# K-Theory of C*-algebras

Recall that two projections in a C*-algebra are equivalent, denoted if there is a partial isometry such that and Let be the set of equivalence classes of projections and We can define addition on by

where , , and (see [1] for why this is possible). This gives an abelian monoid structure on with identity . If is unital, then we define where is the Grothendieck group of

To proceed with the nonunital case, first consider that is a covariant functor sending since if is a *-homomorphism, then we can define by

Recall the unitization of where with funky structure. We have a *-homomorphism such that Since is a covariant functor, and by composition induces a covariant functor we have a map

In the nonunital case we then define Now define

If is unital, then the condition is always satisfied, so we have Define a norm on by and let be the connected component (in the sense of the norm topology) containing Then we define

**Definition 1**. Let be a C*-algebra. We define its **suspension**, denoted as

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

**Theorem 2**. There is an isomorphism defined by where defined by and is a path in between and

in the image refers to the constant map We can then define by induction the higher **K-groups**: All of these are clearly covariant functors since the definition can be reduced to of the -th suspension of

**Corollary 3**. Let

be a short exact sequence. Then the induced sequence

is exact in the middle.

Let us define a map by

for some where and be a lift of Then since commutes with since

**Proposition 4**. makes the slightly longer sequence exact at and

**Corollary 5**. By induction we obtain maps and a long exact sequence

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

# Covariant Systems

Recall a topological group is a group and a topological space such that the maps and are continuous. Let be the -algebra generated by the compact subsets of . A measure on is **left-invariant** if for all and . A **left Haar measure** on is a left-invariant Radon measure on .

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

See [2] for its construction. Let be a left Haar measure on . We can define **right translate** Haar measures by . Clearly these are left invariant as they simply changes the input of the original left invariant Haar measure.

**Proposition 2.** If is a left Haar measure, then there is a unique function called the **modular function of **, such that .

Let be a strongly continuous unitary representation of on a Hilbert space . That is, the group homomorphism is continuous with respect to the norm topology of , and is the subset of consisting of unitary linear operators on . Let be a left Haar measure on and , then the operator

in is bounded and in fact defines a nondegenerate representation as a Banach *-homomorphism with . The product on is convolution:

and the involution is defined by

Recall in the GNS construction we started with a positive functional on a C*-algebra and induced a representation . Here we start with one representation on and induce another, , on . We call the **integrated form of **. We can impose another norm (other than the default sup norm ) on defined by

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

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

for all and .

**Definition 4.** We define the **covariance algebra** of the covariant system as the completion of under the norm

where we define

It is a Banach *-algebra.

**Definition 5.** If is a covariant representation of , then there is a nondegenerate representation defined by

called the **integrated form** of . together with a new norm

is a C*-algebra called the **crossed product** of , which is denoted .

[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 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.

# C*-Algebras

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

.

A ***-algebra** is a complex algebra with a unary operation * that satisfies , , , and . A **Banach *-algebra** is a Banach algebra and a *-algebra that satisfies . A **C*-algebra** is a Banach *-algebra that satisfies . A map between two *-algebras is a ***-homomorphism** if .

**Proposition 2.** Any nonunital Banach algebra has a unitization .

*Proof.* Let . Let , and send . Define , , and . Clearly is a Banach space. We also have

The unit is : .

In fact, if is a nonunital Banach *-algebra, then there is a unitization of which is also a Banach *-algebra. We simply use and define . And if is a C*-algebra, then we modify the norm on to .

**Definition 3.** Let be a Banach algebra and . The **spectrum of** is the set

.

Clearly for nonunital , for all .

**Lemma 4**. Let be a unital Banach algebra. If , then exists and is defined by

.

*Proof.* Denote by . Then

Since , the series is absolutely convergent and hence convergent. So the two sums cancel–leaving .

**Corollary 5.** If , then is a unit.

**Proposition 6.** Let be a unital Banach algebra and .

- is a nonempty compact subset of .
- .

*Proof. *(1) By Heine-Borel, it suffices to show is closed and bounded for all . Let denote the set of units of and a scalar be called a **regular point of ** if . Let be a regular point. Then

.

Since is open, there is a neighborhood . Moreover, since the function defined by is continuous, is open in . Thus the set of regular points is open–implying that the spectrum is closed.

Now if , then is a regular point by the previous claim. Hence .

Nonemptiness and (2) are omitted.

**Corollary 7 (Gelfand-Mazur)**. If is a Banach division algebra, then .

**Proposition 8.** Let be a C*-algebra and be self-adjoint (). Then .

*Proof.* From the C*-axiom we have . Iterating this yields . Hence by (2) of Proposition 6 and the C*-axiom,

.

Since is self-adjoint for all , this says that the norm of a C*-algebra is uniquely determined: .

**Theorem 9 (Gelfand-Naimark)**. Let be a commutative unital C*-algebra and be a closed subset of the unit ball of the dual space . Then the map defined by , 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.