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Algebraic probability spaces

28 June, 2014 in expository, math.CA, math.CT, math.DS, math.PR, math.RA | Tags: , , , | by | 14 comments

As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet {(X, {\mathcal X}, \mu)}, where {X} is a set, {{\mathcal X}} is a {\sigma}-algebra of subsets of {X}, and {\mu: {\mathcal X} \rightarrow [0,1]} is a countably additive probability measure on {{\mathcal X}}. Given such a space, one can form a number of interesting function spaces, including

  • the (real) Hilbert space {L^2(X, {\mathcal X}, \mu)} of square-integrable functions {f: X \rightarrow {\bf R}}, modulo {\mu}-almost everywhere equivalence, and with the positive definite inner product {\langle f, g\rangle_{L^2(X, {\mathcal X}, \mu)} := \int_X f g\ d\mu}; and
  • the unital commutative Banach algebra {L^\infty(X, {\mathcal X}, \mu)} of essentially bounded functions {f: X \rightarrow {\bf R}}, modulo {\mu}-almost everywhere equivalence, with {\|f\|_{L^\infty(X, {\mathcal X}, \mu)}} defined as the essential supremum of {|f|}.

There is also a trace {\tau = \tau_\mu: L^\infty(X, {\mathcal X}, \mu) \rightarrow {\bf C}} on {L^\infty} defined by integration: {\tau(f) := \int_X f\ d\mu}.

One can form the category {\mathbf{Prb}} of classical probability spaces, by defining a morphism {\phi: (X, {\mathcal X}, \mu) \rightarrow (Y, {\mathcal Y}, \nu)} between probability spaces to be a function {\phi: X \rightarrow Y} which is measurable (thus {\phi^{-1}(E) \in {\mathcal X}} for all {E \in {\mathcal Y}}) and measure-preserving (thus {\mu(\phi^{-1}(E)) = \nu(E)} for all {E \in {\mathcal Y}}).

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 {({\mathcal A}, \tau)} where

  • {{\mathcal A}} is a unital commutative real algebra;
  • {\tau: {\mathcal A} \rightarrow {\bf R}} is a homomorphism such that {\tau(1)=1} and {\tau( f^2 ) \geq 0} for all {f \in {\mathcal A}};
  • Every element {f} of {{\mathcal A}} is bounded in the sense that {\sup_{k \geq 1} \tau( f^{2k} )^{1/2k} < \infty}. (Technically, this isn’t an algebraic property, but I need it for technical reasons.)

A morphism {\phi: ({\mathcal A}_1, \tau_1) \rightarrow ({\mathcal A}_2, \tau_2)} is a homomorphism {\phi^*: {\mathcal A}_2 \rightarrow {\mathcal A}_1} which is trace-preserving, in the sense that {\tau_1(\phi^*(f)) = \tau_2(f)} for all {f \in {\mathcal A}_2}.

For want of a better name, I’ll denote the category of algebraic probability spaces as {\mathbf{AlgPrb}}. 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 {({\mathcal A}, \tau)^{op}} rather than {({\mathcal A},\tau)}; another suggestive (but slightly inaccurate) notation, inspired by the language of schemes, would be {\hbox{Spec}({\mathcal A},\tau)} rather than {({\mathcal A},\tau)}. However, we will not adopt these conventions here, and refer to algebraic probability spaces just by the pair {({\mathcal A},\tau)}.

By the previous discussion, we have a covariant functor {F: \textbf{Prb} \rightarrow \textbf{AlgPrb}} that takes a classical probability space {(X, {\mathcal X}, \mu)} to its algebraic counterpart {(L^\infty(X, {\mathcal X},\mu), \tau_\mu)}, with a morphism {\phi: (X, {\mathcal X}, \mu) \rightarrow (Y, {\mathcal Y}, \nu)} of classical probability spaces mapping to a morphism {F(\phi): (L^\infty(X, {\mathcal X},\mu), \tau_\mu) \rightarrow (L^\infty(Y, {\mathcal Y},\nu), \tau_\nu)} of the corresponding algebraic probability spaces by the formula

\displaystyle F(\phi)^* f := f \circ \phi

for {f \in L^\infty(Y, {\mathcal Y}, \nu)}. One easily verifies that this is a functor.

In this post I would like to describe a functor {G: \textbf{AlgPrb} \rightarrow \textbf{Prb}} which partially inverts {F} (up to natural isomorphism), that is to say a recipe for starting with an algebraic probability space {({\mathcal A}, \tau)} and producing a classical probability space {(X, {\mathcal X}, \mu)}. 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 {F} and {G} 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 {G}, with details postponed to below the fold.

  1. Starting with an algebraic probability space {({\mathcal A}, \tau)}, form an inner product on {{\mathcal A}} by the formula {\langle f, g \rangle := \tau(fg)}, and also form the spectral radius {\rho(f) :=\lim_{k \rightarrow \infty} \tau(f^{2^k})^{1/2^k}}.
  2. The inner product is clearly positive semi-definite. Quotienting out the null vectors and taking completions, we arrive at a real Hilbert space {L^2 = L^2({\mathcal A},\tau)}, to which the trace {\tau} may be extended.
  3. Somewhat less obviously, the spectral radius is well-defined and gives a norm on {{\mathcal A}}. Taking {L^2} limits of sequences in {{\mathcal A}} of bounded spectral radius gives us a subspace {L^\infty = L^\infty({\mathcal A},\tau)} of {L^2} that has the structure of a real commutative Banach algebra.
  4. The idempotents {1_E} of the Banach algebra {L^\infty} may be indexed by elements {E} of an abstract {\sigma}-algebra {{\mathcal B}}.
  5. The Boolean algebra homomorphisms {\delta_x: {\mathcal B} \rightarrow \{0,1\}} (or equivalently, the real algebra homomorphisms {\iota_x: L^\infty \rightarrow {\bf R}}) may be indexed by elements {x} of a space {X}.
  6. Let {{\mathcal X}} denote the {\sigma}-algebra on {X} generated by the basic sets {\overline{E} := \{ x \in X: \delta_x(E) = 1 \}} for every {E \in {\mathcal B}}.
  7. Let {{\mathcal N}} be the {\sigma}-ideal of {{\mathcal X}} generated by the sets {\bigcap_n \overline{E_n}}, where {E_n \in {\mathcal B}} is a sequence with {\bigcap_n E_n = \emptyset}.
  8. One verifies that {{\mathcal B}} is isomorphic to {{\mathcal X}/{\mathcal N}}. Using this isomorphism, the trace {\tau} on {L^\infty} can be used to construct a countably additive measure {\mu} on {{\mathcal X}}. The classical probability space {(X, {\mathcal X}, \mu)} is then {G( {\mathcal A}, \tau )}, and the abstract spaces {L^2, L^\infty} may now be identified with their concrete counterparts {L^2(X, {\mathcal X}, \mu)}, {L^\infty(X, {\mathcal X}, \mu)}.
  9. Every algebraic probability space morphism {\phi: ({\mathcal A}_1,\tau_1) \rightarrow ({\mathcal A}_2,\tau_2)} generates a classical probability morphism {G(\phi): (X_1, {\mathcal X}_1, \mu_1) \rightarrow (X_2, {\mathcal X}_2, \mu_2)} via the formula

    \displaystyle \delta_{G(\phi)(x_1)}( E_2 ) = \delta_{x_1}( \phi^*(E_2) )

    using a pullback operation {\phi^*} on the abstract {\sigma}-algebras {{\mathcal B}_1, {\mathcal B}_2} that can be defined by density.

Remark 1 The classical probability space {X} constructed by the functor {G} has some additional structure; namely {X} is a {\sigma}-Stone space (a Stone space with the property that the closure of any countable union of clopen sets is clopen), {{\mathcal X}} is the Baire {\sigma}-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 {F: \textbf{Prb} \rightarrow \textbf{AlgPrb}} and {G: \textbf{AlgPrb} \rightarrow \textbf{Prb}} is given by the following assertion:

  1. There is a natural transformation from {F \circ G: \textbf{AlgPrb} \rightarrow \textbf{AlgPrb}} to the identity functor {I: \textbf{AlgPrb} \rightarrow \textbf{AlgPrb}}.

More informally: if one starts with an algebraic probability space {({\mathcal A},\tau)} and converts it back into a classical probability space {(X, {\mathcal X}, \mu)}, then there is a trace-preserving algebra homomorphism of {{\mathcal A}} to {L^\infty( X, {\mathcal X}, \mu )}, which respects morphisms of the algebraic probability space. While this relationship is far weaker than an equivalence of categories (which would require that {F \circ G} and {G \circ F} 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 {G \circ F: \textbf{Prb} \rightarrow \textbf{Prb}} is a little odd: it takes an arbitrary probability space {(X, {\mathcal X}, \mu)} and returns a more complicated probability space {(X', {\mathcal X}', \mu')}, with {X'} being the space of homomorphisms {\iota_x: L^\infty(X, {\mathcal X}, \mu) \rightarrow {\bf R}}. while there is “morally” an embedding of {X} into {X'} using the evaluation map, this map does not exist in general because points in {X} may well have zero measure. However, if one takes a “pointless” approach and focuses just on the measure algebras {({\mathcal X}, \mu)}, {({\mathcal X}', \mu')}, 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 {{\mathcal A}} may be identified with a proper subset of {L^\infty} that describes the “regular” functions (or random variables) of the space. For instance, starting with the unit circle {{\bf R}/{\bf Z}} (with the usual Haar measure and the usual trace {\tau(f) = \int_{{\bf R}/{\bf Z}} f}), any unital subalgebra {{\mathcal A}} of {L^\infty({\bf R}/{\bf Z})} that is dense in {L^2({\bf R}/{\bf Z})} will generate the same classical probability space {G( {\mathcal A}, \tau )} on applying the functor {G}, namely one will get the space {({\bf R}/{\bf Z})'} of homomorphisms from {L^\infty({\bf R}/{\bf Z})} to {{\bf R}} (with the measure induced from {\tau}). Thus for instance {{\mathcal A}} could be the continuous functions {C( {\bf R}/{\bf Z} )}, the Wiener algebra {A({\bf R}/{\bf Z})} or the full space {L^\infty({\bf R}/{\bf Z})}, but the classical space {G( {\mathcal A}, \tau )} will be unable to distinguish these spaces from each other. In particular, the functor {F \circ G} 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 {X} 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 {{\mathcal A}} to be something other than an algebra {C(X)} 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 {F, G} is as follows. Suppose one has a classical probability space {(X, {\mathcal X}, \mu)} with a measure-preserving action of an uncountable group {\Gamma}, which is only defined (and an action) up to almost everywhere equivalence; thus for instance for any set {E} and any {g, h \in \Gamma}, {T^{gh} E} and {T^g T^h E} might not be exactly equal, but only equal up to a null set. For similar reasons, an element {E} of the invariant factor {{\mathcal X}^\Gamma} might not be exactly invariant with respect to {\Gamma}, but instead one only has {T^g E} and {E} equal up to null sets for each {g \in \Gamma}. One might like to “clean up” the action of {\Gamma} to make it defined everywhere, and a genuine action everywhere, but this is not immediately achievable if {\Gamma} 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 {F}, each shift {T^g: X \rightarrow X} defines a morphism {T^g: L^\infty(X, {\mathcal X}, \mu) \rightarrow L^\infty(X, {\mathcal X}, \mu)} on the associated algebraic probability space (i.e. the Koopman operator), and then applying {G}, we obtain a shift {T^g: X' \rightarrow X'} on a new classical probability space {(X', {\mathcal X}', \mu')} which now gives a genuine measure-preserving action of {\Gamma}, and which is equivalent to the original action from a measure algebra standpoint. The invariant factor {{\mathcal X}^\Gamma} now consists of those sets in {{\mathcal X}'} which are genuinely {\Gamma}-invariant, not just up to null sets. (Basically, the classical probability space {(X', {\mathcal X}', \mu')} contains a Boolean algebra {\overline{\mathcal B}} with the property that every measurable set {A \in {\mathcal X}'} is equivalent up to null sets to precisely one set in {\overline{\mathcal B}}, allowing for a canonical “retraction” onto {\overline{\mathcal B}} that eliminates all null set issues.)

More indirectly, the functors {F, G} 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.

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