| Résume | A paper by Fritz, Netzer, and Thom from 2012, titled "Can you compute the operator norm?", in essence initiated the computable structure theory for C*-algebras. The primary question involved is: given some "standard" presentation of a C*-algebra, is there an algorithm for computing the norm on the algebra? In this talk, I will focus on the case of C*-algebras arising from groups. In fact, most of the talk will focus on the case of the following four group C*-algebras: C^*(F_2), C^*(F_2\times F_2), C^*_r(F_2), C^*_r(F_2\times F_2). Here, for a group G, C^*(G) and C^*_r(G) denote the universal and reduced group C*-algebras associated to G, and F_2 denotes the free group on two generators. The answers to the question about computability of the norm for these four group C*-algebras turn out to be: yes, no, yes, and yes. The first is a result of FNT, the second a result due to myself and Thomas Sinclair from a year ago (answering a question of FN,T using recent results from quantum complexity theory), and the third and fourth recent results of mine (answering a question of Sinclair). Time permitting, I will discuss a potential generation of the latter results to wider classes of groups.
No prior knowledge of C*-algebra theory will be assumed. |
