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 − 2 T ‖ ⩽ 1 . So, the numerical range of the elements of our contractive approximate identity is contained in the closed disk center 1 2 and radius 1 2 . 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.