Abstract. We survey the most recent results on the extension of isometries between special subsets of the unit spheres of $C^*$-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$-Schatten--von Neumann spaces, with special interest on Tingley's problem.

DOI: 10.14232/actasm-018-255-0

AMS Subject Classification (1991): 47B49; 46A22, 46B20, 46B04, 46A16, 46E40

Keyword(s): Tingley's problem, extension of isometries, von Neumann algebra, $p$-Schatten--von Neumann, trace class operators

Received January 5, 2018 and in final form March 7, 2018. (Registered under 5/2018.)