LVOV-KAPLANSKY CONJECTURE ON UTm+ WITH THE TRANSPOSE INVOLUTION

Let UTm be the algebra of all m x m upper matrices with entries in a field F. Let us consider UTm equipped with the transpose involution *. Under a mild technical assumption on F, we will show that the image of any multilinear Jordan polynomial in three variables evaluated on UTm+ = {U is an element of UTm vertical bar U* = U} is a vector space. In particular, we will determine a basis for such image. As an application, we will describe the set of values of some multilinear Jordan polynomials in four variables.