C*-algebra Positive Element Invertibility Criteria in Terms of L<inf>1</inf>-norms Equivalence and L <inf>∞</inf>-norms Equivalence

Abstract

© 2019, Pleiades Publishing, Ltd. We prove that the L1-norms associated with a positive elements aα is equivalent to the L1-norm associated with a positive element aβ if and only if it is equivalent to the natural norm of a C*-algebra, which means the invertibility of a in the case of unital C*-algebra. The same propositions are proved for L∞-norms associated with positive element aα and aβ.