States and Representations

Oct 13, 2025 by windailys.com 24 views

Introduction

In the realm of operator algebras, the concept of non-equivalent representations plays a crucial role in understanding the structure and behavior of $C^*$ -algebras. These algebras, which are defined as norm-closures of certain spaces of linear operators, have far-reaching implications in various areas of mathematics and physics. In this article, we will delve into the world of $C^*$ -algebras and explore the notion of non-equivalent representations, with a focus on the specific case of states $\omega$ on a given $C^*$ -algebra $\mathcal{A}$ .

Background

To begin with, let us recall the definition of a $C^*$ -algebra. A $C^*$ -algebra is a Banach algebra $\mathcal{A}$ over the complex numbers, equipped with a norm $\|\cdot\|$ that satisfies the following properties:

$\|a\|^2 = \langle a, a \rangle$ for all $a \in \mathcal{A}$ , where $\langle \cdot, \cdot \rangle$ denotes the inner product on $\mathcal{A}$ .
$\|ab\| \leq \|a\| \|b\|$ for all $a, b \in \mathcal{A}$ .
$\|a^*a\| = \|a\|^2$ for all $a \in \mathcal{A}$ , where $a^*$ denotes the adjoint of $a$ .

A $C^*$ -algebra is said to be separable if it has a countable dense subset. In this article, we will be dealing with a specific type of $C^*$ -algebra, namely the norm-closure of $\cup_n \mathcal{B}(\mathbb{H})$ , where $\mathbb{H}$ is a Hilbert space and $\mathcal{B}(\mathbb{H})$ denotes the space of bounded linear operators on $\mathbb{H}$ .

States and Representations

A state on a $C^*$ -algebra $\mathcal{A}$ is a positive linear functional $\omega: \mathcal{A} \to \mathbb{C}$ that satisfies the normalization condition $\omega(1) = 1$ , where $1$ denotes the unit element of $\mathcal{A}$ . In other words, a state is a linear functional that assigns a complex number to each element of $\mathcal{A}$ in such a way that the number assigned to the unit element is 1, and the number assigned to any element is non-negative.

A representation of a $C^*$ -algebra $\mathcal{A}$ is a homomorphism $\pi: \mathcal{A} \to \mathcal{B}(\mathbb{H})$ from $\mathcal{A}$ to the space of bounded linear operators on a Hilbert space $\mathbb{H}$ . In other words, a representation is a way of embedding the $C^*$ -algebra $\mathcal{A}$ into the space of bounded linear operators on a Hilbert space.

Non-Equivalent Representations

Two representations $\pi_1$ and $\pi_2$ of a $C^*$ -algebra $\mathcal{A}$ are said to be equivalent if there exists a unitary operator $U$ on the Hilbert space $\mathbb{H}$ such that $\pi_2(a) = U \pi_1(a) U^*$ for all $a \in \mathcal{A}$ . In other words, two representations are equivalent if they are related by a unitary transformation.

However, in some cases, two representations may not be equivalent. This is where the concept of non-equivalent representations comes in. Non-equivalent representations are representations that cannot be related by a unitary transformation.

As mentioned earlier, we are given a $C^*$ -algebra $\mathcal{A}$ defined as the norm-closure of $\cup_n \mathcal{B}(\mathbb{H})$ , where $\mathbb{H}$ is a Hilbert space and $\mathcal{B}(\mathbb{H})$ denotes the space of bounded linear operators on $\mathbb{H}$ . We are also given two different states $\omega$ on this $C^*$ -algebra.

The Question

The question that arises here is whether these two states $\omega$ are equivalent or not. In other words, are these two states related by a unitary transformation, or are they non-equivalent?

The Answer

To answer this question, we need to delve deeper into the properties of the $C^*$ -algebra $\mathcal{A}$ and the states $\omega$ on it. We need to examine the behavior of these states under different representations of the $C^*$ -algebra and see if they are equivalent or not.

Conclusion

In conclusion, the question of whether two states $\omega$ on a $C^*$ -algebra $\mathcal{A}$ are equivalent or not is a complex one that requires a deep understanding of the properties of the $C^*$ -algebra and the states on it. By examining the behavior of these states under different representations of the $C^*$ -algebra, we can determine whether they are equivalent or not.

References

[1] R. V. Kadison and J. R. Ringrose, "Fundamentals of the Theory of Operator Algebras," Academic Press, 1983.
[2] E. C. Lance, "Hilbert C*-Modules: A Toolkit for Operator Algebraists," Cambridge University Press, 1995.
[3] M. Takesaki, "Theory of Operator Algebras I," Springer-Verlag, 1979.

Appendix

The following appendix provides a brief overview of the mathematical notation used in this article.

Banach algebra: A Banach algebra is a complete normed vector space over the complex numbers, equipped with a multiplication operation that satisfies certain properties.
C-algebra*: A C*-algebra is a Banach algebra that satisfies the C*-identity, which states that the norm of an element is equal to the norm of its adjoint.
Hilbert space: A Hilbert space is a complete inner product space over the complex numbers.
Operator algebra: An operator algebra is a Banach algebra of linear operators on a Hilbert space.
Representation: A representation of a C*-algebra is a homomorphism from the C*-algebra to the space of bounded linear operators on a Hilbert space.
State: A state on a C*-algebra is a positive linear functional that assigns a complex number to each element of the C*-algebra in such a way that the number assigned to the unit element is 1, and the number assigned to any element is non-negative.
Unitary operator: A unitary operator is a linear operator on a Hilbert space that preserves the inner product.
Q&A: Doubt on Non-Equivalent Representations - $C^*$ -algebras ===========================================================

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

A: A $C^*$ -algebra is a Banach algebra $\mathcal{A}$ over the complex numbers, equipped with a norm $\|\cdot\|$ that satisfies the following properties:

$\|a\|^2 = \langle a, a \rangle$ for all $a \in \mathcal{A}$ , where $\langle \cdot, \cdot \rangle$ denotes the inner product on $\mathcal{A}$ .
$\|ab\| \leq \|a\| \|b\|$ for all $a, b \in \mathcal{A}$ .
$\|a^*a\| = \|a\|^2$ for all $a \in \mathcal{A}$ , where $a^*$ denotes the adjoint of $a$ .

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

A: A state on a $C^*$ -algebra $\mathcal{A}$ is a positive linear functional $\omega: \mathcal{A} \to \mathbb{C}$ that satisfies the normalization condition $\omega(1) = 1$ , where $1$ denotes the unit element of $\mathcal{A}$ .

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

A: A representation of a $C^*$ -algebra $\mathcal{A}$ is a homomorphism $\pi: \mathcal{A} \to \mathcal{B}(\mathbb{H})$ from $\mathcal{A}$ to the space of bounded linear operators on a Hilbert space $\mathbb{H}$ .

Q: What is a non-equivalent representation?

A: Two representations $\pi_1$ and $\pi_2$ of a $C^*$ -algebra $\mathcal{A}$ are said to be non-equivalent if there does not exist a unitary operator $U$ on the Hilbert space $\mathbb{H}$ such that $\pi_2(a) = U \pi_1(a) U^*$ for all $a \in \mathcal{A}$ .

**Q: How do I determine if two states on a $C^*$ -algebra are equivalent or not?**

A: To determine if two states on a $C^*$ -algebra are equivalent or not, you need to examine the behavior of these states under different representations of the $C^*$ -algebra. If the states are equivalent, then they will be related by a unitary transformation under any representation. If the states are not equivalent, then they will not be related by a unitary transformation under any representation.

**Q: What are some common applications of $C^*$ -algebras?**

A: $C^*$ -algebras have far-reaching implications in various areas of mathematics and physics, including:

Operator algebras
Functional analysis
Quantum mechanics
Quantum field theory
Representation theory

**Q: What are some common tools used in the study of $C^*$ -algebras?**

A: Some common tools used in the study of $C^*$ -algebras include:

Banach algebras
Hilbert spaces
Operator algebras
Representation theory
Functional analysis

**Q: What are some common challenges faced in the study of $C^*$ -algebras?**

A: Some common challenges faced in the study of $C^*$ -algebras include:

Understanding the structure and behavior of $C^*$ -algebras
Developing new tools and techniques for studying $C^*$ -algebras
Applying $C^*$ -algebras to real-world problems
Dealing with the complexity and abstract nature of $C^*$ -algebras

**Q: What are some common resources for learning about $C^*$ -algebras?**

A: Some common resources for learning about $C^*$ -algebras include:

Textbooks on operator algebras and functional analysis
Online courses and lectures on $C^*$ -algebras
Research papers and articles on $C^*$ -algebras
Conferences and workshops on $C^*$ -algebras
Online communities and forums for discussing $C^*$ -algebras