1
$\begingroup$

Motivated by classical Gelfand Naimark duality, the correspondence between the category of commutative $C^{*}$ algebras and the category of locally compact Hausdorff spaces, we define a "continum algebra", as follows:

A "continum algebra" is a (separable) unital algebra without non trivial idempotent. The reason for this definition is that for a classical space $X$, the algebra $C(X)$ satisfies the above conditions if and only if $X$ is a continum.

Motivated by this post, we have the following definition:

A pair $(A,B)$ of non isomorphic continum $C^{*}$ algebras is a "Consecutive" pair if there is a surjective morphism $\phi:A \to B$ such that for every continum algebra $C$ and surjective morphisms $A\to C \to B$ we have either $C \simeq A, \text{or}\;\; C \simeq B$.

A consecutive resolution of $A$ to a simple algebra $A_{0}$ is a finite sires of continum algebras $A=A_{n}, A_{n-1}, A_{n-2},\ldots,A_{0}$ such that every pair $(A_{i}, A_{i-1})$ is a consecutive pair.

Question: To what extent, all continum algebras which possess such resolutions, are classified? Is there an already known name for such algebras? Does every commutative algebra $A=C(M)$ admit such resolution, where $M$ is a compact manifold?

The later can be restated as follows:

Assume that $M$ is a compact manifold: Are there compact connected subsets $M=X_{n}\supset X_{n-1}\supset \ldots\supset X_{0}=\{pt\} $ such that each pair $(X_{i},X_{i-1})$ is a consecutive pair?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.