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 ,
Proof: By definition of the semi-norm, there exists a sequence with and . Then, as ,
As a consequence of lemma 1, we can show that -continuous homomorphisms are isometries. Recall that a linear map between seminormed spaces and is norm-continuous (or norm-bounded) iff there exists a such that for all .
Lemma 2 A homomorphism of *-probability spaces is -bounded if and only if it is an -isometry. That is,
for all .
Letting go to infinity gives , so lemma 1 gives the result. ⬜
Most homomorphisms of interest to us will be -continuous and, so, will be isometries. There are many cases where this is guaranteed, without having to impose continuity as an explicit requirement. For example, this is always the case for commutative algebras, and for tracial states. Recall that a state is tracial if it satisfies the identity and, in particular, all states on a commutative *-algebra are tracial.
Lemma 3 Let be a homomorphism of *-probability spaces. If is commutative or, more generally, if is tracial, then is an -isometry.
Proof: If is tracial, then applying lemma 9 of the previous post,
We are also guaranteed continuity in the case where the image of the homomorphism is dense.
Lemma 4 Let be a homomorphism of *-probability spaces such that is either or dense in . Then, is an -isometry.
Proof: First, by lemma 2 of the post on states, convergence is stronger than convergence so, in either case, we have that is -dense in . We need to show that , which is trivial in the case that . So, suppose that is finite. For any , writing gives
This needs to be extended to all . First, by the assumption that is -dense, there exists a sequence converging in to . Then, for ,
So, using ,
from which we obtain inequality (3), giving as required. ⬜
Yet another case in which we are guaranteed continuity is where is -complete and separated. To say that is separated, we mean that every with is zero or, equivalently, is nondegenerate. By taking the completion, it is always possible to extend to a complete and separated *-algebra.
Lemma 5 Suppose that is a homomorphism of *-probability spaces and that is complete and separated under the topology. Then, is an -isometry.
Proof: We first extend to a homomorphism ,
Choosing any , with finite -norm, set . Then, , which is sufficient to show that has a square root. That is, for some self-adjoint . This fact can be shown using functional calculus. However, I provide a direct proof of the existence of the square root in lemma 6 below. Then,
Multiplying on the right by and on the left by ,
So, applying to both sides,
and as required. ⬜
A C*-algebra is a *-algebra together with a C*-norm , with respect to which is complete. For example, given a bounded *-probability space which is complete and separated under the norm, then it is a C*-algebra. More generally, if it is not bounded, then the collection of for which is finite will form a C*-algebra. The following result can then be used to take square roots, as required in the proof just given for lemma 5.
Lemma 6 Let be a C*-algebra, be self-adjoint, and be real. Then, for some self-adjoint .
Proof: By scaling, we can suppose that , so that , and show that has a square root in . For real , the power series expansion
converges absolutely when . In fact, this converges absolutely for all . Letting tend to -1, the right hand side goes to 0, whereas all the terms in the summation are negative. Hence,
So, the power series expansion converges absolutely for all . Hence, we can set
which, as , the sum is absolutely convergent in . Since is complete and separated, this has a unique limit . Furthermore, as involution is -continuous, any limit of self-adjoint elements must be self-adjoint. In particular, is self-adjoint. Next, by squaring the power series expansion,
the coefficients of powers of on both sides must agree. So, by comparing the coefficients of powers of in the expansion of , we obtain . ⬜
I finish off this post by showing that it is possible for homomorphisms of *-probability spaces to be unbounded with respect to the norm.
Example 1 A homomorphism of *-probability spaces and such that
This example builds on example 4 of the previous post. There, we constructed a *-probability space, which I will denote here by , and a self-adjoint element such that , but for all . For example 1 above, let be the unitial subalgebra of generated by , be the restriction of to , and be the inclusion.
In practice, we will be concerned primarily with bounded *-probability spaces — so that is finite for every . Even in this case, it is possible for homomorphisms to be unbounded.
Example 2 A homomorphism of bounded *-probability spaces and such that
This, again, builds on an example from the previous post (example 5). There, we constructed a bounded *-probability space, denoted here by , and a self-adjoint element satisfying , but for all . Example 2 follows, as above, by letting be the unitial subalgebra generated by , be the restriction of to , and be inclusion.