Which are the Maximal Ideals ?
Ideals of continuous functions which satisfy the {\it off diagonality} condition (1.1) below proved to be important connected with the solution of large classes of nonlinear PDEs, and more recently, in General Relativity and Quantum Gravity.
Maximal ideals within those which satisfy that off diagonality condition are important since they lead to differential algebras of generalized functions which can handle the largest classes of singularities. The problem of finding such maximal ideals satisfying the off diagonality condition is formulated as follows.
Find out the structure of {\it maximal ideals} ${\cal I}$ in the algebra
$( {\cal C} ( \mathbb{R}^n ) )^\Lambda$
among all those ideals which satisfy the {\it off diagonality} condition
(1.1) $~~~ {\cal I} ~\bigcap~ {\cal U}_{\,\Lambda} ( \mathbb{R}^n ) ~=~ {~ 0 ~} $
Here ${\cal C} ( \mathbb{R}^n )$ is the set of all real valued continuous functions on $\mathbb{R}^n$, while $\Lambda$ is an arbitrary infinite set.
Consequently, $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$ is the Cartesian product of $\Lambda$ copies of ${\cal C} ( \mathbb{R}^n )$, thus it can be identified with
${\cal C} ( \Lambda \times \mathbb{R}^n )$
that is, with the set of real valued continuous functions on
$\Lambda \times \mathbb{R}^n$
where $\Lambda$ is taken with the discrete topology.
Clearly, $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$ is a commutative unital algebra over $\mathbb{R}$, and we have the algebra embedding
(1.2) $~~~ {\cal C} ( \mathbb{R}^n ) \ni \psi ~\longmapsto~ u ( \psi ) \in ( {\cal C} ( \mathbb{R}^n ) )^\Lambda $
where $u ( \psi ) = (~ \psi_\lambda ~|~ \lambda \in \Lambda ~)$, with $\psi_\lambda = \psi$, for $\lambda \in \Lambda$.
In this way, the unit element in $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$ is $u ( 1 )$, where $1 \in {\cal C} ( \mathbb{R}^n )$ denotes the constant function with value 1 defined on $\mathbb{R}^n$.
Finally, ${\cal U}_{\,\Lambda} ( \mathbb{R}^n )$ denotes the image of
${\cal C} ( \mathbb{R}^n )$
in
$( {\cal C} ( \mathbb{R}^n ) )^\Lambda$
through the algebra embedding (1.2), thus
(1.3) $~~~ {\cal U}_{\,\Lambda} ( \mathbb{R}^n ) ~=~ {~ u ( \psi ) ~|~ \psi \in {\cal C} ( \mathbb{R}^n ) ~} $
is a subalgebra in $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$, and through (1.2), it is isomorphic with
${\cal C} ( \mathbb{R}^n )$.
With the above, the meaning of (1.1) becomes clear, recalling that ${~ 0 ~}$ in its right hand term denotes the trivial zero ideal in $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$.
In this way ${\cal U}_{\,\Lambda} ( \mathbb{R}^n )$ is in fact the {\it diagonal} in the Cartesian product $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$. Thus (1.1) is indeed an {\it off diagonality} condition on the respective ideals ${\cal I}$ in $( {\cal C} ( \mathbb{R}^n ) )^\Lambda$.
