Abstract: This chapter has three main goals. First, to examine Hilbert C*-modules (and their W*-algebra variant, W _*-modules) as operator modules. It aims to show that the theory of C _*-modules fits comfortably into the operator module framework. Second, to consider space X. In particular, it will discuss the noncommutative Shilov boundary (X) of X. TRO methods and this Shilov boundary provide important insights into the structure of X. Third, to illustrate how C*-module and TRO methods can lead to interesting results about operator spaces. Notes and historical remarks are presented at the end of the chapter.

Keywords: Hilbert C*-modules, noncommutative Shilov boundary, W*-modules, C*-module maps, operator space multipliers, operator modules,