Sylvester Rank Functions on Crossed Products

Baojie Jiang ½¯±¨½Ý

10:00-11:00, September 8, 2020   Zoom 467 658 1686

Abstract:

Sylvester rank functions for a given unital ring $R$ are numerical invariants for matrices or modules over $R$, describing the rank or dimension of such objects. Let $\mathcal{A}$ be a unital $C^*$-algebra and let $\tau$ be any tracial state on $\mathcal{A}$. Set $\mathrm{rk}_\tau(B)=\lim_{k\to\infty}\tau(|B|^{1/k})$ for every rectangular matrices over $\mathcal{A}$. Then $\mathrm{rk}_\tau$ is a Sylvester rank function defined on rectangular matrices over $\mathcal{A}$. In this talk, based on joint work with Prof. Hanfeng Li, we focus on amenable group which admits a tracial preserving action on a unital $C^*$-algebra, we show that two natural Sylvester matrix functions on the algebraic crossed product constructed out of the tracial state coincide.