YWC*A Mini-Course II

Mini-course title: Quantized Functional Analysis, Tensor Norms and the Grothendieck Program

Speaker: Magdalena Musat (University of Copenhagen)

Lecture II: Grothendieck's inequalities---from classical to noncommutative

Abstract: The highlight of Grothendieck's celebrated "Résumé", published in 1956, is a highly non-trivial factorization result for bounded bilinear forms on $C(K_1)\times C(K_2)$, where $K_1$ and $K_2$ are compact sets,  which is now referred to as the Grothendieck Theorem (or, Grothendieck Inequality). The "Résumé" contains several equivalent formulations of it, all describing fundamental relationships between Hilbert spaces (e.g., $L_2$), and the Banach spaces $L_\infty$, respectively, $ C(K)$, and $L_1$. It ends with a remarkable list of six problems, one of which is the conjecture that an analogue factorization for bounded bilinear forms on the product of (noncommutative) C$^*$-algebras holds. This was later proven by Pisier (under an approximability assumption), and by Haagerup (in full generality).

I will survey Grothendieck's inequalities, from classical to noncommutative, including extensions to the setting of completeley bounded bilinear forms on C$^*$-algebras and operator spaces, due to Pisier-Shlyakhtenko (2002), and joint work of Haagerup and myself (2008). I will also explain the connection with Tsirelson's result, announced in the first lecture. This makes crucial use of various tensor norms, and their interconnections.  A brief background on operator spaces, so-called quantized Banach spaces, will be provided along the way.

This is part of the Young Women in C*-Algebras, August 5-6.