Unit-free contractive projection theorems for C⁎-, JB⁎-, and JB-algebras

Abstract

We prove that, if A is a (possibly non-unital) non-commutative JB⁎-algebra, and if π:A→A is a positive contractive linear projection, then π(A), endowed with the product (x,y)→π(xy), becomes naturally a non-commutative JB⁎-algebra, and moreover the equality($)π(π(a)•π(b))=π(a•π(b)) holds for all a,b∈A. The appropriate variant of this result, with ‘JB-algebra’ instead of ‘non-commutative JB⁎-algebra’, is also obtained. In the non-commutative JB⁎-case, the requirement of positiveness for π can be relaxed to the one that the equality ($) holds. In general, this relaxing is strict, but it is not strict if π is bicontractive. Actually, positive bicontractive linear projections on non-commutative JB⁎-algebras are fully described, and a structure theorem for bicontractive linear projections (without any extra requirement) on non-commutative JBW⁎-algebras is proved. Finally, bicontractive linear projections on C⁎-algebras are studied in detail.