Absence of characters on an operator algebra with isomorphic complementary summands
= Absence of characters on an operator algebra with isomorphic complementary summands
If $X=Y\oplus Z$ and the Banach spaces $Y$ and $Z$ are isomorphic, then the operator algebra $\mathcal B(X)$ has no <character of an algebra>[characters]. The two complementary projections have unequal character values, while mutually inverse off-diagonal maps force those values to be equal.