Crossed product of operator algebra
Takasi Turumaru · Tohoku Mathematical Journal · 1958
Among various notions of the modern ring theory the idea of the crossed product of an algebra by its group of automorphisms is seemed to be not yet explicitly introduced.J.von Neumann's method of construction of the example of factor shows us the possibility and the way of introducing this notion.In this paper, we define this notion in a C*-algebra ( § 1), in a unitary algebra ( § 2), and in a special TF*-algebra ( § 3).We shall mainly concern with the representation of the crossed product, and finally show that some of the examples of factors by von Neumann can be considered as our just defined crossed product ( §4).We only interpretate the von Neumann's example from the view-point of the crossed product, and we don't discuss further problems, for example, "For what kind of T7*-algebra and its group of automorphisms, does the crossed product produce the unknown new factor ?" (For some of these problems, cf.N.Suzuki [4]).