Why do Maximal Abelian Self-adjoint Algebras necessarily have unit?

Question

A Maximal Abelian Self-adjoint Algebra (MASA) is defined as a maximal element (with respect to inclusion) in the set of all Abelian self-adjoint operator algebras acting on a Hilbert sapace $H$.

In a remark (remark 4.1.1 to be exact) given in Arveson's book 'A short course on spectral theory' it is mentioned that if a MASA does not contain an identity, it can be enlarged by adjoning an identity to it, without giving a source or argument. It seems like I am kinda lost, because I cannot see any argument why this is true and could not find and source for this result. I will appreciate if someone can explain/give me a source on this result.

Also since I have no clue how this process actually works, what happens after one adjoins an identity to a MASA? After this process will it be again a MASA or you have to take a maximal that contains it?

Answer 1

Suppose $A\subseteq B(H)$ is any abelian self-adjoint algebra. Let $B$ be the subalgebra of $B(H)$ generated by $A\cup\{I\}$. Then $B$ is again abelian (since $I$ commutes with everything) and self-adjoint (since $I$ is self-adjoint). If $I\not\in A$, then $B$ strictly contains $A$, and so this proves $A$ could not have been maximal.