4 The inner product for the multi-matrix algebra
4.1 On
Given a linear functional
Given a linear functional
Given C
As any matrix
Given a linear functional
Here,
Given a linear functional
Here,
If
Given a positive definite matrix
A linear functional
Given a linear functional
Again,
Given a linear functional
is positive and faithful, is pos-def and , defines an inner product on .
4.2 On
For each
Let
By Theorem 4.1.10, we define the inner product on each
for all
We define the inner product on
for all
The adjoint of
Recall that, given an orthonormal basis
4.3 The modular automorphism
Let
Given
For any
In other words,
For any
For any
For any
For any
We clearly get
For any
From Proposition 3.1.4, we see that our Hilbert space on
For any
4.4 Multiplication composed with co-multiplication
Given