Norm of Sums of Peirce Projections

Abstract

SlovenijaPeirce projections P 1 , P 1/2 and P 0 are one of the fundamental technical tools in the theory of JB*-triples. It is well known that all three of them are contractive. We show that the sum of two Peirce projections need not be contractive. We also give the upper estimate for the norm of such a sum, valid in all JC*-triples, and present a conjecture of the exact value of this norm, based on some numerical experiments. 521 Asian-European J. Math. 2009.02:521-531. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/21/15. For personal use only. 522 B. ZalarIt is possible to develop a theory of JBW*-factors, which is somewhat richer than the theory of von Neumann algebras. There exist 6 families of type I factors. Four of them are infinite. Factors of type B(H, K) are called rectangular. Factors of type {x ∈ B(H) : x T = ±x} are called Hermitian and symplectic respectively. Every Hilbert space with dimension ≥ 3 gives rise to a construction of a spin factor, but we do not need to go into details here. JB*-triples for which all type I representations are of one of the above four types, are called JC*-triples. Every JC*-triple can be imbedded into B(H) for a suitable Hilbert space H. There exist also two exceptional type I factors, constructed from octonion matrices. They are both finite dimensional, their dimensions being 16 and 27, and are therefore more interesting from algebraic viewpoint than a functional-analytic. It is known that neither of them is embeddable into B(H).An element u ∈ W is called a tripotent if {uuu} = u. These elements play a similar role in JB*-theory as orthogonal projections play in C*-theory. In the case of the triple B(H, K) it is not difficult to see that tripotents are precisely partial isometries from H into K. If we consider the operator T (x) = {uux}, it can be shown that the Jordan identity implies T (2T − 1)(T − 1) = 0 which shows that W decomposes asOf course this decomposition, which is called Peirce decomposition, depends on the choice of u. The corresponding Peirce projections P i : W → W , whose ranges are W i , can all be given with an explicit algebraic formula P 1 (x) = {u{uxu}u},Peirce projections are one of the most widely used technical tools in research about JB*-triples. It is known that all three Peirce projections are contractive. Moreover P 1 + P 0 is also contractive. The purpose of this paper is to show that the remaining combinations P 1 + P 1/2 and P 1/2 + P 0 are in general not contractive and to give a reasonable estimate for their norms in JC* case.The reader can find proofs of the above explained facts in classical papers [3,9,10,11,12,13,16] and surveys [7,18,19,23]. For those interested in various modern trends in JB*-theory, a starting sample is [1,2,4,5,6,8,14,17,20,21,22]. Example of a lower estimateIn this section we consider the special situation when W = B(H, K) is a JB*-triple of bounded linear operators between two complex Hilbert spaces H and K. We use a rather standard notation β ⊗ α, where α ∈ H and β ∈ K, f...