Direct Sum of Algebra
Point of post: In this post we shall discuss a natural way to build new algebras out of a collection of algebras. This is the direct sum of algebras which, unsurprisingly, mimics the construction of the direct sum of groups or direct sum of vector spaces.
Direct Sum of Algebras
Namely, if is a collection of -algebras and we define the support of to be the set
We then define the direct sum of , denoted , to be the set of all elements of with finite support with coordinate wise operations. More explicitly if we denote in by respectively (where we evidently mean and ) then we define addition in by , multiplication by , and scalar multiplication for . In other words, we just give the ring structure of the direct sum of rings and just multiply scalars coordinate wise. It’s evident that this does, in fact, define an -algebra on with unity equal to .
It’s clear that if is in fact finite that this reduces to defining coordinate wise operations on the set .
For now it will suffice to know just the definition of the direct sum of algebras, but just because they are easy we prove a fundamental results. Namely,
Theorem: Let be a field and a collection of -algebras then
Proof: Let be an element of then evidently for any in we have that so that is a member of . Conversely, if is an element of then we evidently have for each and that where for and is an element of and thus so that for every and in particular . Since was arbitrary it follows that is an element of and thus since was arbitrary we have that is an element of , but since all but since is finite we must have that it, in fact, an element of . The conclusion follows.
Roman, Steven. Advanced Linear Algebra. New York: Springer, 2005. Print.