**Fact 4**. If and are -algebras and then

*Proof*. Since we have Now, let be a finitely generated subalgebra of with a frame Since there exist finite dimensional subspaces of respectively, such that and Let be the algebras generated by respectively. Now, for all and so

Therefore and hence, taking limsup, will give us

Since the above holds for any finitely generated subalgebra of we have

**Fact 5**. If is a -algebra, then

*Proof*. By Fact 2, Now the result follows from and Fact 4.

**Fact 6**. Let be a -algebra and let be an ideal of If for some then

*Proof*. Let be any finitely generated subalgebra of and let be a frame of which contains Let Clearly is a frame of If is an integer, then, as -vector spaces, for some finite dimensional -vector space Note that

Also, since and we have for all Therefore, since the sum is direct for all Clearly for all because both and are in Thus

Hence Since every finitely generated subalgebra of is in the form for some finitely generated subalgebra of the inequality holds for any finitely generated subalgebra of Thus

**Corollary**. Let be a -algebra which is also a domain. Let be a simple -subalgebra of If then is simple too.

*Proof*. Let be a nonzero ideal of If then because is simple, and so Suppose now that Then the natural homomorphism would be injective and so

by Fact 6, which is a contradiction.