Non-commutative standard polynomials applied to matrices

The Amitsur–Levitski Theorem tells us that the standard polynomial in 2n   non-commuting indeterminates vanishes identically over the matrix algebra Mn(K)Mn(K). For K=RK=R or CC and 2≤r≤2n−12≤r≤2n−1, we investigate how big Sr(A1,…,Ar)Sr(A1,…,Ar) can be when A1,…,ArA1,…,Ar belong to the unit ball. We privilege the Frobenius norm, for which the case r=2r=2 was solved recently by several authors. Our main result is a closed formula for the expectation of the square norm. We also describe the image of the unit ball when r=2r=2 or 3 and n=2n=2.