Kazhdan Property (T)

Definition

Kazhdan Property (T) is a property of groups that states every unitary representation of the group on a Hilbert space has a non-zero invariant vector if the representation is not nearly trivial. This property implies strong rigidity in the group and plays a crucial role in various areas such as operator algebras and noncommutative geometry. It connects to spectral theory, representation theory, and has implications for the classification of C*-algebras.

5 Must Know Facts For Your Next Test

1. Groups with Kazhdan Property (T) have rigid representations, meaning their unitary representations exhibit strong structural characteristics.
2. Kazhdan Property (T) is an essential condition for the existence of nontrivial actions on von Neumann algebras, particularly in relation to free group actions.
3. The presence of Kazhdan Property (T) often leads to the conclusion that certain types of algebras, like C*-algebras, are simple or have very few ideals.
4. Kazhdan Property (T) has applications in various fields including geometric group theory and the theory of operator algebras, enhancing our understanding of their structures.
5. Many important groups in mathematics, such as SL(n, Z) for n ≥ 2, exhibit Kazhdan Property (T), making them significant in the study of noncommutative geometry.

Review Questions

• How does Kazhdan Property (T) influence the structure of unitary representations of groups?
  • Kazhdan Property (T) significantly influences unitary representations by ensuring that any nontrivial unitary representation must have a non-zero invariant vector. This means that such representations cannot become arbitrarily close to trivial ones without retaining some invariant properties. This rigidity leads to deeper insights into how groups act on various spaces and shapes our understanding of their representation theory.
• Discuss the implications of Kazhdan Property (T) on von Neumann algebras and C*-algebras.
  • Kazhdan Property (T) has profound implications for von Neumann algebras and C*-algebras, particularly in determining their simplicity. When a group possesses this property, it suggests that any action on a von Neumann algebra must exhibit certain structural features, leading to strong conclusions about the algebra's ideals. This connection has been pivotal in classifying algebras associated with groups and contributes to the broader understanding of operator algebras.
• Evaluate how Kazhdan Property (T) connects geometric group theory with noncommutative geometry.
  • Kazhdan Property (T) serves as a bridge between geometric group theory and noncommutative geometry by providing a framework for analyzing groups through their actions on spaces. This connection allows mathematicians to use geometric methods to understand algebraic structures associated with groups, leading to insights about their representation properties. Furthermore, it helps classify operator algebras through geometric means, highlighting the interplay between algebraic and topological aspects in modern mathematics.

Related terms

Unitary Representation: A way to represent a group as a set of unitary operators on a Hilbert space, preserving the group structure and operations.

C*-Algebra: A type of algebra of bounded operators on a Hilbert space that is closed under taking adjoints and under the norm topology.

Rigidity: A property of a mathematical object that implies it cannot be deformed into other objects without losing its structure, often related to groups or spaces.