On the quest for positivity in operator algebras |
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Authors: | CJ Read |
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Institution: | Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, United Kingdom |
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Abstract: | 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. |
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Keywords: | Operator algebra Banach algebra Contractive approximate identity Positive operator Numerical range |
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