If a abelian subalgebra of a matrix algebra in $\M_{n}(\mathbb{C}) has linear dimension n. Then Why is this subalgebra is maximal abelian in $\M_{n}(\mathbb{C}) ?
Functional Analysis, Von Neumann Algebra, Banach Algebra
Sketch of an idea:
1) Try to prove that under a change of basis, such an abelian subalgebra transforms into the subalgebra of diagonal matrices.
2) Try to show that the centralizer of the subalgebra of diagonal matrices is itself.
In any algebra every abelian subalgebra can be embedded into a maximal abelian subalgebra. It is known that in Mn(C) maximal abelian subalgebras are isomorphoc to Cn, i.e. have the dimension n. Therefore if an abelian subalgebra has dimension n then it must coincide with the maximal abelian subalgebra in which it is embedded.
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