norm one approximate identities in separable C* algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:29:30Z http://mathoverflow.net/feeds/question/73194 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73194/norm-one-approximate-identities-in-separable-c-algebras norm one approximate identities in separable C* algebras Ray Curran 2011-08-19T00:30:28Z 2011-08-19T00:30:28Z <p>I'm trying to prove Corollary 1.4.9 in K. Davidson's book (Exercise 1.5):</p> <p>If A is a separable C* algebra, then there is an increasing sequence $E_i, i=1,...,\infty$ of positive norm-one elements which form an approximate identity for A. </p> <p>The hint suggests to choose $E_n$ successively so that $||E_n A_k - A_k|| &lt; \frac{1}{n}$ for all $k$ from $1$ to $n$. </p> <p>It is clear from how approximate identities are constructed in this book that this can be done with $||E_i||&lt;1$. Why can the $E_i$ be chosen to be increasing and norm = 1? </p>