As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet , where is a set, is a -algebra of subsets of , and is a countably additive probability measure on . Given such a space, one can form a number of interesting function spaces, including
- the (real) Hilbert space of square-integrable functions , modulo -almost everywhere equivalence, and with the positive definite inner product ; and
- the unital commutative Banach algebra of essentially bounded functions , modulo -almost everywhere equivalence, with defined as the essential supremum of .
There is also a trace on defined by integration: .
One can form the category of classical probability spaces, by defining a morphism between probability spaces to be a function which is measurable (thus for all ) and measure-preserving (thus for all ).
Let us now abstract the algebraic features of these spaces as follows; for want of a better name, I will refer to this abstraction as an algebraic probability space, and is very similar to the non-commutative probability spaces studied in this previous post, except that these spaces are now commutative (and real).
Definition 1 An algebraic probability space is a pair where
- is a unital commutative real algebra;
- is a homomorphism such that and for all ;
- Every element of is bounded in the sense that . (Technically, this isn’t an algebraic property, but I need it for technical reasons.)
A morphism is a homomorphism which is trace-preserving, in the sense that for all .
For want of a better name, I’ll denote the category of algebraic probability spaces as . One can view this category as the opposite category to that of (a subcategory of) the category of tracial commutative real algebras. One could emphasise this opposite nature by denoting the algebraic probability space as rather than ; another suggestive (but slightly inaccurate) notation, inspired by the language of schemes, would be rather than . However, we will not adopt these conventions here, and refer to algebraic probability spaces just by the pair .
By the previous discussion, we have a covariant functor that takes a classical probability space to its algebraic counterpart , with a morphism of classical probability spaces mapping to a morphism of the corresponding algebraic probability spaces by the formula
for . One easily verifies that this is a functor.
In this post I would like to describe a functor which partially inverts (up to natural isomorphism), that is to say a recipe for starting with an algebraic probability space and producing a classical probability space . This recipe is not new – it is basically the (commutative) Gelfand-Naimark-Segal construction (discussed in this previous post) combined with the Loomis-Sikorski theorem (discussed in this previous post). However, I wanted to put the construction in a single location for sake of reference. I also wanted to make the point that and are not complete inverses; there is a bit of information in the algebraic probability space (e.g. topological information) which is lost when passing back to the classical probability space. In some future posts, I would like to develop some ergodic theory using the algebraic foundations of probability theory rather than the classical foundations; this turns out to be convenient in the ergodic theory arising from nonstandard analysis (such as that described in this previous post), in which the groups involved are uncountable and the underlying spaces are not standard Borel spaces.
Let us describe how to construct the functor , with details postponed to below the fold.
- Starting with an algebraic probability space , form an inner product on by the formula , and also form the spectral radius .
- The inner product is clearly positive semi-definite. Quotienting out the null vectors and taking completions, we arrive at a real Hilbert space , to which the trace may be extended.
- Somewhat less obviously, the spectral radius is well-defined and gives a norm on . Taking limits of sequences in of bounded spectral radius gives us a subspace of that has the structure of a real commutative Banach algebra.
- The idempotents of the Banach algebra may be indexed by elements of an abstract -algebra .
- The Boolean algebra homomorphisms (or equivalently, the real algebra homomorphisms ) may be indexed by elements of a space .
- Let denote the -algebra on generated by the basic sets for every .
- Let be the -ideal of generated by the sets , where is a sequence with .
- One verifies that is isomorphic to . Using this isomorphism, the trace on can be used to construct a countably additive measure on . The classical probability space is then , and the abstract spaces may now be identified with their concrete counterparts , .
- Every algebraic probability space morphism generates a classical probability morphism via the formula
using a pullback operation on the abstract -algebras that can be defined by density.
Remark 1 The classical probability space constructed by the functor has some additional structure; namely is a -Stone space (a Stone space with the property that the closure of any countable union of clopen sets is clopen), is the Baire -algebra (generated by the clopen sets), and the null sets are the meager sets. However, we will not use this additional structure here.
The partial inversion relationship between the functors and is given by the following assertion:
- There is a natural transformation from to the identity functor .
More informally: if one starts with an algebraic probability space and converts it back into a classical probability space , then there is a trace-preserving algebra homomorphism of to , which respects morphisms of the algebraic probability space. While this relationship is far weaker than an equivalence of categories (which would require that and are both natural isomorphisms), it is still good enough to allow many ergodic theory problems formulated using classical probability spaces to be reformulated instead as an equivalent problem in algebraic probability spaces.
Remark 2 The opposite composition is a little odd: it takes an arbitrary probability space and returns a more complicated probability space , with being the space of homomorphisms . while there is “morally” an embedding of into using the evaluation map, this map does not exist in general because points in may well have zero measure. However, if one takes a “pointless” approach and focuses just on the measure algebras , , then these algebras become naturally isomorphic after quotienting out by null sets.
Remark 3 An algebraic probability space captures a bit more structure than a classical probability space, because may be identified with a proper subset of that describes the “regular” functions (or random variables) of the space. For instance, starting with the unit circle (with the usual Haar measure and the usual trace ), any unital subalgebra of that is dense in will generate the same classical probability space on applying the functor , namely one will get the space of homomorphisms from to (with the measure induced from ). Thus for instance could be the continuous functions , the Wiener algebra or the full space , but the classical space will be unable to distinguish these spaces from each other. In particular, the functor loses information (roughly speaking, this functor takes an algebraic probability space and completes it to a von Neumann algebra, but then forgets exactly what algebra was initially used to create this completion). In ergodic theory, this sort of “extra structure” is traditionally encoded in topological terms, by assuming that the underlying probability space has a nice topological structure (e.g. a standard Borel space); however, with the algebraic perspective one has the freedom to have non-topological notions of extra structure, by choosing to be something other than an algebra of continuous functions on a topological space. I hope to discuss one such example of extra structure (coming from the Gowers-Host-Kra theory of uniformity seminorms) in a later blog post (this generalises the example of the Wiener algebra given previously, which is encoding “Fourier structure”).
A small example of how one could use the functors is as follows. Suppose one has a classical probability space with a measure-preserving action of an uncountable group , which is only defined (and an action) up to almost everywhere equivalence; thus for instance for any set and any , and might not be exactly equal, but only equal up to a null set. For similar reasons, an element of the invariant factor might not be exactly invariant with respect to , but instead one only has and equal up to null sets for each . One might like to “clean up” the action of to make it defined everywhere, and a genuine action everywhere, but this is not immediately achievable if is uncountable, since the union of all the null sets where something bad occurs may cease to be a null set. However, by applying the functor , each shift defines a morphism on the associated algebraic probability space (i.e. the Koopman operator), and then applying , we obtain a shift on a new classical probability space which now gives a genuine measure-preserving action of , and which is equivalent to the original action from a measure algebra standpoint. The invariant factor now consists of those sets in which are genuinely -invariant, not just up to null sets. (Basically, the classical probability space contains a Boolean algebra with the property that every measurable set is equivalent up to null sets to precisely one set in , allowing for a canonical “retraction” onto that eliminates all null set issues.)
More indirectly, the functors suggest that one should be able to develop a “pointless” form of ergodic theory, in which the underlying probability spaces are given algebraically rather than classically. I hope to give some more specific examples of this in later posts.
— 1. Details —
We now flesh out the construction of that was sketched above. The arguments here are drawn from these two previous blog posts, with some minor simplifications coming from the commutativity of the algebraic probability space.
We begin with an algebraic probability space . As indicated, we then give an inner product on by the formula
By construction we see that this is a positive semi-definite inner product. We let be the associated completion of after quotienting out by null vectors, thus is a real Hilbert space and we have an isometry with dense image. We use to denote the norm on , thus
From the Cauchy-Schwarz inequality, we see that
for all . In particular, since , we have
which implies that is a non-decreasing function of . As each is assumed to be bounded, we thus have a well-defined spectral radius
Also, from many applications of Cauchy-Schwarz we have
for and with summing to a power of two; in particular
which by the binomial theorem gives
thus is an algebra norm on ; a similar argument also gives the inequality.
From this, we see that for each , the multiplication operator on induces a self-adjoint bounded linear operator on ; of operator norm at most ; in fact, from the definition of we see that this operator cannot have norm strictly less than . Thus we have identified as a commutative normed algebra with a subalgebra of the space of bounded linear operators on , with the operator norm.
We now define to be the space of functions in which are limits (in ) of sequences in of uniformly bounded spectral radius. The associated multiplication operators are then uniformly bounded in operator norm, and converge in for fixed ; thus defines a multiplication operator that is a self-adjoint bounded linear operator on , which one can check to be independent of the choice of sequence. This identifies each element of with a self-adjoint element of ; we then define the norm of an element to be its operator norm in , thus this extends the spectral radius on . Specialising to we see that this identification of with a subset of is injective, and this gives the structure of a commutative Banach algebra, with being a module over . From construction, we also see that every closed ball in is also closed in .
Define an idempotent element of to be an element such that ; we let denote an index set for the set of idempotents . We have the following basic density result, which ensures an ample supply of idempotents:
Proof: Let ; our task is to approximate to arbitrary accuracy in norm by a finite linear combination of idempotents.
We view as a bounded self-adjoint linear operator on , which contains the unit vector . By the spectral theorem, we can find a Radon probability measure on such that
for all , and in particular that
for any polynomial . Also we see that
where denotes the -essential supremum. From this and a density argument, we have an functional calculus: given any bounded Borel function , we can find such that
and such that the map is a homomorphism. In particular, if is an indicator function, then is an idempotent. Approximating the identity function in by a finite combination of indicator functions, we obtain the claim.
Now, we can give the structure of an abstract Boolean algebra by defining intersection
and then defining union by de Morgan’s law . One can verify (somewhat tediously) that obeys the axioms of an abstract Boolean algebra, and acquires an ordering in the usual manner, with minimal element and maximal element . If , then a short computation shows that
In particular, if is a decreasing sequence in , then the are Cauchy in , and thus converge to another idempotent ; we write , and observe that this is the greatest lower bound of the . Similarly, any increasing sequence has a least upper bound .
Now we consider the Boolean homomorphisms from to the two-element Boolean algebra, or equivalently the space of finitely additive Boolean measures on . We index this space by , thus is the space of Boolean homomorphisms. From Lemma 2, every such Boolean homomorphism uniquely determines a algebra homomorphism , and conversely every homomorphism comes from exactly one such homomorphism, thus is the space of algebra homomorphisms from to .
One can identify with a closed subspace of the product space , and so by Tychonoff’s theorem is a compact space. Every element of the abstract Boolean algebra induces a subset of defined by
The map is easily seen to be a Boolean homomorphism. Let be the -algebra generated by the sets . Define a basic null set to be a subset of of the form with in such that , and let be the collection of countable unions of basic null sets. This is a -ideal of , so we may form the quotient -algebra . The map can be easily verified to be a -algebra homomorphism (not just a boolean algebra homomorphism). We claim that this homomorphism is bijective (and thus an isomorphism). Surjectivity is clear from construction. For injectivity, suppose for contradiction that there was with such that was in , that is to say that could be covered by a countable sequence of intersections with and .
By induction, we may find such that is not covered by for each . If we let , we thus see that with each , but non-empty for all . But from the ultrafilter lemma, is non-empty for each , and is also closed, so we obtain a contradiction from compactness.
From the above isomorphism, we see that every element of differs (up to an element of ) by a unique set in . We then define the measure of by the formula
One can check that this gives a countably additive probability measure. We may now associate to each finite linear combination of idempotents, an element of in such a way that the map is an algebra homomorphism with
which implies that is the essential supremum of , and is the norm of . This and Lemma 2 allow us to define maps from to and to , which one easily verifies to be isomorphisms of Banach algebras and Hilbert spaces respectively.
We have now constructed the action of the functor on algebraic probability spaces. To finish the construction of , we have to describe the classical probability morphism associated to an algebraic probability space morphism . The pullback map preserves the norm and spectral radius, and thus also extends to a Hilbert space isometry and a Banach algebra isometry . As a consequence, we also have a -algebra homomorphism . We then define by the formula
for all and ; one verifies that this indeed defines as an element of (i.e., is a Boolean algebra homomorphism. It is then a routine but tedious matter to check that is a classical probability morphism and that is a functor.
Finally, the homomorphism can viewed as a morphism from the abstract probability space to . Given a morphism , the pullback maps and are intertwined by these morphisms, so we have a natural transformation from to the identity functor, as claimed.