From Groups to Quantum Groups and Their Operator Algebras

Every group admits a faithful unitary representation on some Hilbert space. In other words, every group can be realized concretely as symmetries on a Hilbert space. From these representations we can construct certain operator algebras known as C*-algebras. These group C*-algebras enable an extremely fruitful interaction between group theory and the theory of operator algebras. Celebrated developments of the past decade in the operator algebras community include group dynamical characterizations of the so-called unique trace property and C*-simplicity. Quantum groups for us are generalizations of groups in the sense that they are objects that "act quantumly on Hilbert spaces" and are defined via the associated C*-algebras. We will discuss the construction of group C*-algebras (including a discussion of Hilbert spaces) with an eye towards understanding what a quantum group is, as well as a recent development regarding the unique trace property of quantum groups.

Part of this talk will be based on joint work with Fatemeh Khosravi. This talk will be accessible to undergraduate students.

Zoom information: https://ubc.zoom.us/j/68285564037?pwd=R2ZpLy9uc2pUYldHT3laK3orakg0dz09

Meeting ID: 682 8556 4037

Passcode: 636252