thank you very much for the kind invitation to this workshop I had kind of crude difficulties for will while providing preparing this talk because it is actually a promotional talk so I’m advertising a book project that we are about to finish with co-authors Tanya I’m Ballon d’Or sitting Justin in the back end hina Naga and so this talk is more or less than a promotional talk that you all should buy this book if it when it appears and I hope so we are now in good hope that it appears so but the problem is which community to address do i address the functional analyst the operator theorists do i address the ergodic theorists now the ERM since I’m myself from function lytic background i’m actually a newcomer to a gothic theory although i learned this now for several years but i’m still feeling myself as a newcomer so my goal with is talk and with my well the learning of a gothic series to to spot the the functional analytic aspects of a gothic theory and actually reformulate a lot of existing ergodic theory in a purely functional lytic way if that when it’s possible of course and in this way trying to lure other functional analysts into that area so who may be abhorred by some very comunitaria arguments and whatever where they are where they could be abhorred by but so to make the me to operate a theory and so I hope that other people join and also see a gothic theory as an actually actual field of an application of functional analysis so this is the this is the background so my goal now for the next ten minutes is to explain the main thrust of our approach to to a gothic theory which of course is an old approach but the difference is that well at least on my side from my side I want to apply this approach quite consequently so quite determined to follow this approach and not switch to other approaches which of course is can also be an obstacle but it could also be quite interesting whether the such an approach is possible ok so here is a a bit of setting the stage so maybe I need this I don’t know so this is my first my first section I am writing usually a capital X for a probability space and if the measure is actually mentioned I would just call the new eggs because everything is like Act the probability spaces is the underlying object and it has a state space a Sigma algebra and the probability measure and of course there’s a measure preserving transformation so measurable and preserves the probability measure and from this from this setting one device an operator that’s the koopman operator usually I abbreviate this with just with T so T is the generic symbol for equipment operator T is T Phi which this is just the composition operator and Wolfgang Argon’s language on X now this is an operator that apps l 1 of X 2 itself also at Maps L P of X 2 itself and andel infinity to itself and has a nice as a lot of properties for example it’s a so it preserves of course the one function and it preserves absolute absolute values just because everything is defined point wise and it preserves the integral so that it preserves the integrals just

that fires measure preserving so this is the the measure of a setting so we I usually take the notation for that let’s say X Phi so excess excess identify with the whole probability space in la Taylor would make a difference in the notation here at the board is a bit difficult but I think you will get it so this is the measure preserving situation of course there’s a topological analog that’s the topological dynamics and I’m introducing this tool so this is the other interesting case so you have a compact space and you have a continuous map here and then we denote this by this pair k phi and then you also have the coke-man operator here and that works on continuous functions in the same way so the f is if after five now one of the first decisions that we had to make in the book and which comes from our actually background from semi group theory is that we do not restrict to invertible transformation so we would like we see semigroups everywhere and and groups are of course special cases important special cases of semigroups but we are interested in what whether one can do theory just without an invertibility assumption so this is so let me put this here that’s kind of our EF hn philosophy where i’m not i’m okay i’m a bit patronizing here maybe maybe it’s just my philosophy but so so i tried to force enforce that in the book and i think i know was also in favor and you can ask out my two co-authors whether they actually liked it on so the one of the the first decisions is that we do not restrict to invertible system so non-invertible systems is so where possible aware we are reasonable of course so just looking at which theory can be can be done there so okay so the next point so that’s the these are the object of study now the next observation here is actually it’s about so in a gothic theory on topological dynamics very often one would restrict to metric compact spaces my tricycle compact spaces or to probability spaces where the Sigma algebra is countably generated are essentially countably generated so we do not make this restriction so my question is it’s also in developing the theory what can be said about non meat rice of case in the come for non meat rice of a compact spaces or for non-separable so to speak probability spaces so actually a non separate aside non-separable x which is which by which i mean actually that the l1 space is separable but i will i will come to this later now that the next important important the next important topic is what is an isomorphism or actually a homomorphism and how do i see this notion as an as a functional analyst now in passing from from probability spaces to the operators we we change the category so to speak so there’s a change of category here namely the change so one cast is from measure preserving

system to to the l1 space or the idea to space if you like and and ending the operator now there’s a natural notion of so actually know okay the first thing to remark is that that there is a natural notion of homomorphism here which is not present here because because he even has much more structure linear structure and so so I introduce the following class of operators which has no direct analogue on on the on the state space leather that’s what we call a mark of operator so so suppose x and y are probability spaces and so t an operator from el of x 1 of x two L one of Y is Markov and so actually when maybe by Markov or by stochastic would be more appropriate but somehow the terminology is borrowed from glasses from from illy glassons book he uses also the term mark of operators and we we thought we should not deviate from from a standard textbook talent book on that monograph on that topic so it’s mark off if T is a positive operator that means it maps positive functions to positive functions it is integral preserving and and it preserves the constant one function so that’s a mark of operators out the mark of operators form of course a convex set is a closed in the week operator topology that’s mark of operator and we call T a mark of an embedding so t is a mark of embedding and usually I would only save em embedding if if it is actually if it’s isometric which is actually equivalent in this case to being a lattice hormone office and that means that it preserves absolute values so this is actually the same as to say that tf1 sf-1 so for all F I mean every everything is quantified 0 for all ethanol but that’s that’s not so important so this is the important property here now then we see that actually so the change of category goes to the pairs at one of X with T being a mark of embedding and now we can say what a homomorphism of dynamical system should be so the classical the classical play space notions that of a factor so what is a factor so classical we would have a mapping tall of let’s say 1 probability space to the other measurable measure preserving and that’s a factor if if it commutes with the dynamic so that means that so I write Phi X for the dynamics on x + 5 y for the dynamics of Y so this is a this is the classical notion of a factor now a factor of course when we cognize it that means when we pass to the Koopman operators of this underlying map top then then we arrive at an at a mapping s which maps one of Y 2 1 of X and that’s actually a mark of embedding because it’s a good man operator and and of course it satisfies the commutation property so the tea so the TX here so this is a intertwines the two copeman operators okay so factors factors are in in one-one correspondence so you can go write that

here and that also factors are in one-one correspondence okay now know before i say this i should i should now under underpin what what’s now the difference I mean we are when we are working in this category we would apply this notion of a factor but when we apply in this category we would apply that notion of a factor and that is not linked to that actually s is a common operator of an underlying map we’re just saying okay if we have a mark of embedding into another probability space and satisfying this this would be then we would call l1 of why why would be a factor of X so so so this is already like generalizing the notion of a factor here and now in a second i will show you how general data that actually is so in this sense in this sense of a factor that means in this operator theoretic sense of a factor we have the following the following characterization the factors are 11 are in one-one correspondence with either of the following class so the closed t invariant bama sub lattices bunneh sub lattices of a1 the factors of a system are in one-one correspondence with close t marion banner sub lattices of l1 that includes in include that include the constant one function so take a bun of subtleties of Avenel ones but that is a contains the constants then it’s actually a one of a sub Sigma algebra if you like ok and of because of this grass pendants are also with a with the so called mark of projections so that’s mark of operators that are projections the square is the same the operator mark of projections beyond on a one such that well the range of P is invariant the range of P is a banner sub-clause vanassa blood is containing one and and it should be teen variants or the teen variance is this operator theoretic equality so both both collections of options are in one-one correspondence with factors now not only Sophie please turn on the on the Beamer so I am trying now I I now ask the question how actually what do we lose here do we do we lose to do I be more general here actually and that on that side and the first theorem here to state is now okay 30 seconds have to talk another 30 seconds so the first theorem to state here is saying the compact the compact case so when when you actually when when you don’t don’t you’re not in measure of the dynamics button topological dynamics you pass from a topological dynamical system to the ck dynamics you don’t lose anything so you can go back and forth it’s just the same so here the two categories are isomorphic so now the picture comes maybe you know let’s say that comes something but not a picture which it should come so beamers on but ok so so in the I’m summarizing what I will show you in a second on the slide I’ll hear this wonderful okay so the first tier in here it’s actually classical theorem so in our book mostly but it’s of course come copied from like for example Schaeffer surface book so if you have two compact spaces and you have a linear operator that preserves one then all these four statements here are equivalent so it is multiplicative that means it distributes over over a products of functions see the lattice homomorphisms that that is it satisfies this commuting with the it commutes with taking the absolute value it’s also a it’s not so important here from now it’s an extreme point of this

set of operators it’s a convex set and he’s an extreme point if and only if it’s a lot of selling morphism and the most important thing is that that T is actually a common operator of a continuous function so in if you think of passing here from from the from the dynamic situation to the operator dynamic situation you don’t lose anything so you can go back and forth files uniquely determined by T so there’s a 11 this is an isomorphism of categories so so on the topological side we’re good now what is how about the measure theoretic side and here it’s actually not true so here actually the it’s not the same so passing you cannot go back from if you have let’s say such a situation a mark of embedding on another one space it’s not always induced by an underlying state space map and there is a there’s an important subclass of of probability spaces where that actually is true and that’s due to fun Normand so the first so the phenomena theorem is actually the equivalence with three so and that is involving the involving the notion of a standard probability space which is modeled on a Polish measure space like a Polish topological space with probability measure on it and and that’s actually I think that’s the reason why one of the reasons why I got a theorists like standard probability space is so much actually there is a this one one correspondence between mark of embeddings and an underlying state space maps via the koopman procedure actually we looked at the literature we were not so convinced about the proofs we could find so we met have a new one and I can hack and highly recommend this proof in our book good so that’s the that’s the point so when we are in standard on summer probability spaces then then we are good but now since we do not want standard probability spaces the restriction to this we have to we have to look what we can actually say in general and here and here is the the next here’s the next point and that’s about models so so that’s the first point so when we start in an abstract in an abstract situation then one actually can always find an isomorphic abstract situation which is actually induced by equipment operator so so in a sense module isomorphism the classes are the same and so this is the the next theorem and I call this the section about models here is here’s the theorem so if you have an abstract measure preserving system that means you have an l1 space of a probability space you have a mark of embedding on that space and you have a sub algebra and infinity which contains the constants and is an invariant seastar subalgebra and infinity then you are the following you can find a topological system right that is a compact of logical space a measure a continuous map by an invariant probability measure on that on that space and you can find a mark of embedding of l1 of k2 l 1 of x that intertwines the copeman operators and map cfk onto a so because the mark of embedding so this is our the measure has full support so Casey of K really sits as a subspace in L infinity and fires is then injective so that’s a by jection between cfk and a the proof is actually very simple if you know the girl front nymag theorem galvin imac theorem provides you first with a with a mapping file on the level of continuous functions and then you transfer the measure and then you transfer the operator because the operator is multiplicative or lattice homomorphism you find an underlying underlying dynamics and and then you close everything so the cks 1001 and then to transfer Phi you extend phi by density 20 so it’s actually is a straightforward proof if you know the girl from my mac

theorem the range of phi here is the l1 closure of a that means if if a is l1 dense is very dense in a one that’s the case even only a Phi is actually an isomorphous good so this is the important this is the important theorem and our philosophy in our book and also in my subsequent endeavors is actually to use this to reduce measure measure preserving systems to are the study of measure preserving systems to the study of topological systems by choosing a good sub algebras here and modeling the system on a on a good space and i will show you examples for an app for applications now ok so now we have to switch off this again and so this is now my fourth my fault so I’m I want to show you some applications of this of this theorem how you can work with this and reduce measure preserving situation 22 topological situations actually because of time reasons I I don’t want to say my first point was to prove the disintegration theorem in the general case by choosing models and and showing that that actually the proof becomes very natural and easy if you if you choose compact models but but for time reasons I skip this I want to talk about joinings first so the what is it Jonah ng actually you’ll eat a meal off said this is a seems to be a difficult notion well sorry ubi we’re not friends but actually i find it also a difficult notion so but when you look at it in the right way from a function at the point it becomes it becomes for also for functional analysis it becomes a natural notion that’s what i want to want to tell so what is it joining so adjoining is suppose you have two systems one on x1 and x2 with copeman operators or with the form equipment operators what is joining well joining is actually something that where you can embed this whole situation in in one in one larger l1 space in such a way so that means so here you have embeddings mark of embeddings that’s the first one let’s say j2 and that’s j1 that maps map as mark of embeddings into into that larger space and you have commutation of the Koopman operators that means so maybe I should stick to that notation so you have j is 1 & 2 so that just means you embed both both systems in a larger system and in such a way that actually this is minimal that means if you take the the range of j 1 and the range of J 2 then this is actually the generating everything so you cannot find so this is the smallest sub lattice close-set lattice that contains these two so then that means that if you take the lattice the clothes sub lattice that contains the range of j 1 and the range of J 2 then this is actually everything so if you think of the the lattice of factors of a system that just this means that within within the lattice of factors of a one these two systems so this is it’s the supremum of these two systems the smallest factor that contains the move okay now if you have such a joining oh by the way you always have one namely the direct product so if you justjust if you just take xx1 10 0 x 1 cartesian product with x2 with the product measure that is always and then canonical like embeddings that’s always a joining because okay so here’s an example so y is x 1 time obviously I’m talking to the functionalists here because the Gothic theorists know this stuff of course very

well so if you take this with a product measure and and you take the embedding j j 1 of f is just F with projection onto the first coordinate and j 2 f this F projection onto the second coordinate so that’s this is a and the dynamics of product dynamics is is adjoining so this is a standard way of forming adjoining now if you have such a joining then there’s an operator this is operator s which is going from here to here and then taking the adjoint of this operator down i forgot to say that the adjoint of a mark of such a mark of operators i defined is also a mark of operator works on alpha L infinity first but then you can extend it as the same properties it’s preserves one it preserves the integral it’s positive and then you can extend it to all over one so if you do this so this is the first one and then you take the adjoint this is a mark of operator from purely on the systems that you started with and and actually there’s a theorem that says you can go the other way around well this this microphone where has the has the this property so this is the one on X 1 and this is the 1 X 2 so if you just you can compute easily and then so if you start with such a situation that means you have a mark of operator from one system to the other that satisfy this this equation then you can construct adjoining and essentially unique joining such that this operator becomes this operator and now this is done easily by choosing choosing topological models so here here is just the sketch of proof well proof little sketch how do I do this by choosing topological models well well the first place this this whole situation is actually categorical so I don’t need base base representations it’s just with about operators so if I pass to an a an isomorphic situation i’m still getting such picture I mean I’m just and just apply an isomorphism and so without loss of generality we can assume that the XJR compact and actually not just that the koopman operators actually actually leave see XJ invariant and the mark of operator also maps to continuous functions of one space into the other I mean the cheapest were the cheapest way to do this is just to take as the sub algebras just the Infiniti algebras then you get enormously enormous big compact spaces but at least you can do it but in certain situations you can actually take different compact sets here I mean you can take smaller sub alger algebras the only thing you have to make sure is that the sub algebras that you take I just are invariant under under the operators under the dynamics and are so s maps one into the other so then then of course I’m just taking the compact product space now we are in a compact situation that’s a compact space and I’m taking I’m defining the operator Q&As from from the prince the continuous functions of the product to the continuous functions of the second factor in the following way if I have a function of two variables what I’m doing is I’m fixing I’m fixing the second variable then it becomes a continuous function of the first one then I can apply s which maps into the continuous functions on the second and then I can evaluate again in x2 a moment’s reflection is shows you that this this construction this definition is perfectly well defined

because this again is a cons is a function I can evaluate and this becomes again a continuous functions in x2 continuous function next to so on if I’m looking at elementary tensors here that like functions that are just of the form f of X 1 times G of X 2 then if i plug this in that just means that I’m getting this s of F times G now this Q Maps 121 it’s a positive operator on x2 I have a probability measure I take the adjoint I transfer this to this to this space so I it’s a Q prime of MU X 2 so nuoc stew is the probability measure on X 2 this is a the opera opera take the the adjoint that map’s measures two measures by the resource entation theorem so this is a probability measure on why so it was a P measure on on why and and now actually we define the embeddings of l1 of x1 into l1 of Y by well the canonical embeddings and so on the same for the other side and the rest is really like just exercise of computation that in this way you get adjoining and actually you get that SS is really the embedding here and joint of the embedding here so you get adjoining and and it satisfies this this equation so this is a this is a way of constructing joinings by using topological models now of course I gotta curious say yeah but why I mean when I start with a let’s say at standard probability spaces then I actually would like to model my joining on the on the product of these two spaces and not just on some other whatever space well actually in the case that that you are on standard probability space you can choose your models in such a way that they are metric because because ok you have dense countable sets and then you generate it sub C star sub algebras are separable now if you have a ck which is a separable space than K as compact space is meet risible so that means if you if you choose your subject so it’s your sub algebras carefully then you can arrange it in such a way that these two spaces compact spaces metra are mate risible so also the product is mate risible now you have a joining modeled on net product you still have models but then for nine months theorem comes and then says okay well this situation is isomorphic to the original one so it isn’t used by base base mappings so you can transfer everything back to your original product system so you don’t have to mess around with null set or whatever in boreal category passing to compact spaces there and then just invoking for Norman’s theorems of foreknowledge theorems than just the universal tool to to arranged in such a way that one doesn’t stay on on some unknown models but comes back to the original polish spaces let’s say our standard probability spaces okay so this is the this is one application there’s much more to this because this whole this whole construction can be used also for infinite joinings I mean this is just to joinings of two systems but you can do infinite joinings and actually what is called infinite joining and in a gothic theory is it called dilation in an operator theory so you get with the same technique you can construct a mark of dilations of mark of operators and well this has been done also by my nagaland parliaments on okay so this was the this was my story about joinings just to show you that this notion is actually natural and the the technique is it’s helpful okay so here now my last part of the talk I want to give you a different application actually you will not see proofs here because they would require a bit more time but just to show you also an application here where a classical

theorem can be considered the proof of a classical theorem can be considerably simplified although I don’t show the proof you will not see how simple it becomes you have to look into the paper but just I want to inform you about this so this is a joint work with nikita marie Yakov and this is about systems with quasi discrete spectrum so in our book we have the situation of discrete spectrum and our way to prove the homage for Norman theorem that allegoric system with discrete spectrum is isomorphic to a group rotation rotation on a compact grew a monophyletic group this is done in the following way so you first you pass to a nice model double-ought gical model and then you actually are in the topological discrete spectrum situation where you have all the methods of topology and then you there it is very easy to see that this is a group rotation and then because you somehow from the eigenvalues you just construct the group directly and the ice of the homeomorphism done is easy and then when you are there by so it’s the ideas also here the measure theoretic situation is a corollary of the topological situation and and so I thought okay this should this should be true with quasi discrete spectrum systems too and actually that works in the same way it’s just a bit more complicated to to handle these notions so let me just say what is this I’d not a system with crazy discrete spectrum for people who haven’t seen it so suppose you have a such a system and you suppose that is totally a gothic that’s a technical assumption one can also say something about none totally erotic systems but I restrict to these that means just that if you that every operator that means they become the fixed basis are all just the constants so every power of the operator has just the constants as fixed vectors that means totally a gothic so to use our gravity t square and so on now define the group G tilde that’s all the if the l1 functions that have constant modulus one that’s a group you can you come out apply the inverse is just a multiplication with a with the adjoint I with a conjugated function and on that group you consider the homo morphism detailer to G tilde that lambda f is f bar T of F so actually DF / FF you lie so this is a group homomorphism of this group now the next step is okay I’m writing it here so you define the GN to be the kernel of lambda n you have to be a bit patient you will see in this in a second why where this leads to so GN is the kernel of lambda N and G is just the it’s just the union of all these kernels so on G it’s clear that that that lump G is lambda invariant and actually lambda is Neil potent on G because every in every point is in some GN and then for n iterations of lambda you are here to the constant so this is a new potent so Jesus lambda is Neil potent on G ok now what has this to do with spectrum and eigenvalues well this is the following ok g0 is obviously just the constant one function because that’s the kernel of lambda is used to Colonel off the identity so that’s just one but what is g1 well g1 is all the constants with modulus constant functions with constant modulus one that’s the fix trays of tea intersected with G tilde so why because ok what is g one g one is the kernel of lambda so when it’s lambda f equal to one well that means that TF is f so f is the fixed space and since we are a gothic situation it’s

a constant so this is you one but g2 is the effin in G tilde such that lambda F is in g1 so lambda F is a constant so that so if lambda F is a constant then TF is a constant times F but that means F is an eigenvector and this constant is an eigenvalue so so this is those lambda in the Toros where TF is lambda F so there’s a there’s a London the Toros at TF is lambda if so that means so and if I met this then I’m getting exactly that constant so this is just the the constant functions that have a value as if their value is an eigen an eigen value of T and I put this embed this into T by let’s say ETA 1 so lambda 1 is mapped to lambda this is G tour so this is the group page 1 and this is the group of eigenvalues ok now hm is the image of GN plus 1 and this is khadija called quasi ghen values for for this operator and now i’m taking the union of all these then this is just image of G on the lambda and I collect this in a in a tribe triple which I call a signature the signature of of the system so this is the group of quasi I ghen values that’s this homomorphism lambda which is neat and and either one is just an embedding of h1 into the Taurus this is a signature now we say that the system has quite a discrete spectrum if if the linear span of these squares are eigen vector so the linear span of G is is full so everything or is dense in L 2 so X so this L 1 xt as quasi discrete spectrum if the span of G is everything so we are to let’s say is energy or in a one it’s them Spanish and who won’t dance ok so now we need another slide here and I hope I can let’s see ok I have to put this down that is the wrong one this is the right one so so here is the theorem and now the point is this theorem is old but this serum was first proved by Abramoff but somehow in an awkward way it was modeled on the proof of the original how much for nine month um which to might has has also proved in an awkward way well ok whatever was also proved in an awkward way because what for no man and how much did that what they prove it and what Abramoff did is first he showed that if I have to dynamical systems with quasi discrete spectrum and their signatures are isomorphic in an obvious in an obvious way then the two dynamic systems are isomorphic and second he showed that to every signature I have a quasi discrete spectrum that has this signature as signature so with every abstract signature so to speak I have a concrete dynamical system and then of course that means every dynamical system with quasi discrete spectrum is isomorphic to such a concrete one but this is somehow a bit weird I mean first you prove everything two are isomorphic then you have to meet you need a realization theorem that you can realize every signature and then you have the representation so so actually I think it’s much more natural to go for the representation directly and then the other the item of physics theorem actually comes for free not that not the realization though but the isomorphism comes for free that’s is this is the theorem I know that I have to stop so just showing you that it’s the classical theorem that if you have a totally our God exists the quasi discrete spectrum that you can represent it as an affine automorphism system on the dual group of

this quasi quasi I ghen values and now this proof there’s also a topological analogue which was proved by hard in Perry a couple of years later and it was proved in the same way also first isomorphism theorem then realization and then you get the representation as a corollary so actually so with a with a notion of topological models you can do this much easier so the major theoretic situation is a complete corollary of the topological one because you just take you just take this span of the span of G which is an algebra because geez group so the span of it is an algebra you take the l a– finicky closure this is your is a good sister sub algebra you represented on a compact space as a compact spacey of k and then there you are you have topological quite a discrete spectrum because of the total ago density here there it is uniquely a gothic that’s you have to of course you have to work somewhere they took in the topological case you have to work but my point is you only have to work once named in the topological situation the the measure theory is then measure the heretic situation is then just a corollary okay so I’m about to finish just one more word so this paper will be tomorrow on archive are just somebody yesterday we had some last amendment to do there’s also a second section which completely independent of this representation results where we also cover in an operator completely operate a theoretic way the other results of Harlan Perry about these systems in particular like a really straightforward proof that these systems always have zero entropy without that you don’t have to put in a lot of knowledge about entropy to see this here for the future there are of course much more many more things to say and so one of the challenging topics are Co cycle extensions and the function oolitic treatment of co cycle extensions up to now we don’t see a cop man approach to co cycle extensions but so this is on the list classification of abstract isometric extensions just for the a gothic theorists here and then also exploiting the Hilbert module structure in any in an in a purely functional analytic way actually there I would like to speak to the experts because they know put it may be more the literature than I do when may be many things are already done but so we will do this in private conversation answer thank you very much for your attention you you