Norm (group)

Abstract: By Carl D Schlichting and Massimo Pigliucci Sunderland, MA: Sinauer 1998 387 pp ISBN 0-87893-799-4 $3895 (paper)

Abstract: In the present paper, we will study on the projection of norm one from any W*-algebra onto its subalgebra. By a projection of norm one we mean a projection mapping from any Banach space onto its subspace whose norm is one. At first, we find some properties of a projection of norm one from a C*-algebra to its C*-subalgebra. These properties turn out to have some interesting applications to the recent theory of W*-algebras, which we shall show in the following. Through our discussions we denote the dual of a Banach space M and the second dual by M’ and M", respectively. Theorem 1. Let M be a C*-algebra with a unit and N its C*subalgebra. If r is a projection of norm one from M to N, then 1. r is order preserving, 2. r(axb)--ar(x)b for all a, beN, 3. r(x).r(x) r(x.x) for all x e M. Proof. Consider the second dual of M and N, M" and N". M" is a W*-algebra containing M as a a-weakly dense C*-subalgebra by Sherman’s theorem (cf. [14, 15), and N" may be considered as a W*-subalgebra of M", for it is identified with the bipolar of N in M". The second transpose of r, the extension of r to M", is a projection of norm one from M" to N". Thus, it suffices to prove the theorem when M is a W*-algebra and N a W*-subalgebra of M. As in [5, Lemrna 8 we can show that r is ,-preserving and order preserving, which one can easily see since r is of norm one. Next, take a projection e of N and a eM, positive and ilai[_<1. We have e_> eae, whence e_> r(eae), so that r(eae)er(eae)e. Thus, we have r(exe)-er(exe)e for all xeM. Take an element xM, ltxlt_l. Put r(ex(1--e))-x’. Then il ex(1-e)+ne ]i-ll {ex(1-e)+ne}[(1-e)x.e+ne} il ]i ex(1-e)x.e/ne ]]/ <_ (1 +n) for all integers n. On the other hand, if ex’e+ ex’*e 0 we may suppose without loss of 2 generality that this element has a positive spectrum >0. Then, il x’Ene ]]ii ex’e+ ne +ex’(1--e)E(1--e)x’eE(1--e)x’(1--e) [[