Let X and Y be two nxn Hermitian matrices. In the article "Proof of a conjectured exponential formula" (Linear and Multilinear Algebra (19) 1986, 187-197) R. C. Thompson proved that there exist two nxn unitary matrices U and V such that $$ e^{ i X}e^{ i Y}=e^{ i (UXU^*+VBV^*)}. $$ In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II$_1$ factor.
Categories: math.FA