"Operator Algebras, Quantum Information Theory and the Grothendieck Program"
I will discuss recently emerged interconnections between operator algebras and quantum information theory, leading to reformulations of one of the most important open questions in operator algebras: the Connes embedding problem. The lectures are based on joint work with Uffe Haagerup, and are dedicated to his memory.
April 5, 2016
Classical and Quantum Correlations, Operator Algebras, and Asymptotic Behaviour of Quantum Channels
In 1980, Tsirelson showed that Bell’s inequalities—that have played an important role in distinguishing classical correlations from quantum ones, and that were used to test, and ultimately disprove the Einstein-Podolski-Rosen postulate of “hidden variables", coincide with Grothendieck’s famous inequalities from functional analysis. Tsirelson further studied sets of quantum correlations arising under two different assumptions of commutativity of observables. While he showed that they are the same in the finite dimensional case, the equality of these sets was later proven to be equivalent to the most famous still open question in operator algebras theory: the Connes embedding problem. In recent joint work with Haagerup, we establish a different reformulation of the Connes embedding problem in terms of an asymptotic property of quantum channels possessing a certain factorizability property (that originates in operator algebras). Several concrete examples will be discussed.
Following Tuesday's lecture, there will be a reception in honor of Professor Musat at the Koenig Alumni Center, 1202 University Avenue (the SE corner of Broadway and University).
April 6, 2016
Grothendieck Inequalities — From Classical to Noncommutative
The highlight of Grothendieck’s celebrated “Résumé", published in 1956, is a highly nontrivial factorization result for bounded bilinear forms on C(K1) C(K2), where K1 and K2 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., L2), and the Banach spaces L1, respectively, C(K), and L1. 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 completely bounded bilinear forms on C -algebras and operator spaces, due to Pisier-Shlyakhtenko, and joint work of Haagerup and myself. I will also point out connections with quantum information theory and reformulations of the Connes embedding problem.
| Magdalena Musat is an Associate Professor in Mathematics at the University of Copenhagen, Denmark. A native of Romania, she earned her undergraduate degree from the University of Bucharest in 1993, and her Ph.D. from the University of Illinois at Urbana-Champaign in 2002, under the joint supervision of Donald Burkholder and Marius Junge. Before her appointment in 2009 at the University of Copenhagen, she was an S.E.Warschawski Visiting Assistant Professor at the University of California, San Diego, an Assistant Professor at the University of Memphis, and a Lektor (Associate Professor) at the Southern Denmark University (SDU).
Musat’s mathematical interests lie at the interface between operator algebras, noncommutative probability (in particular, Lp-theory of noncommutative martingales), and some of the analytic, geometric and probabilistic aspects of group theory related to operator algebras. A postdoctoral position at SDU in 2006 marked the beginning of a fruitful collaboration with Uffe Haagerup. Musat has regularly been an invited speaker at international conferences and workshops, and enjoys sharing her enthusiasm for mathematics with colleagues and her many students. She is the organizer of the Distinguished Harald Bohr Lecture Series in Mathematics at the University of Copenhagen.
This Lecture Series is funded by an endowment given by Professor Ira M. DeLong, who came to the University of Colorado in 1888 at the age of 33. Professor DeLong essentially became the mathematics department by teaching not only the college subjects but also the preparatory mathematics courses. Professor DeLong was a prominent citizen of the community of Boulder as well as president of the Mercantile Bank and Trust Company, organizer of the Colorado Education Association, and president of the charter convention that gave Boulder the city manager form of government in 1917. After his death in 1942 it was decided that the bequest he made to the mathematics department would accumulate interest until income became available to fund DeLong prizes for undergraduates and DeLong Lectureships to bring outstanding mathematicians to campus each year. The first DeLong Lectures were delivered in the 1962-63 academic year.