann algebra $A$ on a separable Hilbert space $H$ such that $B$ is a von Neumann subalgebra of $A'$.  Then there exists a measure space $(X,\Sigma,\mu)$, a collection of Hilbert spaces $\{H_x\}_{x\in X}$, and a  unitary map $U:H\to H_X^\oplus$ such that $U\left(\rho(B)(s)\right)=D(H_X^\oplus)(s')$ for all $s\in H$ and corresponding $s'\in H_X^\oplus$ and

$\displaystyle UTU^*=\int_X^\oplus T_x\,d\mu\in L(H_X^\oplus)$

for all $T\in\rho(B')$.

Thus if we let $B=A'$ from above, then $L(H_x)$ is a factor and we also write

$\displaystyle\rho(A')=\int_X^\oplus\rho(A')_x\,d\mu=\int_X^\oplus\rho(A_x)'\,d\mu.$

This is called the central decomposition of $A$.  This also gives a representation of $A$ on $H_X^\oplus$ defined by $a\mapsto U\rho(a)U^*$.

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

# 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.