Crossed product c*-algebra

Definition

A crossed product c*-algebra is a type of algebraic structure that arises from a dynamical system, specifically when a group acts on a C*-algebra. This construction allows one to capture the interactions between the algebra and the group action, leading to a new C*-algebra that encodes both the algebraic and topological properties of the original system. Crossed products are particularly useful for studying noncommutative geometry and are fundamental in understanding various applications in operator algebras.

5 Must Know Facts For Your Next Test

1. Crossed product c*-algebras are denoted as $C^*(A times_ heta G)$, where $A$ is the original C*-algebra, $G$ is the group, and $ heta$ describes the action of the group on the algebra.
2. These algebras can generalize traditional constructions like the group algebra by incorporating the structure of both the group and the C*-algebra.
3. When dealing with compact groups, crossed products can lead to important results such as Morita equivalence, which indicates that certain algebras share essential properties.
4. The construction involves quotienting by certain relations that reflect how elements of the group act on elements of the C*-algebra.
5. Crossed products play a key role in various areas like quantum mechanics, where symmetries and observables can be modeled using noncommutative algebras.

Review Questions

• How does the action of a group on a C*-algebra influence the structure of the crossed product c*-algebra?
  • The action of a group on a C*-algebra introduces new relations and interactions between elements of the algebra and the group. This influence shapes the structure of the crossed product c*-algebra by combining properties from both entities, leading to a new algebra that retains information about how the group symmetries affect the algebra. Essentially, it allows one to examine both aspects simultaneously, capturing their interplay within the new algebraic framework.
• What role do crossed product c*-algebras play in understanding symmetries in quantum mechanics?
  • Crossed product c*-algebras are essential for modeling symmetries in quantum mechanics, particularly in situations where observables are affected by group actions. By constructing a crossed product, one can encode how quantum states transform under various symmetries, providing insight into the dynamics of quantum systems. This connection helps to analyze phenomena such as symmetry breaking and invariance, making crossed products a powerful tool in theoretical physics.
• Evaluate how crossed product c*-algebras contribute to noncommutative geometry and provide an example where this application is evident.
  • Crossed product c*-algebras significantly contribute to noncommutative geometry by allowing mathematicians to explore spaces where classical geometric notions fail. They help formalize the idea that spaces can be represented as algebras, facilitating an understanding of geometric structures through operator algebras. An example is in the study of quantum groups and their actions on function spaces; here, crossed products reveal how geometric features can emerge from algebraic properties when analyzing quantum symmetry actions on spaces.

Related terms

C*-algebra: A C*-algebra is a complex algebra of bounded linear operators on a Hilbert space that is closed under taking adjoints and contains an identity element.

Dynamical System: A dynamical system consists of a space and a group acting on that space, capturing how points evolve over time according to specific rules.

Group Action: A group action describes how elements of a group correspond to transformations or symmetries of a space, allowing for the study of invariant properties under those transformations.