Algebraic Ideals and the Span of Squares

Oct 13, 2025 by miodeya.com 15 views

Does an Algebraic Ideal $I$ of a $C^*$ -Algebra Satisfy $I=\mathrm{span}(I^2)$ ?

In the realm of operator algebras, specifically $C^*$ -algebras, ideals play a crucial role in understanding the structure and properties of these algebras. A $C^*$ -algebra is a Banach algebra with an involution operation that satisfies the $C^*$ -identity. An ideal $I$ of a $C^*$ -algebra $A$ is a closed subspace of $A$ such that for any $a \in A$ and $x \in I$ , both $ax$ and $xa$ are in $I$ . In this article, we will explore the relationship between an algebraic ideal $I$ of a $C^*$ -algebra and the span of its square, denoted by $\mathrm{span}(I^2)$ .

Algebraic Ideals and the Span of Squares

An algebraic ideal $I$ of a $C^*$ -algebra $A$ is a subspace of $A$ that is closed under multiplication by elements of $A$ . In other words, if $x \in I$ and $a \in A$ , then both $ax$ and $xa$ are in $I$ . The span of the square of an ideal $I$ , denoted by $\mathrm{span}(I^2)$ , is the set of all linear combinations of elements of the form $xy$ and $yx$ , where $x, y \in I$ . The question of whether an algebraic ideal $I$ satisfies $I = \mathrm{span}(I^2)$ is a natural one, as it relates to the structure and properties of the ideal.

Counterexamples and Open Questions

The question of whether an algebraic ideal $I$ satisfies $I = \mathrm{span}(I^2)$ is still an open one. However, there are some counterexamples and partial results that shed light on this question. For instance, it is known that if $I$ is a closed two-sided ideal of a $C^*$ -algebra $A$ , then $I = \mathrm{span}(I^2)$ . However, if $I$ is a non-closed two-sided ideal, the situation is more complicated.

Non-Closed Two-Sided Ideals

A non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ is an ideal that is not closed in the norm topology of $A$ . In this case, the question of whether $I = \mathrm{span}(I^2)$ is still open. In fact, it is not even clear whether there exists a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I \neq \mathrm{span}(I^2)$ .

Motivation and Related Questions

The question of whether an algebraic ideal $I$ satisfies $I = \mathrm{span}(I^2)$ is motivated by the study of operator algebras and their properties. In particular, it is related to the study of ideals and their structure. The question also has implications for the study of $C^*$ -algebras and their classification.

In conclusion, the question of whether an algebraic ideal $I$ of a $C^*$ -algebra satisfies $I = \mathrm{span}(I^2)$ is still an open one. While there are some counterexamples and partial results, the situation is more complicated for non-closed two-sided ideals. Further research is needed to fully understand the relationship between algebraic ideals and the span of their squares.

Open Questions and Future Directions

Existence of Non-Closed Two-Sided Ideals: Does there exist a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I \neq \mathrm{span}(I^2)$ ?
Classification of Ideals: Can we classify ideals of $C^*$ -algebras based on their relationship with the span of their squares?
Properties of Ideals: What are the properties of ideals that are preserved under the operation of taking the span of their squares?

[1] David, "Answer to a related question"
[2] "Operator Algebras and Their Applications"
[3] "C* Algebras and Their Representations"

For the sake of completeness, we provide a brief overview of the necessary background and definitions.

**Definition of a $C^*$ -Algebra**

A $C^*$ -algebra is a Banach algebra with an involution operation that satisfies the $C^*$ -identity.

Definition of an Ideal

An ideal $I$ of a $C^*$ -algebra $A$ is a closed subspace of $A$ such that for any $a \in A$ and $x \in I$ , both $ax$ and $xa$ are in $I$ .

Definition of the Span of Squares

The span of the square of an ideal $I$ , denoted by $\mathrm{span}(I^2)$ , is the set of all linear combinations of elements of the form $xy$ and $yx$ , where $x, y \in I$ .
Q&A: Does an Algebraic Ideal $I$ of a $C^*$ -Algebra Satisfy $I=\mathrm{span}(I^2)$ ?

**Q: What is an algebraic ideal $I$ of a $C^*$ -algebra?**

A: An algebraic ideal $I$ of a $C^*$ -algebra $A$ is a subspace of $A$ that is closed under multiplication by elements of $A$ . In other words, if $x \in I$ and $a \in A$ , then both $ax$ and $xa$ are in $I$ .

Q: What is the span of the square of an ideal $I$ ?

A: The span of the square of an ideal $I$ , denoted by $\mathrm{span}(I^2)$ , is the set of all linear combinations of elements of the form $xy$ and $yx$ , where $x, y \in I$ .

Q: Does every algebraic ideal $I$ satisfy $I = \mathrm{span}(I^2)$ ?

A: No, not every algebraic ideal $I$ satisfies $I = \mathrm{span}(I^2)$ . In fact, it is not even clear whether there exists a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I \neq \mathrm{span}(I^2)$ .

Q: What is the relationship between closed and non-closed two-sided ideals?

A: A closed two-sided ideal $I$ of a $C^*$ -algebra $A$ satisfies $I = \mathrm{span}(I^2)$ . However, the situation is more complicated for non-closed two-sided ideals.

Q: What are some open questions related to this topic?

A: Some open questions related to this topic include:

Existence of Non-Closed Two-Sided Ideals: Does there exist a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I \neq \mathrm{span}(I^2)$ ?
Classification of Ideals: Can we classify ideals of $C^*$ -algebras based on their relationship with the span of their squares?
Properties of Ideals: What are the properties of ideals that are preserved under the operation of taking the span of their squares?

Q: What are some related topics in operator algebras?

A: Some related topics in operator algebras include:

Operator Algebras and Their Applications: This is a broad field that includes the study of operator algebras and their applications to various areas of mathematics and physics.
C* Algebras and Their Representations: This is a field that studies the representation theory of $C^*$ -algebras and their properties.
Ideal Theory: This is a field that studies the properties and classification of ideals in operator algebras.

Q: What are some resources for further reading?

A: Some resources for further reading on this topic include:

[1] David, "Answer to a related question"
[2] "Operator Algebras and Their Applications"
[3] "C* Algebras and Their Representations"

Q: What are some open problems in this area?

A: Some open problems in this area include:

Classification of Ideals: Can we classify ideals of $C^*$ -algebras based on their relationship with the span of their squares?
Properties of Ideals: What are the properties of ideals that are preserved under the operation of taking the span of their squares?
Existence of Non-Closed Two-Sided Ideals: Does there exist a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I \neq \mathrm{span}(I^2)$ ?