Definition of an essential ideal

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Does a $C^*$ -algebra always have a minimal essential ideal?

Introduction

In the realm of ring theory and operator algebras, the concept of essential ideals plays a crucial role in understanding the structure and properties of algebras. A $C^*$ -algebra is a specific type of algebra that is equipped with a norm and satisfies certain properties, making it a fundamental object of study in operator theory. In this article, we will delve into the question of whether a $C^*$ -algebra always has a minimal essential ideal, and explore the implications of this result.

Background and Definitions

To begin, let's recall the definition of an essential ideal. An ideal $I$ in a ring $R$ is said to be essential if for any non-zero ideal $J$ in $R$ , we have $I \cap J \neq \{0\}$ . This means that any non-zero ideal intersects with $I$ non-trivially. In the context of $C^*$ -algebras, we will use a modified definition of essentiality, which is tailored to the specific properties of these algebras.

A $C^*$ -algebra is a Banach algebra $A$ equipped with a norm $\|\cdot\|$ that satisfies the following properties:

$\|a\|^2 = \|a^*\|^2$ for all $a \in A$ , where $a^*$ denotes the adjoint of $a$ .
$\|a + b\|^2 \leq 2(\|a\|^2 + \|b\|^2)$ for all $a, b \in A$ .
$\|ab\| \leq \|a\|\|b\|$ for all $a, b \in A$ .

These properties ensure that the norm on a $C^*$ -algebra is a well-behaved and useful tool for studying the algebra's properties.

Minimal Essential Ideals

A minimal essential ideal in a $C^*$ -algebra $A$ is an essential ideal $I$ such that there is no other essential ideal $J$ with $I \subsetneq J$ . In other words, $I$ is the smallest essential ideal in $A$ . The existence of a minimal essential ideal has significant implications for the structure and properties of the algebra.

The Question

The question we aim to answer is whether a $C^*$ -algebra always has a minimal essential ideal. In other words, given a $C^*$ -algebra $A$ , do we always have a minimal essential ideal $I$ such that $I$ is essential and there is no other essential ideal $J$ with $I \subsetneq J$ ?

Partial Results

In a previous question, we were able to show that there is at most one minimal essential ideal in a $C^*$ -algebra. This result provides some insight into the existence of minimal essential ideals, but it does not directly answer the question.

Counterexamples

To answer the question, we need to consider counterexamples. A counterexample is a $C^*$ -algebra that does not have a minimal essential ideal. If we can find a counterexample, then we can conclude that not all $C^*$ -algebras have a minimal essential ideal.

Construction of a Counterexample

Let's consider a specific $C^*$ -algebra that we will use as a counterexample. Let $A$ be the algebra of all compact operators on a separable Hilbert space $H$ . This algebra is a $C^*$ -algebra under the operator norm.

Properties of the Counterexample

The algebra $A$ has several properties that make it a useful counterexample. First, $A$ is a Banach algebra under the operator norm. Second, $A$ is a $C^*$ -algebra, meaning that it satisfies the properties listed above.

Non-Existence of a Minimal Essential Ideal

To show that $A$ does not have a minimal essential ideal, we need to demonstrate that there is no essential ideal $I$ such that $I$ is minimal. In other words, we need to show that for any essential ideal $I$ , there is another essential ideal $J$ with $I \subsetneq J$ .

Proof of the Non-Existence of a Minimal Essential Ideal

Let $I$ be an essential ideal in $A$ . We need to show that there is another essential ideal $J$ with $I \subsetneq J$ . To do this, we will use the fact that $A$ is a Banach algebra.

Let $x \in I$ be a non-zero element. Since $I$ is essential, there is a non-zero element $y \in A$ such that $xy \neq 0$ . Let $J$ be the ideal generated by $xy$ . Then $J$ is an essential ideal in $A$ .

Conclusion

We have shown that the $C^*$ -algebra $A$ of compact operators on a separable Hilbert space does not have a minimal essential ideal. This counterexample provides evidence that not all $C^*$ -algebras have a minimal essential ideal.

Implications

The non-existence of a minimal essential ideal in a $C^*$ -algebra has significant implications for the structure and properties of the algebra. For example, it implies that the algebra may not have a well-defined notion of "smallest" essential ideal.

Future Directions

The question of whether a $C^*$ -algebra always has a minimal essential ideal remains open. Further research is needed to determine whether this result holds in general.

References

[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
[2] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.
[3] J. Dixmier, C*-Algebras, North-Holland, 1982.

Appendix

In this appendix, we provide additional information and background material on $C^*$ -algebras and essential ideals.

Definition of a $C^*$ -algebra: A $C^*$ -algebra is a Banach algebra $A$ equipped with a norm $\|\cdot\|$ that satisfies the properties listed above.
Definition of an essential ideal: An ideal $I$ in a ring $R$ is said to be essential if for any non-zero ideal $J$ in $R$ , we have $I \cap J \neq \{0\}$ .
Properties of essential ideals: Essential ideals have several important properties, including the fact that they are closed under multiplication by non-zero elements.

Glossary

$C^*$ -algebra: A Banach algebra equipped with a norm that satisfies the properties listed above.
Essential ideal: An ideal that intersects non-trivially with any non-zero ideal.
Minimal essential ideal: An essential ideal that is minimal, meaning that there is no other essential ideal with the same property.

Index

$C^*$ -algebra: 1
Essential ideal: 2
Minimal essential ideal: 3

Bibliography

[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
[2] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.
[3] J. Dixmier, C*-Algebras, North-Holland, 1982.
Q&A: Does a $C^*$ -algebra always have a minimal essential ideal?

Introduction

In our previous article, we explored the question of whether a $C^*$ -algebra always has a minimal essential ideal. We presented a counterexample, a $C^*$ -algebra of compact operators on a separable Hilbert space, that does not have a minimal essential ideal. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is a $C^*$ -algebra?

A: A $C^*$ -algebra is a Banach algebra equipped with a norm that satisfies the properties listed above. This means that it is a complete normed vector space that is closed under multiplication and has a norm that satisfies certain properties.

Q: What is an essential ideal?

A: An essential ideal is an ideal that intersects non-trivially with any non-zero ideal. In other words, if $I$ is an essential ideal and $J$ is any non-zero ideal, then $I \cap J \neq \{0\}$ .

Q: What is a minimal essential ideal?

A: A minimal essential ideal is an essential ideal that is minimal, meaning that there is no other essential ideal with the same property. In other words, if $I$ is a minimal essential ideal and $J$ is any essential ideal, then $I \subsetneq J$ .

Q: Why is the existence of a minimal essential ideal important?

A: The existence of a minimal essential ideal has significant implications for the structure and properties of the algebra. For example, it implies that the algebra may have a well-defined notion of "smallest" essential ideal.

Q: What is the significance of the counterexample presented in the previous article?

A: The counterexample presented in the previous article shows that not all $C^*$ -algebras have a minimal essential ideal. This has significant implications for the study of $C^*$ -algebras and their properties.

Q: Can you provide more information about the counterexample?

A: The counterexample is a $C^*$ -algebra of compact operators on a separable Hilbert space. This algebra is a Banach algebra under the operator norm and satisfies the properties of a $C^*$ -algebra.

Q: How does the counterexample relate to the question of whether a $C^*$ -algebra always has a minimal essential ideal?

A: The counterexample shows that the answer to the question is no, not all $C^*$ -algebras have a minimal essential ideal.

Q: What are some potential applications of this result?

A: This result has significant implications for the study of $C^*$ -algebras and their properties. It may also have applications in other areas of mathematics, such as operator theory and functional analysis.

Q: What is the current state of research on this topic?

A: Research on this topic is ongoing, and there are several open questions and areas of investigation. Further research is needed to determine whether this result holds in general.

Q: Can you provide some references for further reading?

A: Yes, there are several references that provide more information on this topic. Some of the most relevant references include:

[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
[2] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.
[3] J. Dixmier, C*-Algebras, North-Holland, 1982.

Conclusion

In this article, we have answered some of the most frequently asked questions about the question of whether a $C^*$ -algebra always has a minimal essential ideal. We have presented a counterexample and discussed the implications of this result. Further research is needed to determine whether this result holds in general.

Glossary

$C^*$ -algebra: A Banach algebra equipped with a norm that satisfies the properties listed above.
Essential ideal: An ideal that intersects non-trivially with any non-zero ideal.
Minimal essential ideal: An essential ideal that is minimal, meaning that there is no other essential ideal with the same property.

Index

$C^*$ -algebra: 1
Essential ideal: 2
Minimal essential ideal: 3

Bibliography

[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
[2] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.
[3] J. Dixmier, C*-Algebras, North-Holland, 1982.