# Grothendieck Inequality, Random matrices and Quantum Expanders

Lecture 1: Grothendieck's inequality in the XXIst century

In a famous 1956 paper, Grothendieck proved a fundamental inequality involving the scalar products of sets of unit vectors in Hilbert space, for which the value of the best constant KG(called the Grothendieck constant) is still not known. Surprisingly, there has been recently a surge of interest on this inequality in Computer Science, Quantum physics and Operator Algebra Theory. The first talk will survey some of these recent developments.

Lecture 2: Non-commutative Grothendieck inequality

Grothendieck conjectured a non-commutative version of his “Fundamental theorem on the metric theory of tensor products”, which was established by the author (1978) and Haagerup (1984). This gives a factorization of bounded bilinear forms on C*-algebras. More recently, a new version was found describing a factorization for completely bounded bilinear forms on C*-algebras, or on a special class of operator spaces called “exact”. We will review these results, due to the author and Shlyakhtenko (2002) and also Junge, Haagerup-Musat (2012) , Regev-Vidick (2014).

Lecture 3: The importance of being exact

The notion of an exact operator space (generalizing Kirchberg's notion for C*-algebras) will be discussed in connection with versions of Grothendieck's inequality in Operator Space Theory. Random matrices (Gaussian or unitary) play an important role in this topic.

Lectures 4 and 5: Quantum expanders

Quantum expanders will be discussed with several recent applications to Operator Space Theory. They can be related to “smooth” points on the analogue of the Euclidean unit sphere when scalars are replaced by N×N-matrices. The exponential growth of quantum expanders generalizes a classical geometric fact on n-dimensional Hilbert space (corresponding to N=1). Miscellaneous applications will be presented:

--to the growth of the number of irreducible components of certain group representations in the presence of a spectral gap,

--to the metric entropy of the metric space of all n-dimensional normed spaces for the Banach-Mazur distance or its non-commutative (matricial) analogues,

--to tensor products of C*-algebras.