On certain binary functions in operator algebras (Q1815653)

As seen by the authors [Math. Ann. 239, 97-110 (1979; Zbl 0417.34097), J. Differ. Equation 78, No. 1, 136-143 (1989; Zbl 0671.34013) and ibid. 130, No. 2, 356-367 (1996; review below)], certain binary expressions \[ \lceil p,q\rceil_a,\quad\lceil p,q\rceil_b, \quad\lceil p,q\rceil_c,\quad\lceil p,q\rceil_d \] play a fundamental role in the theory of differential equations in operator algebras. Their usefulness depends on equivalences such as \[ \lceil p,q\rceil_b=\lceil p,q\rceil_d,\quad\lceil p,q\rceil_a=\lceil p,q\rceil_c \] that are by no means obvious. The first of these is used in the first cited paper and is proved there. The second, which is needed in the last cited paper, is more subtle and the proof was postponed with a view to presenting it separately.