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
,
(1)