On the quest for positivity in operator algebras

We show that in every nonzero operator algebra with a contractive approximate identity (or c.a.i.), there is a nonzero operator T such that ‖I−T‖⩽1. In fact, there is a c.a.i. consisting of operators T with ‖I−2T‖⩽1. So, the numerical range of the elements of our contractive approximate identity is contained in the closed disk center and radius . This is the necessarily weakened form of the result for C⁎-algebras, where there is always a contractive approximate identity consisting of operators with 0⩽T⩽1 – the numerical range is contained in the real interval [0,1]. So, if an operator algebra has a c.a.i., it must have operators with a “certain amount” of positivity.