Tag Archives: representations

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.

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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 xth 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 Ax 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.