32.4 Ideals

Definition 32.4.1 (Ideal).label Let $A$ be a Banach algebra and $I \subset A$, then $I$ is a left ideal if:

1. (1)

   $I$ is a subspace of $A$.

2. (2)

   For each $x \in A$, $aI \subset I$.

and $I$ is a two-sided ideal if in addition to the above,

1. (3)

   For each $x \in A$, $Ia \subset I$.

In this document, the ”sidedness” of ideals are omitted. Within each block, the statement should hold as long as the interpretation is consistent.

Definition 32.4.2 (Proper Ideal).label Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is proper if $I \subsetneq A$.

Definition 32.4.3 (Maximal Ideal).label Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is maximal if:

1. (1)

   $I$ is proper.

2. (2)

   For any proper ideal $J \subsetneq A$ with $J \supset I$, $I = J$.

The set $\cm(A)$ of maximal two-sided ideals of $A$ is the maximal ideal space of $A$.

Proposition 32.4.4.label Let $A$ be a unital Banach algebra and $I \subset A$ be a proper ideal, then:

1. (1)

   $I$ contains no invertible elements.

2. (2)

   $\ol I$ is also a proper ideal.

3. (3)

   $I$ is contained in a maximal ideal.

4. (4)

   If $I$ is maximal, then $I$ is closed.