The internal tensor product of Hilbert -modules is a generalisation of the algebraic notion of a tensor product of modules.
Let and be -algebras, and let be a right Hilbert -module and be a right Hilbert -module, and finally let be a -homomorphism.
Form the algebraic tensor product which is the quotient of the vector space tensor product by the subspace spanned by elements of the form for . We define a -valued inner product by
It is easy to check that this respects the quotient by . Less obvious is:
If is such that then .
To do.
It follows that we have a -valued inner product on which turns into a Hilbert C-module over , which is denoted by . This is the internal tensor product of and (using ).
To do.
Lance’s book, Chapter 4.
Created on March 27, 2018 at 16:23:23. See the history of this page for a list of all contributions to it.