I previously introduced the concept of a *-probability space as a pair consisting of a state on a *-algebra . As we noted, this concept is rather too simplistic to properly capture a noncommutative generalisation of classical probability spaces, and I will later give conditions for to be considered as a true probability space. For now, I continue the investigation of these preprobability spaces, and will look at homomorphisms in this post.
A *-homomorphism between *-algebras and is a map preserving the algebra operations,
for all and . The term `*-homomorphism’ is used to distinguish it from the concept of simple algebra homomorphisms which need not preserve the involution (the third identity above). Next, I will say that is a homomorphism of *-probability spaces and if it is a *-homomorphism from to which preserves the state,
for all .
Now, recall that for any *-probability space , we define a semi-inner product on and the associated seminorm, . Homomorphisms of *-probability spaces are clearly -isometries,
For each , the seminorm is defined as the operator norm of the left-multiplication map on , considered as a vector space with the seminorm. Homomorphisms of *-probability spaces do not need to be -isometric.
Lemma 1 If is a homomorphism of *-probability spaces then, for any ,