SAMS Subject Classification Number: 11

Characterizations of the extreme points (of the unit balls) of various Banach function spaces have often proved useful in obtaining structural descriptions of isometries on those spaces (see for example, [1] and [2]). In this talk we look at how this technique can be employed in the setting of the noncommutative space $L^1+L^{\mathrm{\infty }}(\mathcal{A})$, where $\mathcal{A}$ is a semi-finite von Neumann algebra.

[1] N.L. Carothers, S.J. Dilworth and D.A. Trautman, On the geometry of the unit spheres of the Lorentz spaces $L^{w,1}$, Glasgow Math. J. 34 (1992), 21-25.

[2] R. Grzaślewicz and H.H. Schaefer Surjective isometries of $L^1\cap L^{\mathrm{\infty }}[0,\mathrm{\infty })$ and $L^1+L^{\mathrm{\infty }}[0,\mathrm{\infty })$, Indag. Math. (N.S.) 3(2) (1992), 173-178.