# 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.

# 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.

# von Neumann Algebras

Let be a magma and . The **commutant** of is defined as:

for all . We have that and for and that iff is abelian. We can also refer to the commutant of the whole structure as the **center** of (i.e. ).

**Definition 1**. A **von Neumann algebra** is a C*-algebra such that .

A **projection** in a *-algebra is an element such that . A **partial isometry** is an element such that is a projection. Recall an element is positive, denoted , if and , and that if . We will also say that two elements are **orthogonal**, denoted , if .

**Definition 2.** Two projections are **Murray-von Neumann equivalent**, denoted , if there is a partial isometry such that and . We say is **subordinate** to , denoted , if there is a projection such that and .

**Proposition 3**. Let be a *-algebra and be a sequence of pairs of projections such that and for and for all . Then Also, if for all then

*Proof. *If , then and Hence and . Now define and Then we have

where the last equality follows from orthogonality. Hence

Now suppose Then there are such that and Then since by the previous claim, it remains to show that But since for all So we have the result.

**Proposition 4 (Schroder-Bernstein).** Let and be projections in a *-algebra such that and Then .

It turns out that if a unital von Neumann algebra is a **factor** (), then is a total order on the projections.

**Definition 5.** A projection is

**abelian**if is commutative;**finite**if where implies that ;**infinite**if it is not finite;**properly infinite**if and where and

**Lemma 6.** If is an infinite projection in a von Neumann algebra , then there is a projection such that is nonzero and properly infinite.

This allows for a somewhat complicated decomposition (see [1] for details) of

where is a *discrete central projection*, is the largest *finite continuous central projection*, is the largest *properly infinite semifinite continuous projection*, and is a purely infinite projection. The algebra is said to be of **pure type** if for all

[1] Blackadar, Bruce. *Operator Algebras*. Encyclopedia of Mathematical Sciences. 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.