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)
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,
(2) for all
.
Proof: If is bounded then
for some fixed real
. If
is infinite, then (2) follows immediately from (1). So, suppose that
is finite. From the C*-norm properties (corollary 6 of the post on *-algebras),
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
(3) |
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.