Generalized bicircular projections on JB∗-triples

Let A be a JB∗-triple and let P:A→A be a linear projection. It is proved that P+λ(Id-P) is an isometry for some modulus one complex number λ≠1 if and only if either λ=-1, or P is hermitian. It is also proved that every rank one bicontractive projection on A is hermitian. The particular case when A is a C∗-algebra is discussed through several examples.