The study of crossed product $C^$-algebras provides a powerful tool for understanding the interaction between group actions and operator algebras. These algebras arise naturally in various areas of mathematics, including functional analysis, topological dynamics, and quantum field theory. One fundamental aspect of analyzing crossed product $C^$-algebras is understanding the nature of their elements. This article delves into the structure and properties of elements in crossed product $C^*$-algebras, exploring how they relate to the underlying group action and the original $C^*$-algebra.
Constructing Crossed Product $C^*$-algebras
To understand the elements of a crossed product $C^$-algebra, we must first understand its construction. Let $A$ be a $C^$-algebra and $G$ be a locally compact group acting on $A$ by automorphisms. The crossed product $C^$-algebra, denoted by $A \rtimes_\alpha G$, is formed by considering a specific type of function from $G$ to $A$. These functions, called covariant representations, satisfy certain conditions that capture the interplay between the group action and the $C^$-algebra.
Covariant Representations
A covariant representation of the pair $(A,G)$ consists of a pair $(\pi, U)$, where:
- $\pi: A \to B(H)$ is a non-degenerate representation of $A$ on a Hilbert space $H$.
- $U: G \to U(H)$ is a strongly continuous unitary representation of $G$ on the same Hilbert space $H$.
These representations are subject to the crucial covariance condition:
$\pi(\alpha_g(a)) = U_g \pi(a) U_g^*$
for all $a \in A$ and $g \in G$. This condition essentially ensures that the representation of $A$ is compatible with the group action $\alpha$.
The Crossed Product Construction
The crossed product $C^$-algebra $A \rtimes_\alpha G$ is constructed using the universal property of $C^$-algebras. It is the completion of the space of continuous, compactly supported functions from $G$ to $A$, equipped with a specific convolution product and involution that reflect the group structure and the covariance condition.
Elements in Crossed Product $C^*$-algebras
The elements of the crossed product $C^*$-algebra $A \rtimes_\alpha G$ can be viewed as "twisted" functions from $G$ to $A$. They are not simply functions, but rather formal linear combinations of the form:
$\sum_{g \in G} a_g u_g$
where $a_g \in A$ and $u_g$ are unitary elements in $A \rtimes_\alpha G$ representing the group action. The multiplication and involution are defined using the group action and the original operations in $A$:
- Multiplication: $(a_1 u_{g_1}) (a_2 u_{g_2}) = a_1 \alpha_{g_1}(a_2) u_{g_1 g_2}$
- Involution: $(a u_g)^* = \alpha_{g^{-1}}(a^*) u_{g^{-1}}$
Important Properties of Elements
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Compactly Supported Elements: The elements of the form $\sum_{g \in F} a_g u_g$, where $F$ is a finite subset of $G$, are called compactly supported elements. These elements play a crucial role in understanding the structure of the crossed product $C^*$-algebra.
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Spectral Properties: The spectral properties of elements in $A \rtimes_\alpha G$ are closely related to the spectral properties of elements in $A$ and the representation theory of $G$.
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Regular Representations: The crossed product $C^$-algebra $A \rtimes_\alpha G$ admits a natural regular representation on the Hilbert space $L^2(G,H)$, where $H$ is the Hilbert space associated with a faithful representation of $A$. This representation provides a crucial tool for analyzing the structure of the crossed product $C^$-algebra.
Examples of Crossed Product $C^*$-algebras
Several important examples showcase the relevance and applications of crossed product $C^*$-algebras.
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Group $C^*$-algebras: When $A = \mathbb{C}$ (the complex numbers) and the action is trivial, the crossed product $C^$-algebra reduces to the group $C^$-algebra $C^*(G)$. This algebra captures the harmonic analysis of the group $G$.
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Cuntz Algebras: These algebras are related to the action of the free group on a finite number of generators. Cuntz algebras play a significant role in the study of non-commutative geometry and quantum field theory.
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Quantum Groups: Crossed product $C^*$-algebras are essential in the study of quantum groups. The quantum group structure can be encoded using a crossed product construction.
Applications and Significance of Crossed Product $C^*$-algebras
The study of elements in crossed product $C^*$-algebras plays a vital role in various areas of mathematics and theoretical physics.
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Functional Analysis: Crossed products provide a framework for studying the representation theory of $C^*$-algebras and their interactions with group actions.
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Topological Dynamics: Crossed product $C^*$-algebras are instrumental in analyzing the dynamics of actions of groups on topological spaces.
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Quantum Field Theory: They appear in the construction of quantum field theories with non-trivial symmetries.
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Operator Algebras: Crossed product $C^*$-algebras play a crucial role in the study of non-commutative geometry and the classification of operator algebras.
Conclusion
Understanding the structure and properties of elements in crossed product $C^*$-algebras is crucial for comprehending the interplay between group actions and $C^*$-algebras. The representation theory of these algebras, along with the spectral properties of their elements, opens doors to diverse applications in various areas of mathematics and theoretical physics. As we delve deeper into the intricacies of crossed products, we uncover a rich tapestry of mathematical connections and insights that enhance our understanding of these fundamental algebraic structures.
