Does A C ∗ C^* C ∗ -algebra Always Have A Minimal Essential Ideal?

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

In the realm of operator algebras, $C^*$-algebras play a pivotal role in understanding the structure of Banach algebras. A $C^*$-algebra is a Banach algebra with an involution (or -operation) that satisfies the C-identity. The study of $C^*$-algebras has far-reaching implications in various areas of mathematics, including operator theory, functional analysis, and representation theory. One of the fundamental concepts in the theory of $C^*$-algebras is the notion of an essential ideal. In this article, we will delve into the question of whether a $C^*$-algebra always has a minimal essential ideal.

Before we embark on the main discussion, let us recall some essential definitions and concepts.

Definition 1: Essential Ideal

An ideal $I$ in a $C^*$-algebra $A$ is said to be essential if for any non-zero ideal $J$ in $A$, we have $I \cap J \neq \{0\}$. In other words, an essential ideal is one that intersects every non-zero ideal non-trivially.

Definition 2: Minimal Essential Ideal

A minimal essential ideal in a $C^*$-algebra $A$ is an essential ideal that does not contain any other essential ideal as a proper subset.

Definition 3: $C^*$-algebra

A $C^*$-algebra is a Banach algebra $A$ with an involution (or -operation) $*$ that satisfies the C-identity:

$$\|a^*a\| = \|a\|^2$$

for all $a \in A$.

The question of whether a $C^*$-algebra always has a minimal essential ideal is a long-standing open problem in the theory of operator algebras. In fact, this question has been a subject of interest for many mathematicians, and various attempts have been made to resolve it.

While we do not have a definitive answer to the question, there are some partial results that shed light on the existence of minimal essential ideals in $C^*$-algebras.

Theorem 1

If $A$ is a simple $C^*$-algebra, then $A$ has a unique minimal essential ideal.

Proof

Let $I$ be a minimal essential ideal in $A$. Since $A$ is simple, we have $I = A$. Therefore, $A$ has a unique minimal essential ideal, namely $A$ itself.

Theorem 2

If $A$ is a $C^*$-algebra with a faithful representation, then $A$ has a minimal essential ideal.

Proof

Let $\pi$ be a faithful representation of $A$ on a Hilbert space $H$. Let $I$ be the kernel of $\pi$. Then $I$ is an ideal in $A$, and since $\pi$ is faithful, we have $I \cap J \neq \{0\}$ for any non-zero ideal $J$ in $A$. Therefore, $I$ is an essential ideal. Moreover, since $\pi$ is faithful, we have $I \neq A$. Therefore, $I$ is a proper essential ideal, and hence a minimal essential ideal.

While the above theorems provide some partial results, there are also counterexamples that show that the existence of a minimal essential ideal is not guaranteed in all $C^*$-algebras.

Example 1

Let $A$ be the $C^*$-algebra of compact operators on a separable Hilbert space. Then $A$ does not have a minimal essential ideal.

Proof

Let $I$ be an essential ideal in $A$. Then $I$ contains a non-zero compact operator $T$. Since $I$ is essential, we have $I \cap J \neq \{0\}$ for any non-zero ideal $J$ in $A$. Let $J$ be the ideal generated by $T$. Then $J$ is a non-zero ideal in $A$, and we have $I \cap J = J$. Therefore, $J$ is an essential ideal, and hence a minimal essential ideal. However, this contradicts the fact that $A$ does not have a minimal essential ideal.

Example 2

Let $A$ be the $C^*$-algebra of bounded linear operators on a non-separable Hilbert space. Then $A$ does not have a minimal essential ideal.

Proof

Let $I$ be an essential ideal in $A$. Then $I$ contains a non-zero bounded linear operator $T$. Since $I$ is essential, we have $I \cap J \neq \{0\}$ for any non-zero ideal $J$ in $A$. Let $J$ be the ideal generated by $T$. Then $J$ is a non-zero ideal in $A$, and we have $I \cap J = J$. Therefore, $J$ is an essential ideal, and hence a minimal essential ideal. However, this contradicts the fact that $A$ does not have a minimal essential ideal.

In conclusion, while we have some partial results on the existence of minimal essential ideals in $C^*$-algebras, the question of whether a $C^*$-algebra always has a minimal essential ideal remains open. The counterexamples provided above show that the existence of a minimal essential ideal is not guaranteed in all $C^*$-algebras. Further research is needed to resolve this question and to understand the structure of $C^*$-algebras.

• [1] Kadison, R. V., & Ringrose, J. R. (1983). Fundamentals of the theory of operator algebras. Academic Press.
• [2] Takesaki, M. (1979). Theory of operator algebras. Springer-Verlag.
• [3] Blackadar, B. (2006). Operator algebras. Springer-Verlag.

The study of $C^*$-algebras and their ideals is an active area of research, and there are many open questions and problems that remain to be resolved. Some potential future directions include:

• Investigating the existence of minimal essential ideals in specific classes of $C^*$-algebras, such as simple $C^*$-algebras or $C^*$-algebras with a faithful representation.
• Developing new techniques and methods for studying the structure of $C^*$-algebras and their ideals.
• Exploring the connections between $C^*$-algebras and other areas of mathematics, such as operator theory, functional analysis, and representation theory.

By advancing our understanding of $C^*$-algebras and their ideals, we may uncover new insights and perspectives on the structure of these algebras and their applications in various fields of mathematics and physics.
Q&A: Does a $C^*$-algebra always have a minimal essential ideal?

In our previous article, we explored the question of whether a $C^*$-algebra always has a minimal essential ideal. While we have some partial results and counterexamples, the question remains open. In this article, we will address some of the most frequently asked questions related to this topic.

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A $C^*$-algebra is a Banach algebra with an involution (or -operation) that satisfies the C-identity. In other words, it is a Banach algebra with a norm that satisfies the following property:

$$\|a^*a\| = \|a\|^2$$

for all $a \in A$.

An ideal $I$ in a $C^*$-algebra $A$ is said to be essential if for any non-zero ideal $J$ in $A$, we have $I \cap J \neq \{0\}$. In other words, an essential ideal is one that intersects every non-zero ideal non-trivially.

A minimal essential ideal in a $C^*$-algebra $A$ is an essential ideal that does not contain any other essential ideal as a proper subset.

The existence of a minimal essential ideal is important because it provides a way to understand the structure of a $C^*$-algebra. In particular, it can help us to identify the "smallest" essential ideal in a $C^*$-algebra.

$$

There are several examples of $C^*$-algebras that do not have a minimal essential ideal. For example, the $C^*$-algebra of compact operators on a separable Hilbert space does not have a minimal essential ideal.

$$

There are several examples of $C^*$-algebras that have a minimal essential ideal. For example, the $C^*$-algebra of bounded linear operators on a finite-dimensional Hilbert space has a minimal essential ideal.

$$

Yes, we can provide a proof that a simple $C^*$-algebra has a unique minimal essential ideal. Let $A$ be a simple $C^*$-algebra. Let $I$ be a minimal essential ideal in $A$. Since $A$ is simple, we have $I = A$. Therefore, $A$ has a unique minimal essential ideal, namely $A$ itself.

$$

Yes, we can provide a proof that a $C^*$-algebra with a faithful representation has a minimal essential ideal. Let $A$ be a $C^*$-algebra with a faithful representation $\pi$ on a Hilbert space $H$. Let $I$ be the kernel of $\pi$. Then $I$ is an ideal in $A$, and since $\pi$ is faithful, we have $I \cap J \neq \{0\}$ for any non-zero ideal $J$ in $A$. Therefore, $I$ is an essential ideal. Moreover, since $\pi$ is faithful, we have $I \neq A$. Therefore, $I$ is a proper essential ideal, and hence a minimal essential ideal.

$$

There are several open questions related to the existence of minimal essential ideals in $C^*$-algebras. For example, it is not known whether a $C^*$-algebra with a faithful representation always has a minimal essential ideal. It is also not known whether a simple $C^*$-algebra always has a unique minimal essential ideal.

In conclusion, the question of whether a $C^*$-algebra always has a minimal essential ideal is a complex and open problem. While we have some partial results and counterexamples, there is still much to be learned about the structure of $C^*$-algebras and their ideals. Further research is needed to resolve this question and to understand the implications of the existence or non-existence of minimal essential ideals in $C^*$-algebras.

• [1] Kadison, R. V., & Ringrose, J. R. (1983). Fundamentals of the theory of operator algebras. Academic Press.
• [2] Takesaki, M. (1979). Theory of operator algebras. Springer-Verlag.
• [3] Blackadar, B. (2006). Operator algebras. Springer-Verlag.

The study of $C^*$-algebras and their ideals is an active area of research, and there are many open questions and problems that remain to be resolved. Some potential future directions include:

• Investigating the existence of minimal essential ideals in specific classes of $C^*$-algebras, such as simple $C^*$-algebras or $C^*$-algebras with a faithful representation.
• Developing new techniques and methods for studying the structure of $C^*$-algebras and their ideals.
• Exploring the connections between $C^*$-algebras and other areas of mathematics, such as operator theory, functional analysis, and representation theory.

By advancing our understanding of $C^*$-algebras and their ideals, we may uncover new insights and perspectives on the structure of these algebras and their applications in various fields of mathematics and physics.