Quadratic forms in unitary operators

Let u(1),...,u(n) be unitary matrices on l(2). Denote by Sigma(k=1)(n) u(k) x (u) over bar(k) the matrix A defined by A[(i,i'), (j,j')] = Sigma(k=1)(n) u(k)(i,j)<(u(k)(i',j'))over bar>, acting as a bounded operator on l(2)(N X N). In other words, A is the sum of the Kronecker products of u(k) with their complex conjugates. We show the following sharp inequality: \\Sigma(k=1)(n) u(k) x (u) over bar(k)\\ greater than or equal to 2 root n -1. As an application, we show that the natural representation rho of U(N) (N greater than or equal to 1), acting on L-2 of the unit sphere in C-N and restricted to mean zero functions, satisfies for any choice omega(1),...,omega(n) in U(N) the lower bound \\Sigma(1)(n) rho(omega(k))\\ greater than or equal to 2 root n-1. This extends a result due to Lubotzky, Phillips, and Sarnak, who proved this with SO(3) in the place of U(N). (C) 1997 Elsevier Science Inc.