so so I want to begin by quickly recapitulating some of the things that we then I said last time for which so let me begin by setting up a little notation is sometimes we just as useful so both X is a Banach space and suppose e is some again just be a subset but it will never use it again you be just a set some subset of the dual dual space the space of continuous linear functionals okay then one defines there I this one means the smallest apology with respect to which everything in s is continuous so what that means is X alpha converges to X if and only if FX alpha and moisture fix for all they finish the way you define the week topology week start apology and so on for appropriate choices of the set is now it is I’m being silly by saying you take a set s in fact one can see that if you take the this really the same as if things are lesser containers then linear combination of those things or what be continuous so it might as well assume that the that of the set reducing the topology is a vector space vector subspace of this X term and in general okay so here’s a little proposition lemma whatever fact which you will find for example it is not hard to prove you will find this proof act in that those notes of strategy though the first few pages that i gave you on what do they think ah the first thing that i gave you ok so it is not hard to see that if if what should I call it if B is a bounded norm bonded subset of X so for example I can and I we say or equivalently a bounded set index if you have a norm bonded subset of X then I am saying that a sigma x be this topology with and no I don’t want to be Sigma X okay okay sorry no I don’t mean end and if let me say if e contained in X star is a vector space i will assume that is a vector space now then i would say that sigma x e the subspace topology I’ll say and Sigma X eclosing on on e are the same give the same topology on e this is the kind of argument that we

discussed in the end you ha anh anh be sorry see these are apologies on X so if i take be with the substrate apology coming from from either E or e closure then I get the same same topology so this is improved by exactly the same motion that we went through during the tutorials on Monday ok ok so they thought or equivalently what this says is that so what this says is that ayee if x alpha converges to a fix when they say this is implicitly I’m assuming that X alpha the net index this holds for all if in some septum said is if and only if the same thing holds for all laugh in spanish if a non leave rohloff in Spanish closure they really the absurd of what I what I said before I’ll sorry for X alpha s coming from some non bonded said if you have a net which is uniformly long bounded to say that you have converged second everything in s the same is saying your conveyance against everything in the norm closure of this panel face okay endo so let me see I know me two more minutes of the generality before i get to be of each okay and the next remark i want to make is that if and okay I say if a closure equals f is separable so if I am looking the subspace they give the topology separable or equivalently I look at a set s which is countable and they look at the corresponding Sigma access on bounded sets then the statement is then the subspace topology on be again and then be bounded for bounded be is material ie in fact with respect to Mike and I’ll even tell you what the metric is d of XY so I did I take some so what does this mean this means there that you have some set FN which is dense in any clothes no no no okay so so there is this thing that you would have to say that if you given any subset a of X star one can define the travel the smallest apology with respect to which so honest with this phase topology with respect which all those things are continues so i say this i think you should have you

seen all of this in the week topology no i do not want to go good I am trying to minimize points at apology so that is why I want to just use your the language of red Sandra did everything that they expect to happen will happen so this you can say sum over N absolute FN x minus FN way I is it safe n of X minus way I have normally if I do this then this this will this will be a convergent some and to say that this this zero to say that d x and y 0 is exactly the D X alpha to say that FN x alpha converges pen for every n that means exactly the topology Sigma X F restricted to the board it said be so yeah this was just some sub silly or jeans that I want to get out of the way before we get to it out using these files okay now of course we will be interested in hearing in the case where X equal B of H and F efore me like a run for unfortunate rotation efore me will be I so Sigma X efore me will be to be cooperated topology that is e is things of the form and an element of e looks like looks like f of an X equal to summation 1 to N X say a ETA so these are these are the weekly continuously those are the linear functionals on on be a way that I want to be continuous so notice the rivet okay and then let me actually retain rotation let me call this as if there is a reason for this so i will read f4 all functionals which look like this 48 f of an a for all finite sums of this kind okay and so by what what we have seen the topology Sigma X F unbounded Seth is the same as Sigma of XF closure this F is clearly an element of B of H star that for every for every F in of this form if it definitely matters when you vector n data x squared X I ETA be continued so they with respect to the operator norm on be a waste these are continuous function is and so I i I’m interested in this topology right and so so now this will be called I i mumbled some words yesterday this will be the so called Sigma week topology I will have more to say about this before too long okay so what are you going on here so stir so let me back up so i will tell you l i’ll get to where they’ll use all this stuff so they you have seen in in in pathos course so you’re wrong but Cecil Oliver’s there is there there there is an abstract definition versus concrete he defined the sea and I

see Sir algebra abstractly as some Banach algebra as a star algebra which satisfies that sister identity that was there is an abstract definition with Cecily by concrete see sir I will mean a norm crossed Saba zebra starts abaza bra be of age and there is a theorem the the so-called elf enamel theorem which will tell you that any abside Cecil algebra is isomorphic to a concrete see serology similarly in phenomenal zebras there is an abstract definition and a concrete definition the abstract definition is so let me just say in an abstract object phenomenal do bruh is M is first sister algebra such that n 2 there exists in lower star as I e.m as a Banach space is the dual space or some other normed space in fact a Banach space so a sista algebra which admits a pre dual in the sense is an upside for nominal gibran and on the other hand a concrete for now an algebra is an M sitting inside a be of age said such that as the star subalgebra satisfying the conditions of the double comma Tron theorem the dimensionless straight or do they say I’m saying this is SOT closed or wot clothes or whatever okay so maybe I should have said something no no no not yet have come today so I guess I should have said something else when I was saying these earlier things about the Sigma X F and so on so so i define the sigma week topology as the topology as this topology where it is a crosser Sigma X be of age of closure ok so then jumping back and forth because they didn’t follow the order correctly ok so let me get a foot type either the end of this string so here also there is a girl phenom our theorem which will say that any upside wanna have an algebra is isomorphic to a concrete one of an algebra ok we say something that we will see as we go along ok and I in all this what is it oh ok this is if you take this ok let me give some ok so so examples of abstract one an example of a concrete 10 in algebra is easy to get you take you take Bo weight and you take any star closed sot closed subalgebra that is 1am in algebra unitl okay but but a at the upside level so for example you have L infinity chemical it Omega mu I take some measures faisal infinity and look at the space of essentially bounded measurable functions

with the essential supremum norm then you can see that it is a sister algebra of the soup norm and it has a pre dual namely l1 second then and this is the fact that i will tread a little time talking born in fact be of age itself is a is a nap psychic phenomena it is clear that it is a seesaw algebra so i just have to tell you what its pre doing this so it is what it is true that in fact what i called with b 08 start is what I called F closures but F was all those finite linear combinations of those inner product functionals you remember this this F that I wrote down before either functional zombie of weights that look like this there are sub space of B of 8th star so I can take the closer in boah so that is some subspace of boah star and I get take Sigma B of H with respect to that closure so you forget now now just take that F closure they have closed it a certain norms face so I am saying that be of age can be identified with the dual space of that F closure there is something that one has to vary fail but we will I hope eventually you I will convince you of this fact okay but but right now let let me not get into the proof of this I will get back to this okay and and so and there is there is a theorem by Sakai which says that this pre duel of a phenomenal zebra is in fact unique meaning as a Banach space up to isometric isomorphism if you have to Banach spaces my x1 and x2 so that M is identifiable with the dual space of these two then m1 and m2 are isometrically isomorphic Banach spaces that means the space of functional that they give on em see see in yet transgender if you take this M so I’m saying m Lord char is unique so this occurs theorem so what this tells you is that a one-armed and algebra almost by definition a nap site 1 i’m an algebra comes with a god-given pre duel so that is unique and there is this other theorem that you study in early bond a theory that any dual space is if you take the unit ball of a dual space it is compact in the so-called weak start apology as the so-called alia hablas theorem so I’m just saying earlier blue says that if I take the unit ball of em is compared in the sigma week topology so so i am using the same what do you mean different things so what I mean by Sigma V topology is just Sigma M by this MN Sigma M em starve the topology on em that you get from M lower star so see I

called earlier so this is an abstract thing there is but there is said earlier about what this priedu nov 08 is but I am saying that that that once you identify that pre double as a pre door if you believe suck eyes and that that thing is unique so there’s no matter which way you came to it there’s only one thing it can be okay so the okay so let me say this thing that i forgot to say earlier namely it is so one of an algebra so summer in summary a phenomenal algebra has a god-given topology a canonical topology which is this week start apology coming from the unique creed or so it has this topology with respect to which its unit ball is compact and whenever you have compactness anyway then you have a good thing that is the reason for for many many things things that that happen in this volume in algebra and so it is important to look at so i will use the word sigma week apology for a general Donovan algebra for an abstract 1 i’m an algebra to denote that apology coming from that pre do okay okay now the point is that if if and once one has shown that the what i called as f closure earlier is a pre dual that will tell you that this Sigma week topology and it will say that if m is a concrete wall nominal dobrze and then it will say that here the Sigma V apology and the week operator topology I let me not say let me say it on boah agree on bounded subsets oboh that is there is a point of Magnolia that F&F closure will induce the same topology and so because F gave you exactly the week operate apology and F closure gives you the sigma vector value which I haven’t approved yet for you then it will show that these two things are the same and in the end when you’re dealing with one of an algebra the natural thing to look things look at are things with a continuous in this topology the Sigma V topology the natural object to look at when a one owner and so little let me just say that there is this one thing in the last lecture it because Sigma week topology there is a is a is a mess and a pain to work with in general where the wake operator topology is something very tangible that you can lay hands on and work with and and because of this fact it is good to be able to work on bound norm bonded sets and so the theorem of kaplinski which we will have to use on

occasion which says that if a is a very unitl star algebra in set B of H sorry then Larson what we saw was that from the one of intensity theorem what we saw was it a is strongly dense in a double prime then now Kowalski said something stronger he says they’re in fact we take the unit ball of a predict this the closer the strong operated topology that is equal to the unit ball of a double frame so you just think about for a minute see the Varnum identity theorem says that a is strongly dance in a double prime so anything in a double frame can be approximated from me but this said that if you if you had take anything in the unit ball of a double prime you can approximate by things in the unit ball of a and because the topology Sigma being topology and bake operator topology and everything is the same and everything is material with that earlier thing that I said if I take all see if I take we may mumbling the the object of by saying something earlier about something being Matai zabal was to say that on a separable hilbert space if you take the baked operated apology on bounded subsets of boah the week order topology is realizable so far Troy charge I am being very easy haphazard interpolation so let me say this again so we’re here was the statement that I made now see CFC I said that if you take Sigma of exists or Spanish or Spanish closure they’re the same unbounded seeds then so if a day is to be all operators for the form okay i might as well do the thing properly know si si be of age any operator or a Hilbert space separable Hilbert space so you can you it has a countable orthonormal basis so any operator can be represented by an infinite matrix then so and not every infinite bed because you have honored obey or to worry about which ones to and so on why keep forgetting what if i were to say huh ah exactly so some every every operate operators of the sub matrix and only some may deceive give raise to operators but if I have a fat dick for example is separable if i take thing of it is l2 which means spelling an obsolete Gordon of an orthonormal basis for H then in here again look at operator called EIA which as matrices have one in the edge a spot and 0 elsewhere okay so these are all operators on this and it is it is easy to see that these edges there I would find that linear combinations of these no I only when i get to the closure because fixing the basis so i am

sorry about making a mess of this thing so no no forget forget everything I said the last three minutes let me start again and what I had hoped to talk about today was I want to talk about a one talk about be a witch as a phenomenal gboro I want to talk about the pre do under be of age to get a feeling for that thing and i also want to discuss two fundamental theorems and operated a theory that one will use all the time the so called spectral theorem and the polar decomposition theorem okay so to get to get to first base with these notions let me first discuss all this when H is finite dimensional let us understand what all these things are and then we will see the the passage will become a little more transparent I hope okay so let’s start with spectral theorem to shut out H is finite dimensional so I might as well assume that H is C and once they fix an orthonormal basis assume that H is n dimensional Hilbert space so it is a red feather to be an orthonormal basis of age with the standard orthonormal basis for CN and I by the same token I hadn’t fed boah with MNC m capital NC okay so what does the spectral theorem say it says that so let X in be of age I said in terms of matrices says that then following are equivalent X is normal that means X commutes with this a joint 2x is diagonalizable there exists you in you in a unitary matrix and unban unitary matrix such there you x u star is a diagonal matrix I will just read alpha for the rental diagonal matrix was alpha 1 alpha n on the diagonal ok any matrix can be any normal matrix can be conjugated by unitary midi to give your diagonal mating that’s how the spectral theorem says there is actually something that path or put up on is bored here I need some way norm of any nog normal now of any element of a commutative see seller’s libera is equal to its spiritual radius and there is exactly what I said it said take the spectrum take the right zone so that is the norm ok so there what the spectral theorem says okay and what does the foliar decomposition theorem say ok so before I get to that let me just say already in I think yesterday’s lecture of pathos he has been trying to tell you the importance of the notion of functional calculate calculus you have some algebra of functions and for operators for some operators for some multi Bravo creators you can you get a home off ISM from the function algebra to the algebra of operators which you do not BF going to f

of X where if f is a polynomial then that P of X is the natural polynomial and in when when X is when you have a top the the first example path or talked about the analytic functional calculus or holomorphic if you take any element of any one of algebra then you can get a mapping from the algebra of functions with a holomorphic in neighborhood of the spectrum into the algebra which you call the the functional calc the analytic functional calc is if you have two two elements of the algebra then f g of x is equal to FX times G of X what they good about sea star algebra is there one has not just see the one is getting greedy and greedy initially you have a polynomial function of calculus which any any any algebra will give you and then we throw this contour integration one got this holomorphic function of calculus in any Banach algebra so you’ve expanded the scope from polynomial swarm of few functions now i am now you want to expand even further not only horrible but even just continuous so 11 to have a continuous function of calculus so that is what one will prove a resistor algebra in the sea star algebra he vexes a normal element in a sister algebra then you have a home or film from the continuous functions on this victim of x to the sister has liberal f going to f of X and you can use their availability of lots of continuous functions to say sensible things about this is to algebra now in one of analogous it is you will go one step further that is a race you see Sir algebras are like continuous function C of X one of madrid’s are like L infinity of X this bounded measurable functions I will make sense of the indicator function of an of an element of a novel element function which is not continuous trait so the the so-called measurable functional calculus is what one Hammond algebras will give you that is what we will be doing but but before we get to all that so let us see just here at the baby stage you can see what the continuous function of calculus is for Cecil as the brush in finite dimensions what what will f of X be you will see that this is actually equal to u U star diagonal F alpha because the diagonal or five with you if you conjugate by you you the spectral theorem says you might as well assume that your self adjoint is diagonal so for diagonal things I should tell you what the function of calculus is and if I want if you want to tell if you want to tell you what log of X is and if X is a diagonal matrix then log of that is going to be log alpha 1 log our friend for that makes sense of course all the Alpha is will be positive so the functional calculus will make sense on d where the function they defend on the spectrum ok so for example one can talk about mod X so rotation if X is ready for any X in MN it is not normal so I cannot apply functional calculus but I can look at ecstatic which is self-adjoint and it is in fact positive the function f of t equals square root of t is a continuous function different on the positive leg so again defend this this power half that is something which will make sense but here the exactly that if you mod X if X tarick’s has eigenvalues alpha 1

our fan mod X is the diagonal matrix with square root alpha is so now let’s get to the polar decomposition if you take any X in MN they again the finite dimensional version i will i will later later today i hope get to talk about the infinite dimensional version or the polity composition the supply there exists a you in unitary so it’s then x equals u products so this year was the compared with the polar decomposition of complex numbers you said to the complex number i can write this e to the I theta times what Z so there were the polar decomposition says so notwithstanding all my mumbling in the first half hour of my lecture I hope at least the spectral theorem and polar decomposition finite dimension is clear so now let me get to so I want to understand what MN of I want to understand boah star is over so for finite dimensional H let us see what this is so be that is I want to see what MN star is I will drop the sea or just ate em in what is the pre duel in finite dimensions if some something there is the dual phase of some fun adventures face every finite dimensional Banach space is reflexive solo star and upper store everything is the same so as a vector as a vector space you can identify it with with them in the only thing that is not clear this what is the norm of this thing I want to understand the norm of this pre do and so here is a let me give give away the answer there the hear the beautiful factors you would have seen this in your first course in functional analysis you would have seen that if you take the sequel space c0 one day for usually the sequences which are converging 20 null convergent sequences with the supremum norm so this is abbas draw of l infinity so what I am saying is that c0 star is actually l1 throw this something you have seen and l1 star is l affinity and the analogy to keep in mind is taken in the diagonal matrix with this a spread on the diagonal of made two variants when will we give you a bounded later test paradise in a yep general if we will see that there is this be 08 there are the so-called finite rank of Peters operators fenêtre if you take and they there will be a subspace of V H take the norm closure of this this is usually called compact operators and one

understands it top of this is that k wait stop is let’s see one of each in c1 which star equal to B of each so somewhere the analog of l1 will be the pre do love p of h and to see come we copied it apology you get all operators so there is an exercise and there is a certain other norm that will put on fire truck operators will call the one norm will be this spree do so then till they turn out to be a certain other on the closes the finite ranch we will do not be exactly the c1 okay to understand any of the norms really you the first thing one should notice is that all the norms are really ok so let us trade one from the norm of let us way to understand common stock which is same as a mint pre duel as a Banach space right so you two what are the what are the linear functionals on this so either vector space you can think of MN as just a CN square n square the virtual space and so it’s dual space is just also identified with n sweat dimensional space i can add afraid think of this as identified with MN and suppose it take as either element of the pre dual i will call call you so much like row or out of that the events of MN i will denote as X X and they’re even so Chris the feed will I will I will write with the symbol Rho so how will hold the house the duality given to Rho X so so the row is also one things of this as actually equal to n as a set how much as a vector space i am going to tell you what the norm is so what is the action of Rho onyx general duality in finite dimensional space finite dimensional Hitler’s field will tell you that this looks like summations throw a day bar x IJ that’s exactly what this is now with with a little respecting you will see that this is nothing but the trace of Rho star X did they made this multiplication I take row 40 * is they’re called conjugate complex conjugate matrix if i take roast are there is a medical ID NT ro j a bar and then you tell you take some i did until you verify this so there is what this whole the plead won the duel ads on ma so now i want to find the norm so

what is the norm of such a row so al did with one norm for the norm of row as an element of MN lowest are her them installed or eminence what is the linear functional norm of that room so to see that so this by definition going to be the soup of absolute value trace of roast RX for nor mix the many now floating Roger let me read infinity do not the operator norm so I want to find what the Infinity numbers say they parallel to this thing I have the Infinity norm here i am trying to find out what the one norm is except where matrices now not sequences okay so what is this okay so i want to say that see the spectral theorem will tell you that row can be written as u Mordreaux and polar decomposition sorry to tell you you can read so again read so if I want to look at this quantity what they get is equal to trace of Madra self-adjoint so they just taste more dro you star X so which is really see they did distress the pairing theythey trace of or ocean see what I am trying desperately to say here is that in fact cxn you star X have the same norm operator norm infinity norm and so the other functional if we look at mod row acting on you star X that will be bounded by norm of ma drove times norm of U star x which is equal to norm of X so finally what you will see their home row 1 in fact I am saying will turn out to be nothing but trace of Montreux then they put up a tough enough computation there for you to convince ourselves that this what you can’t care what we get because I’m because of the fact that norm of U star x infinity norm is the same with infinity nowadays so if you multiply by unitary you don’t change the unit ball of the operator algebra right so this is equal to the one novels of Ross a file called PA if Rho equal to mod raw is diagonal PA to deliver the traces the trace so the tears of Madra is there is this the trace of any of oily self adjoint operator is the sum of this which I ghen values so ma draw diagonals is a diagonal from PS then the trace of Madra is just the sum of those pas so what we see that the the pre do love MN is a man with this one no and what I was trying to say earlier is that again the i am leaving gaps have not reading the proof in full detail what i am trying to say that from this fact one will in fact i want to say that

be of age lower star yes okay MA so so so let me just say that this is so if I think of BHS infinite matrices which induced bounded operators I want to identify what is pre do this so if I have an element of the pre do love be of age flows are it acts on infinite matrices so if i have a row in here then so i guess i can take an element X in here and look at only these corners how did look at things which look like zero of the stage in I look at MNC sitting naturally inside be of age every linear functional on be of age will in particular restrict such same to give you a linear first order and MN and what will be the norm of that as a functional on on MN will be exactly the trace norm of that corner and because and it is easy to see that the the matrix inducing the functional is unique in any MN so I’m saying so start rotting with row for each end I will get an in vain matrix and what I will find is that if i look at if I think of row in as a relevant of MN lower stall which is just obtained by resetting to the action of the corner there so given an element of B of age lower I have a sequence of such functionals and if I think of this as sitting inside MN plus 1 lower storage which because identify these with just matrices after all if I think of Rowan as also sitting inside n plus 1 size matrix then what you will see is that in fact the row n will be a corner of row n plus 1 they will not be consistently different so this row will give you the consistently defined sequence row n of matrices and the fact that the whole row is a bounded linear functional will tell you there and the fact that the norms of the Royals is exactly the norm of row receipted that that piece will tell you that in fact limit norm in EM tending to infinity in smaller than m ro n minus Rho M 1 is 0 so this little work but one can see that these things the rowans will be converging in this round so in some sense I am saying that be of a slower start if the completion of all possible I should have said finite rank matrices you take all finitely supported matrices which all give boda operators or be of age and you give the norm given by the one norm the completion of that is exactly we have edged closer no no no I am saying that see see each row and we’ll give you an element of

that what I called s what i call def before and so i am saying that the Rowan will converge to 2 s 2 row as an element in in bh Tov no upper strand that I said earlier that be of age lower star is the closure in be of age up install of the ferret ranks in fact you don’t have deck all finite transit is enough to take on live finite rank with respect to specific basis apparently supported matrices you take the completion with the switch to that and the normal is the one known okay so are you hope students to control toy but but the point is that you know one has to be careful about taking traces you can’t take the trace of any matrix just because not every sequence is convergent sorry mobile you got big matrix and just some all the diagonal entries and hope to get something which is independent of which bases you choose and so so this be a witch so to what is what I’m saying is that be of a slow start so I I just say that say this is a fight there exists a class which are called see one of H which is contained in NB of age in fact in fact the these are contained in compact operators with the bday good property that if you dig at any X in C 1h the if you think of the matrix oh I’m just saying that for all X in this and row in this and X in V of age the pairing see I’m saying that this is the dual space of this that means I’ll have a pairing given something here and something there we had a number out of it ok so I’m saying be of age is the dual spheres of row so if you want pairing like this all makes of row over whatever yes nothing but trace of floors no that is because I earlier add put the start because they use the hilbert space language inner products so the point is that the injunction is also an on preserving thing so as far as it things normal is concerned it does make a difference so so what I’m saying here is that this makes sense so so this is in fright group this says many things are being implicitly ciated hear you say that this is actually an ideal meaning if I take something here and something here and if i multiply at the answer will be here this is an ideal of this algebra and for everything in this class I can make sense of the trace so what does that mean if I take any orthonormal basis and represent that operator with respect to that basis then the diagonal entries will be summable and the sum of the diagonal entries will give you an answer which is independent of the basis that you choose so all this requires some verification and I don’t want to get into detail by you you can see all of this in for example the satellite here’s the thing you’ll tell you where they sing the truth okay so so maybe I will get on with you know got

a functional gags its whistle theorem today hi I’m I’m being released I think it’s better to be slow rather than do too much so let me I think cover a few comments that I made in the notes okay so going back to the thing so the point of the last half NOS rambling was to show that be of age is indeed dual space and just to give you an idea of what the pre do love be of age luckily I haven’t said anything about the uniqueness why it is unique and so on but in order intend to okay so in the in the cattle see in the category of one novel algebras okay see if you look down but Cecil algebras if you’re talking about the category of sea stars in the language of seats where the natural thing would be the objects will be C star algebra these are involute of algebra and what are the morphisms the maps between these this k objects will be there algebra so you would like them to be algebra home or films and you would like that to be star homomorphisms and further ideally you would like them because this thing come with a norm they are cut then you would like the home of them to be bounded this is a theorem that Parker will assume show for you sometime that any home office / sister algebras is automatically bounded he showed you this fact you think the the spectrum is always there quickly inclusion the norm is determined by spectral radius and the sister identity to anyway so in seeds are algebras the natural morphisms in the category of seats are as diverse are just star home officers when you talk about 10 min algebras at least at the abstract level what are the morphisms so so given to one hour now c plus m1 and m2 morphisms would mean star home off isms which are these to the kid the feature one on algebras is there in addition to the serratura structure that pre do that they have the Sigma week topology that they have so there is the Sigma week topology on domain and range show continues with respect to Sigma V topologies on domain and range so rather than reading all these things again and again so one will just abbreviate all this thing to say one will use the word normal I start a map of one ammon address will be called normal if it is continuous with the correct apologies so natural morphisms are the normal Shalom officers okay so so again the days if they fairly I mean not difficult five but if one has to do a little bit of points in topology one I swayze any sort of thing which is the following so if you have four in cesa algebra you know that if the cisa algebra and if a knot is some some star algebra which is norm closed then it is itself is actually so won’t talk about 10 months of algebra so if you if you have em okay so so let me say the funny thing so M concrete one ever algebras huh absolute one anonymous this is the

easy part of the theorem that one wants to prove said afterward concrete and outside but I want to say that any concrete one of another that means a sigma weekly close in fact more generally I would say for generating if n is there Sigma weekly closed stars above the brow offer one of another play em so the other is upset right no upset then n is also an upside for now in algebra that means I just have to I just have to produce the pre do because it is C sahaja bruh so the details are destroyed in that in that remark in here which I will skip town in the mach 2.2 i do the two point two three years of the one says that any such one Norman subalgebra over whenever a sahaja bruh one of an algebra either one of an algebra so in particular what were a concrete one Evan algebras by definition if I use the word varnum and Sabah zebra to be in a sigma weekly closed starts about the blow-up of one Evan algebra then a concrete one now in algebra is nothing but and and abstract one I’m and sahaja bruh of be of age and so by what one is so shown here okay so okay they worth it let me illustrate that comp Lansky’s here so I want to say that any see suppose it take and and so let me prove this fact Clearing Corp landscape suppose M is a confident one of an algebra so it is sitting in set B of each should it wot closed in bh so in view of the remark that have made here it is enough for me to prove that M is actually Sigma V 3 close and be of age okay then it will also be an absolute one of an algebra sam tighe say i am trying to prove that if i have a concrete 1am in algebra which i call em i want to say that it is an absolute one have an algebra so what does this win the swing there it is a WOD close starts off as Abramovich but what the what one has just said within that Ramon either if you take it Sigma weekly close of algebra offer of an after one another that is an upside one have an address I just need verify that this implies Sigma weekly closed in be of age as a sahaja bureau of be of age when all that is Sigma actually closed then it will follow that this thing is indeed true so and is it is sorry for be laboring a trivial point but it is it is good to see an instance of how and where you will need something like the kaplan skin density theorem ok so the order to support x alpha m converges to X or another ex is in M

Sigma V closure then I want to say that X belongs to it without loss of generality nor makes it less than equal to 1 ok then well then what I know is that I’m maybe I should yeah so see what kaplinsky will tell you is c2 then if I take a is equal to M what cap Lansky will tell you is that X is in a double prime in a closure week operate apology big that okay there and I’m make the mess of it again so the point is this on the unit ball the sigma week and baked apologies are the same and if you can approximate something in something in EM closure by the Sigma V topology that then you can approximate it the Sigma V topology by things in the unit ball if something in the unit bod is a proximal but by something in the algebra you can approximate by things in the unit ball of the algebra there ever complains he says but it makes the limit of X alpha so there is a DX limit X alpha with norm X alpha this is equal to one limit variance Sigma V but unbounded set with the Excel first so unfortunately it is a nature that you have to get into these topological skirmishes with radical to get what you want and the odd or semi door okay and then I meet both the same and if something is dictate from embark and which is equally Signori the enclosure of these all those things so that is really what I want to get down to but but to get down to a priori I need a cab Blonsky to say that you could choose the approximate from the unit volume that all the action was happening in a boundary region and they’re the true topologies with the same and so you have actually we convey convergence hope your own grounded got down to so there’s spectral theorem I think one should I think maybe I should

I should forget the fifth in the last lecture and then train to the first cover the first five lectures in the or as much of that as they can so i shall now got down to spectral theorem or polar decomposition in infinite dimension but that will taint quickly run through next okay you you may even go if you want I’ll take I’ll take well getting all the things together from