Counterexamples and Open Questions

Oct 13, 2025 by miodeya.com 76 views

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

In the realm of operator algebras, a fundamental concept is the algebraic ideal, which plays a crucial role in understanding the structure of $C^*$ -algebras. A $C^*$ -algebra is a Banach algebra with an involution that satisfies the $C^*$ -identity. In this context, an algebraic ideal $I$ of a $C^*$ -algebra $A$ is a subset of $A$ that is closed under multiplication by elements of $A$ and satisfies certain properties. The question of whether an algebraic ideal $I$ satisfies $I=\mathrm{span}(I^2)$ is a natural one, as it relates to the behavior of ideals under multiplication.

The question of whether an algebraic ideal $I$ satisfies $I=\mathrm{span}(I^2)$ is motivated by the study of ideals in $C^*$ -algebras. Ideals in $C^*$ -algebras are essential in understanding the structure of these algebras, and their properties have far-reaching implications in operator theory and functional analysis. In particular, the question of whether an algebraic ideal $I$ satisfies $I=\mathrm{span}(I^2)$ is related to the study of the behavior of ideals under multiplication, which is a fundamental aspect of the theory of $C^*$ -algebras.

Algebraic Ideals and the Span of $I^2$

An algebraic ideal $I$ of a $C^*$ -algebra $A$ is a subset of $A$ that is closed under multiplication by elements of $A$ . In other words, if $x \in I$ and $a \in A$ , then $ax \in I$ . The span of $I^2$ is the set of all linear combinations of elements of the form $xy$ , where $x, y \in I$ . In this context, the question of whether an algebraic ideal $I$ satisfies $I=\mathrm{span}(I^2)$ is equivalent to asking whether every element of $I$ can be written as a linear combination of elements of the form $xy$ , where $x, y \in I$ .

Counterexamples and Open Questions

The question of whether an algebraic ideal $I$ satisfies $I=\mathrm{span}(I^2)$ is still an open one, and there are several counterexamples that have been constructed to show that the answer is no. One such counterexample is a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I\neq\mathrm{span}(I^2)$ . This counterexample is motivated by the study of ideals in $C^*$ -algebras and has far-reaching implications in operator theory and functional analysis.

An algebraic ideal $I$ of a $C^*$ -algebra $A$ satisfies several properties, including:

Closure under multiplication: If $x \in I$ and $a \in A$ , then $ax \in I$ .
Closure under addition: If $x, y \in I$ , then $x + y \in I$ .
Closure under scalar multiplication: If $x \in I$ and $\lambda \in \mathbb{C}$ , then $\lambda x \in I$ .

These properties are essential in understanding the behavior of algebraic ideals in $C^*$ -algebras.

There are several examples of algebraic ideals in $C^*$ -algebras, including:

The set of compact operators: Let $A$ be a $C^*$ -algebra of compact operators on a Hilbert space. Then the set of compact operators is an algebraic ideal of $A$ .
The set of finite-rank operators: Let $A$ be a $C^*$ -algebra of finite-rank operators on a Hilbert space. Then the set of finite-rank operators is an algebraic ideal of $A$ .

These examples illustrate the importance of algebraic ideals in understanding the structure of $C^*$ -algebras.

In conclusion, the question of whether an algebraic ideal $I$ of a $C^*$ -algebra satisfies $I=\mathrm{span}(I^2)$ is a natural one, and it has far-reaching implications in operator theory and functional analysis. The study of algebraic ideals in $C^*$ -algebras is essential in understanding the structure of these algebras, and their properties have important implications in various areas of mathematics. While there are several counterexamples that show that the answer is no, the question remains an open one, and further research is needed to fully understand the behavior of algebraic ideals in $C^*$ -algebras.

[1] Davidson, K. R. (1996). C*-algebras by example. American Mathematical Society.
[2] Murphy, G. J. (1990). C*-algebras and operator theory. Academic Press.
[3] Pedersen, G. K. (1979). C*-algebras and their automorphism groups. Academic Press.

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

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

**Q: What are some examples of algebraic ideals in $C^*$ -algebras?**

A: There are several examples of algebraic ideals in $C^*$ -algebras, including:

The set of compact operators: Let $A$ be a $C^*$ -algebra of compact operators on a Hilbert space. Then the set of compact operators is an algebraic ideal of $A$ .
The set of finite-rank operators: Let $A$ be a $C^*$ -algebra of finite-rank operators on a Hilbert space. Then the set of finite-rank operators is an algebraic ideal of $A$ .

**Q: Does an algebraic ideal $I$ of a $C^*$ -algebra satisfy $I=\mathrm{span}(I^2)$ ?**

A: The question of whether an algebraic ideal $I$ satisfies $I=\mathrm{span}(I^2)$ is still an open one, and there are several counterexamples that have been constructed to show that the answer is no. One such counterexample is a non-closed two-sided ideal $I$ of a $C^*$ -algebra $A$ such that $I\neq\mathrm{span}(I^2)$ .

Q: What are some properties of algebraic ideals?

A: An algebraic ideal $I$ of a $C^*$ -algebra $A$ satisfies several properties, including:

Closure under multiplication: If $x \in I$ and $a \in A$ , then $ax \in I$ .
Closure under addition: If $x, y \in I$ , then $x + y \in I$ .
Closure under scalar multiplication: If $x \in I$ and $\lambda \in \mathbb{C}$ , then $\lambda x \in I$ .

**Q: How do algebraic ideals relate to the structure of $C^*$ -algebras?**

A: Algebraic ideals play a crucial role in understanding the structure of $C^*$ -algebras. They are essential in the study of ideals in $C^*$ -algebras, and their properties have far-reaching implications in operator theory and functional analysis.

**Q: What are some open questions related to algebraic ideals in $C^*$ -algebras?**

A: There are several open questions related to algebraic ideals in $C^*$ -algebras, including:

Does an algebraic ideal $I$ of a $C^*$ -algebra satisfy $I=\mathrm{span}(I^2)$ ?
What are the properties of algebraic ideals in $C^*$ -algebras?
How do algebraic ideals relate to the structure of $C^*$ -algebras?

**Q: What are some references for further reading on algebraic ideals in $C^*$ -algebras?**

A: Some references for further reading on algebraic ideals in $C^*$ -algebras include:

Davidson, K. R. (1996). C*-algebras by example. American Mathematical Society.
Murphy, G. J. (1990). C*-algebras and operator theory. Academic Press.
Pedersen, G. K. (1979). C*-algebras and their automorphism groups. Academic Press.

These references provide a comprehensive overview of the theory of $C^*$ -algebras and their ideals, and they are essential in understanding the behavior of algebraic ideals in these algebras.