# Canonical Representation of Finitely Presented Coxeter Groups

**Definition 1**. A **Coxeter group** is a group with presentation

where and for Also says that for any

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

**Proposition 2**. Let be a Coxeter group with generators Then we have

- ;
- If for all distinct then is abelian;
- .

*Proof.* (i) Trivial since (ii) We have We also of course have that Thus

(iii) so

Thus The other inequality is dually shown.

Let be the matrix defined by This is called the **Coxeter matrix** of

Let be a Coxeter group with generators and define an inner product on

Let such that Then define

We then define the representation of the Coxeter group on by

We also assume goes to Let us now show that the relations carry over: First suppose then

For note that is acting on the subspace One can show that has order 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 be a collection of Hilbert spaces such that is a measure space. Now define

where (the th component of ). Then

defines a pre inner product on . Now let . Then is an inner product space, and its completion is a Hilbert space called the **direct integral of** . We denote the direct integral by

Now suppose is a collection of linear operators where such that is uniformly bounded. Then there is an operator where and where we can define

This gives a representation defined by

which is essentially just a component-wise left action of on . is called the **algebra of diagonalizable operators** of , which we will denote .

**Theorem 1**. Let be a representation of von Neumann algebra on a separable Hilbert space such that is a von Neumann subalgebra of . Then there exists a measure space , a collection of Hilbert spaces , and a unitary map such that for all and corresponding and

for all .

Thus if we let from above, then is a factor and we also write

This is called the **central decomposition** of . This also gives a representation of on defined by .

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

# Tensor Products of C*-algebras

Let and be C*-algebras. We can define their *-algebra tensor product as the standard tensor product of algebras with product and involution . There are a variety of norms one can impose on this tensor product to make a Banach *-algebra. For example we may define

.

This seminorm becomes a norm on modulo the appropriate subspace, and its completion is denoted and is called the **projective tensor product** of and . We also have

so is a Banach *-algebra. But it fails to satisfy the C*-axiom ():

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

**Definition 1**. Let and be representations on and . We define the **product representation** on as

.

Since we always have the trivial representations, the set of representations of on and on are never empty. Let us define the **minimal C*-norm** on by

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 with this norm is a C*-algebra called the **minimal (or spatial) tensor product of ** and with respect to and , and is denoted .

**Definition 2.** Let be a representation and be the largest subspace of such that for all and . Then is called the **essential subspace** of , and we will denote it . If , then is said to be **nondegenerate**. is **degenerate** if it is not nondegenerate.

In other words, is nondegenerate if .

**Proposition 3.** If is a nondegenerate representation, then there are unique nondegenerate representations and such that .

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

**Definition 4.** Let be a Hilbert space and be C*-algebras. We define the **maximal C*-norm** on as

where . This is also a C*-norm, and the completion of under this norm is a C*-algebra called the **maximal tensor product** of and and is denoted .

We also have that where is any C*-norm. It follows that .

**Definition 5.** A functional on is **positive** if for all . A **state** on where and are unital is a positive linear functional on such that . We denote the set of states by .

As in the previous post, there is a GNS construction that gives a representation for a positive linear functional , although one must show the left action on is by bounded operators.

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

Hence in such a case, we would have , and thus denote the product C*-algebra by . 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; the group C*-algebra of (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 be a C*-algebra and . is **positive** if it is self-adjoint and . We denote the set of positive elements of by and write for all . We also write if .

It turns out that if is a C*-subalgebra of , then . Hence .

**Definition 2**. A functional is **positive** if (hence ). A **state** is a positive linear functional such that . The set of all states is called the **state space** and is denoted .

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

.

If is a Hilbert space, we can endow a *-algebra structure on , the space of linear operators on where multiplication is composition, and if , then is defined as the adjoint of .

**Definition 3.** Let be a C*-algebra and be a Hilbert space. A **representation** is a *-homomorphism . A **subrepresentation** is a representation where is a closed subspace of which is invariant under action from . A representation is **irreducible** if it has no nontrivial subrepresentations. A representation is **faithful** if .

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

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

.

is a closed left ideal in , and is an inner product space. Let be its Hilbert completion. Now define by action of : . is clearly a representation, and is called the **GNS representation** of associated with .

If is unital, let denote the image of in the completion/quotient composition . Then induces a positive linear functional on defined by

for .

**Theorem 4**. Let be positive linear functionals on a C*-algebra such that (meaning for all ). Then there is a unique operator such that and for all .

This is a generalization of the Radon-Nikodym theorem (which is special case for where is a finite measure space and ).

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

It follows that .

**Proposition 6.** Let be a state on a C*-algebra . Then is irreducible if and only if is pure.

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