*Taken from C*-Algebras and Operator Theory by Gerald J. Murphy.*

Although the principal aim of this section is to construct direct limits of C*-algebras, we begin with direct limits of groups.

If is a sequence of groups, and if for each we have a homomorphism , then we call a *direct sequence of groups. *Given such a sequence and positive integers , we set and we define inductively on by setting

.

If , we have .

If is a group and we have homomorphisms such that the diagram

commutes for each , that is , then for all .

The product is a group with the pointwise-defined operation, and if we let be the set of all elements in such that there exists for which for all , then is a subgroup of (this follows readily from the fact that the are homomorphisms). Let be the identity of . The set of all such that there exists for which for all is a normal subgroup of (again from the homomorphism property), and we denote the quotient group by (the cosets are sets of elements that have equal ‘tails’.). We call the *direct limit *of the sequence , and where no ambiguity can result we sometimes write for .

If , define to be the sequence where for , and for all (tailed by the ). Then , and the map

, ,

is a homomorphism, called the *natural *homomorphism from to . It is straightforward to check that the diagram

commutes for each , and that is the union of the increasing sequence .

## Theorem 6.1.1

*Let be the direct limit of the direct sequence of groups and let be the natural map for each .*

*If and and , then there exists such that (if the tails agree they’ll have to agree in ‘finite time’).**If is a group, and for each there is a homomorphism such that the diagram*

*Proof *: Condition 1. follows directly from the definitions.

Assume we have and as in condition 2. If and and , then by condition 1. there exists such that . Hence,

Thus we can well-define a map by setting . It is easily checked that is a homomorphism, and by definition for all . Uniqueness of is clear

### Remark

From condition 1. of the preceding theorem, if , then there exists such that , a result we shall be using frequently.

Let be a *-algebra. A *C*-seminorm *on is a seminorm on such that for all we have

, and .

If in addition is in fact a norm, we call a *C*-norm.*

If is a *-homomorphism from a *-algebra into a C*-algebra, then the function

, ,

is a C*-seminorm on , and if is injective, is a C*-norm.

If is any C*-seminorm on a *-algebra , the set is a self-adjoint ideal of ()m and we get a C*-norm on the quotient *-algebra by setting . If denotes the Banach space completion of with this norm, it is easily checked that the multiplication and involution operations extend uniquely to operations of the same type on so as to make a C*-algebra. We call the *enveloping *C*-algebra of the pair , and the map

, ,

the *canonical *map from to . Of course, is a dense *-subalgebra of .

If is a C*-norm, we refer to more simply as the *C*-completion *of . In this case is a dense *-subalgebra of .

Let be a sequence of C*-algebras and suppose that for each we have a *-homomorphism . Then we call a *direct sequence of C*-algebras. *The product is a *-algebra with the pointwise-defined operations, and if denotes the set of all elements of such that there is an integer for which for all , then is a *-subalgebra of . Note that if (since the are norm-decreasing), so the sequence is eventually decreasing (and of course bounded below). It therefore converges, and we set . It is straightforward to verify that

, ,

is a C*-seminorm on . We denote the enveloping C*-algebra of by , and call it the *direct limit *of the sequence . If no ambiguity results, we sometimes write for .

Similar to the group case, if , we define in to be the sequence such that are zero and for all . If is the canonical map, then the map

, ,

is a *-homomorphism, called the *natural map *from to . A routine argument show that for all the diagram

commutes, and that the union of the increasing sequence of C*-subalgebras is a dense *-subalgebra of .

is the element which is ‘evolves’ from in the th place. That this diagram commutes is a very simple consequence of this.

Also,

if .

## Theorem 6.1.2

*Let be the direct limit of the direct sequence of C*-algebras , and suppose that is the natural map for each .*

*If , , and , then there exists such that .**If is a C*-algebra and there is a *-homomorphism for each such that the diagram*

*commutes, then there is a unique *-homomorphism such that for each the diagram*

*Proof *: If this implies that converges pointwise to .

Suppose that and are as in condition 2. Let and , and suppose that . If , then by condition 1. there exists such that . Consequently,

.

Letting , we therefore have . This shows that we can well-define a map from to

to

by setting . If is any integer, then

,

and therefore

.

Thus is norm-decreasing, and it it easily seen to be a *-homomorphism. Since is a dense *-subalgebra of , we can extend to a *-homomorphism , and for all . Uniqueness follows from the density of in

### Remark

Retaining the notation of the preceding theorem, if and and , then by condition 1. there exists such that .

### Remark

Let be a C*-algebra and let be an increasing sequence of C*-subalgebras of whose union is dense in . Let be the inclusion map. A straightforward application of Theorem 6.1.2 condition 2., shows that is *-isomorphic to the direct limit of the direct sequence .

## Theorem 6.1.3

*Let be a non-empty set of simple C*-subalgebras of a C*-algebra . Suppose that is upwards-directed (that is if , then there exists a such that ), and is dense in . Then is simple also.*

*Proof *: To show that is simple it suffices to show that if is a surjective *-homomorphism onto a non-zero C*-algebra , then it is injective ( is linear and non-zero, so if it not injective, the kernel is non-zero so that is not simple.). If , then the restriction of to is either , or it is injective, and therefore isometric. Since is not the zero map on , it follows easily from the upwards-directed property of that is not the zero map on any non-zero . Hence, is isometric on , and therefore, by continuity, is isometric on

## Theorem 6.1.4

*Suppose that is a direct sequence of simple C*-algebras. Then the direct limit is simple also.*

*Proof *: Let be the natural map, where . Then the set is an upwards-directed family of simple C*-subalgebras of whose union is dense in , so by Theorem 6.1.3, is simple

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