Operator Theory, Complex Analysis and Applications Seminar 

16/05/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building
Sérgio Mendes, Instituto Universitário de Lisboa, ISCTE-IUL

Noncommutative summands of the $C^{*}$-algebra $C_r^{*}{\mathrm{SL}}_2({𝔽}_2((\varpi )))$

Let ${𝔽}_2((\varpi ))$ denote the Laurent series in the indeterminate $\varpi$ with coefficients over the finite field with two elements ${𝔽}_2$. This is a local nonarchimedean field with characteristic $2$. We show that the structure of the reduced group $C^{*}$-algebra of ${\mathrm{SL}}_2({𝔽}_2((\varpi )))$ is determined by the arithmetic of the ground field. Specifically, the algebra $C_r^{*}{\mathrm{SL}}_2({𝔽}_2((\varpi )))$ has countably many noncommutative summands, induced by the Artin-Schreier symbol. Each noncommutative summand has a rather simple description: it is the crossed product of a commutative $C^{*}$-algebra by a finite group. The talk will be elementary, starting from the scratch with the definition of $C_r^{*}{\mathrm{SL}}_2$.