Groups and Operator Algebras Seminar

Speaker: Ian Thompson (University of Copenhagen)

Title: The Universal Dilation Constant

Abstract: Fix an integer $d \geq 2$. It is well-known that there is a generator-preserving quotient map of $\mathrm{C}^*(\mathbb{F}_d)$ onto $\mathrm{C}^*(\mathbb{Z}^d)$ yet this map is never split, not even as a UCP map. However, there is a natural weakening one can consider: is there a UCP splitting that is generator-preserving up to a constant? If such a map exists, then what is the optimal value for such a constant? Finding the optimal value for such a constant is interwoven with a problem in multivariate operator theory: determining the true value of the so-called universal dilation constant $C_d$. The journey to find the true value of this constant, as well as related problems on different dilation constants, has led to connections from matrix convexity, quantum information theory, free probability, and mathematical physics. In this talk, we will discuss the history of these problems, as well as recent work on the value of the universal dilation constant in the case of $d=2$.