We previously defined the notion of positive linear maps and states on *-algebras, and noted that there always exists seminorms defining the and topologies. However, for applications to noncommutative probability theory, these are often not the most convenient modes of convergence to be using. Instead, the weak, strong, ultraweak and ultrastrong operator topologies can be used. This, rather technical post, is intended to introduce these concepts and prove their first properties.
Weak convergence on a *-probability space is straightforward to define. A net tends weakly to the limit if and only if for all . Before going into details, I note that the operator topologies are usually applied to the algebra of bounded operators on a Hilbert space so, in this post, I work in more generality than just for *-probability spaces. Let us call a pair a *-algebra representation, where is a *-algebra, is a semi-inner product space, and acts on by left multiplication in a fashion consistent with the algebra operations,
for all , and . Define the operator seminorm on ,
We will say that an element is bounded if is finite, and is bounded if is finite for all .
For a *-probability space , then we have the semi-inner product on given by , so that considering as acting on itself by left-multiplication, defines a *-algebra representation. The operator semi-norm on is then the same as the seminorm.
We define weak and strong topologies on . Given a collection of maps from set to topological space , they generate a topology on defined as the weakest topology making each continuous. It can be seen that this has a base consisting of the sets for and open . A net converges to a limit in the topology generated by iff for all . In the following, we use the standard topology on and the (strong) topology on defined by its seminorm.
Definition 1 Let be a *-algebra representation. Then, define the following topologies on .
- The weak topology generated by the maps , , for .
- The strong topology generated by the the maps , , for .
So, a net converges to a limit in the weak topology iff for all , and in the strong topology iff for all . As these are vector topologies, iff , so that the topology is characterised by the nets converging to zero. Convergence in the weak and strong topologies are related in straightforward way.
Lemma 2 A net tends strongly to zero iff tends weakly to zero.
Proof: If strongly then, for all , and tend to zero and, hence,
Therefore, weakly. Conversely, if weakly then, for all ,
and, hence, strongly. ⬜
I look at continuity of the algebra operations with respect to the weak and strong topologies.
Lemma 3 Let be a *-algebra representation. Then, the vector space operations
are jointly continuous, using either the weak or strong topology for . The involution
is weakly continuous. The operator product
is individually continuous in each of , using either the weak or strong topology for . Furthermore, it is jointly strongly continuous if is restricted to a norm-bounded subset of .
Proof: For each , the maps
are jointly continuous using the weak topology on , as multiplication and addition are continuous in . Hence (1) are jointly weakly continuous. Similarly, the maps
are jointly continuous using the strong topology on , as scalar multiplication and addition are continuous in . Hence (1) are jointly strongly continuous. Next, the map
is weakly continuous, as complex conjugation is continuous in . So, (2) is weakly continuous. Also,
is weakly continuous in and in separately, so (3) is weakly continuous in and separately. Similarly,
is strongly continuous in for each fixed , so (3) is strongly continuous in . Finally, if are nets converging strongly to and respectively, with for some real then,
Hence, (3) is jointly strongly continuous when restricted to . ⬜
Notably, involution is not strongly continuous. For example, taking to be the space of sequences of complex numbers with , this has inner product . The shift maps given by converge strongly to zero. However, their adjoints are isometries, , so do not converge strongly. Sometimes, an additional operator topology is used, the *-strong topology, under which a net converges, if and only if it converges in the strong topology and the adjoints converge strongly. This forces the adjoint map to be *-strong continuous. I will not make use of this topology here.
When applied to unbounded sets, the weak and strong topologies are actually slightly too weak for our purposes. So, we define a couple of extra topologies — the ultraweak and ultrastrong topologies — by enlarging the vector space . For any semi-inner product space, define
An element of is a sequence such that is finite. For elements and of , define the semi-inner product,
This sum is absolutely convergent since,
This makes into a semi-inner product space which is, respectively, a true inner product space or a Hilbert space, whenever is. For any bounded linear map , then we can also define to act diagonally on by . It can be seen that the operator norm of acting on is the same as its action on . Hence, if is a bounded *-algebra representation, then also defines a *-algebra representation.
If is a bounded *-algebra representation, then the ultraweak and ultrastrong topologies on are just the weak and strong topologies defined with respect to the representation .
Definition 4 Let be a bounded *-algebra representation. Then, define the following topologies on .
- The ultraweak topology generated by the maps , , for .
- The ultrastrong topology generated by the maps , , for .
So, a net converges ultraweakly to a limit iff
for all , and converges ultrastrongly to iff
for all .
These topologies, and the use of , may seem a bit mysterious at first sight. Other than the simple fact that these definitions turn out to be useful, I will give a brief justification. The collection of maps of the form , for , is not closed under linear combinations. Consider the set of all linear combinations of such maps, denoted for now by . This is a vector space of bounded linear functionals of the form
for finite sequences so, under the operator norm, is a normed space. The weak topology on is then generated by . The completion, , of is a Banach space of bounded linear functions , which can be shown to be of the form for . It is natural to want all functions in to be continuous, for which we would use the topology generated by , but this is precisely the ultraweak topology. See, also, lemma 12 below which, for classical probability spaces, gives a simple description of the maps for and identifies the ultraweak topology with .
Applying lemma 2 to relates the ultraweak and ultrastrong topologies.
Lemma 5 A net tends ultrastrongly to zero iff tends ultraweakly to zero.
Similarly, applying lemma 3 to the action of on extends it to the ultraweak and ultrastrong topologies.
Lemma 6 Let be a bounded *-algebra representation. Then, the vector space operations
are jointly continuous, using either the ultraweak or ultrastrong topology for . The involution
is ultraweakly continuous. The operator product
is individually continuous in each of , using either the ultraweak or ultrastrong topology for . Furthermore, it is jointly ultrastrongly continuous if is restricted to a norm-bounded subset of .
The ordering between topologies is straightforward to establish. Note that, despite the terminology, the ultraweak topology is stronger than the weak topology.
Lemma 7 The topologies on are ordered as follows, where the arrows point from stronger to weaker topologies.
ultrastrong ⇒ ultraweak ⇓ ⇓ strong ⇒ weak
Proof: Strong ⇒ Weak: Suppose that the net tends strongly to . For any , and, hence, , so weakly.
Ultrastrong ⇒ Ultraweak: Apply `strong ⇒ weak’ to .
Ultraweak ⇒ Weak: Suppose that the net tends ultraweakly to . For , define by and . Then,
and, hence, weakly.
The closed unit ball of is defined with respect to the operator seminorm,
which is a closed subset of under the operator topologies.
Lemma 8 The unit ball is closed under the weak, strong, ultraweak, and ultrastrong topologies.
Proof: As the weak topology is weaker than all the others, it is sufficient to show that is weakly closed. Consider a net converging to a limit . Then, for ,
showing that , so . ⬜
On bounded sets, the weak and strong topologies coincide with the `ultra’ topologies. Note that a vector topology restricted to is uniquely determined by the convergence of nets to zero, since converges to a limit iff tends to zero.
Lemma 9 If is a bounded *-algebra representation then,
- the weak and ultraweak topologies coincide on .
- the strong and ultrastrong topologies coincide on .
Proof: Let be a net converging weakly to zero. For any ,
For each , by weak convergence, so, by dominated convergence, . Hence ultraweakly.
We note that, to show that a linear map is of the form for some , it is enough to express it as
for sequences such that is finite. In this case, by scaling, we can find satisfying and . Then,
So, defining and in , then .
If we have a bounded *-probability space then, as previously noted, has semi-innner product . So, is a *-algebra representation, with acting on itself by left-multiplication. Hence, the definitions and results here apply to which is, of course, the whole point of this post.
Lemma 10 Let be a bounded *-probability space. Then, there exists such that .
Proof: If is unitial, then the result is trivial, as we can write . More generally, by lemma 10 of the post on states, there exists a sequence with and , in which case, taking ,
However, again by lemma 10, is -Cauchy, so we can pass to a subsequence with for all . So,
Next, for a *-probability space, the fact that it has a semi-inner product means that we have further topologies on . First, there is the seminorm topology, also called the strong topology, so that tends to a limit iff . There is also a weak topology, where iff for all . These topologies are related to the ones described above as follows.
Lemma 11 Let be a bounded *-probability space. Then,
- the ultraweak topology on is stronger than the weak topology.
- the ultrastrong topology on is stronger than the strong topology.
If is unitial then,
- the weak topology on is stronger than the weak topology.
- the strong topology on is stronger than the strong topology.
If is tracial, so that for all , then,
- the weak topology on is stronger than the weak topology.
- the strong topology on is stronger than the strong topology.
- the weak, ultraweak and weak topologies coincide on .
- the strong, ultrastrong and strong topologies coincide on .
If is tracial and is unitial then, the weak and weak topologies coincide, and the strong and strong topologies coincide.
Proof: Suppose that net tends to zero weakly. If is unitial then this implies that tends to zero for all , so -weakly.
Suppose that strongly. If is unitial then this implies that tends to zero -strongly.
Suppose that ultraweakly, then ultraweakly for any . By lemma 10, is ultraweakly continuous, so , and -weakly.
Suppose that ultrastrongly. Lemma 5 says that ultraweakly and, as is ultraweakly continuous, , so -strongly.
Suppose that in the weak topology. If is tracial then, for ,
showing that weakly.
Suppose that in the strong topology. If is tracial then, for ,
Finally, if is tracial then, from the above, the weak topology lies between the weak and ultraweak so, as lemma 9 says that the weak and ultraweak topologies coincide on , all three topologies coincide on . Similarly, the strong, strong , and ultrastrong topologies coincide on . ⬜
So, for tracial states, the weak, ultraweak and weak topologies coincide on , as do the strong, ultrastrong, and strong topologies. For this reason, when dealing only with tracial *-probability spaces, it is possible to get by with just the topology. On the other hand, for general states, the topologies can be strictly weaker. Example 5 of the post on states gives a *-probability space and self adjoint such that for all . Hence, trivially in the strong (and weak) topologies. However, there exists (unitary) with for all , so that does not tend to zero in the operator topologies.
These additional topologies can be added to the ordering described by lemma 7.
To demonstrate the definitions given above, I apply them to the case of a classical probability space . Use to denote the space of uniformly bounded (and measurable) complex-valued random variables , for the integrable random variables, and for the square-integrable random variables. It is usual to identify any two random variables which are almost surely equal. Then, it is standard theory that is a Banach space under the norm , with dual space under the essential supremum norm, , and under the pairing for and . The topology on is the weakest topology such that is continuous for each . This is also called the weak topology on although, to avoid confusion with the weak topology defined above, I do not use this terminology . On , we have an associated norm .
Lemma 12 Let be a probability space. With respect to the *-probability space , the following are equivalent for any map ,
- , for some .
- , for some .
- the ultraweak topology on coincides with the topology.
- the strong topology on coincides with the norm topology.
- the ultrastrong and strong topologies on the unit ball of coincide with convergence in probability.
Proof: Suppose that for some , with as an inner product space. Then,
Hence, setting , by monotone convergence,
So, and, by dominated convergence,
Conversely, suppose that for some . Write for with . Then, set and . As these are uniformly bounded, they are in and, by monotone convergence,
So, we can define and in . By dominated convergence,
We have shown equivalence of the two statements for . Hence, from the definition of the ultraweak topology, it is the weakest topology on such that is continuous for all . By definition, this is the same as the topology.
Next, as is unitial and commutative, the strong and strong topologies coincide by lemma 11.
Finally, we show that the strong and ultrastrong topologies and convergence in probability agree on the unit ball of . First, the dominated convergence theorem says that, on the unit ball, the -norm topology coincides with convergence in probability and, as we showed above, also agrees with the strong topology. Also, the strong and ultrastrong topologies coincide by lemma 9. ⬜
Lastly, I will note some further basic properties relating the operator topologies, the proofs of which are slightly out of scope here but which will be included in more detail in a later post. Given a bounded *-algebra representation , use to denote the closure of under, respectively, the weak, strong, ultraweak, and ultrastrong topologies. For example, lemma 9 above can be expressed by the identities
for all -bounded sets . Now, for the special case where is a Hilbert space, then it is well-known that the weak and strong topologies are compatible. This means that a linear map is weakly continuous if and only if it is strongly continuous. By the Hahn–Banach theorem, this is equivalent to both topologies having the same closed convex sets, so for convex . Considering the action on , it is similarly true that . Using Hilbert space completions, this last statement can be extended to the general situation where is only a semi-inner product space. So, we have the identity
for all convex . Next, as lemma 9 states that the unit ball is closed under each of the operator topologies, we have where is any of the operator topologies. For the special case where is a Hilbert space, the Kaplansky density theorem states that, if is a *-subalgebra, then the reverse inequality holds for the strong topology and, by considering the action on , for the ultrastrong topology. In this form, the statement generalizes to arbitrary semi-inner product spaces . Combining with (4) and (5), we have the following satisfying string of equalities for any *-subalgebra of ,