Isometries of unital Haagerup–Schultz algebras

Describe the isometries of the unital Haagerup–Schultz algebra L_{\log_+}(\mathcal{M},\tau) associated with a von Neumann algebra \mathcal{M} equipped with a faithful normal semifinite trace \tau, with respect to the F-norm \|x\|_{\log_+}=\int_0^1\log(1+\mu(s;x))\,ds.

Background

The paper completely characterizes not necessarily surjective linear isometries of the Haagerup–Schultz algebra \mathcal{L}{\log}(\mathcal{M},\tau), showing that they are implemented by a partial isometry composed with a trace-preserving normal Jordan *-monomorphism. It then introduces the larger unital Haagerup–Schultz algebra L{\log_+}(\mathcal{M},\tau), defined by the condition \log_+|x|\in L_1(\mathcal{M},\tau)+\mathcal{M}, and equips it with the F-norm |x|{\log+}=\int_0^1\log(1+\mu(s;x))\,ds.

The authors explicitly ask for an analogous description of the isometries of L_{\log_+}(\mathcal{M},\tau). The problem concerns the unital algebra associated with a general semifinite von Neumann algebra and is not resolved elsewhere in the paper; the preceding characterization applies to \mathcal{L}{\log}, not to the larger unital algebra L{\log_+}.