Finsler Module

Let be a -algebra and be its positive part. Suppose that is a complex linear space which is a left -module and , where , , and equipped with a map such that

1. The map is a norm on , and

2. for each and .

Then is called a pre-Finsler -module. If is complete then is called a Finsler module over the -algebra . This definition is a modification of one introduced by Phillips and Weaver (1998; Moslehian 2001).

For example, if is a Hilbert -module over , then together with is a Finsler module because . There are Finsler -modules can not be regarded as Hilbert -modules (Phillips and Weaver 1998).