i was raised a hardcore c-algebraist .my thesis focused on c-dynamical systems , and never once required a weak topology .as a fresh phd my knowledge of von neumann algebras was superficial , at best .i did nt really like von neumann algebras , did nt understand them , and certainly did nt need them to prove theorems .conversations with new w-phds make it clear that this goes both ways ; they often know little about c-algebras , and care less .  today i still know relatively little about von neumann algebras , but i have grown to love them . andthey are an indispensable tool in my c-research .conversely , recent work of narutaka ozawa ( some in collaboration with sorin popa ) has shown that c-techniques can have deep applications to the structure theory of certain von neumann algebras .in other words , there are very good reasons for c-algebraists and von neumann algebraists to learn something about each other s craft . in the old " days ( say 30 or more years ago ) , the previous sentence would have been silly ( indeed , some old " timers may still find it silly ) as the field of operator algebras was small enough for students to become well acquainted with most of it . that s no longer the case .hence , i hope these notes will help my generation , and those that follow , to see the delightfully intertwined theories of c- and w-algebras as an indivisible unit .i do not intend to write an encyclopedia of c- and w-interactions .amenability ( for groups , actions and operator algebras ) is a perfect context for illustrating some of the most important interactions , so these notes are organized around that theme . also ,i assume familiarity with basics such as c- and w-algebras associated to discrete groups , crossed products , and numerous other things . keeping these notes self contained would , i fear , bury the ideas being advertised under a mountain of details .hence , the reader will frequently be referred to ( another advertisement , of the shameless sort ) for more details .* acknowlegment : * these notes are based on lectures given at the summer school on aspects of operator algebras and applications ( palacio de la magdalena , santander ) and the workshop operator algebras , dynamics and classification ( texas a&m ) .the organizers at both places were amazing and i ca nt thank them enough for their hospitality .i distinctly remember my first encounter with a w-fundamentalist . after a day at gpots , we were at a bar impressing locals with exotic vocabulary .out of nowhere came the verbal left hook .  why do you study c-algebras ?" he asked with palpable contempt . they re not interesting , you ca nt do anything with them ... they do nt even have a borel functional calculus ! "i was stunned silent .( no small accomplishment , given my big mouth . ) but later that night , lying awake and trembling with rage , i finally had a piercing come back .   anybody can prove a theorem about von neumann algebras .how hard is that ?look at all the tools you give yourself : weakly compact unit balls , polar decompositions , you even have a borel functional calculus! a child could do something with all those tools . andanother thing : you re a jerk ! "  pleased at having silenced the voice in my head , i fell asleep with a smug smile .;-)  fundamentalism is stupid .trust me , i know , i m a recovering fundamentalist .it took me a long time to see that i was guilty of precisely the sort of tribalism that i now wish to discourage .( and , yes , i occasionally relapse . )these days when i meet a w-fundamentalist , i try to point out what c-algebras have done for von neumann algebras .for example , having complicated representation theory is not a c-defect it s an opportunity . just ask bob powers .his celebrated construction of non - isomorphic type iii factors ( cf . ) depends on the non - triviality of c-representation theory . in the latter half of these noteswe ll see more recent examples ; ozawa s work on solid von neumann algebras is largely c-algebraic .  for my own part ,i no longer denigrate the rich structure of von neumann algebras .i exploit it .i love the fact that they have projections and ( in the finite case ) tracial states .i m green with envy over the compactness of their unit balls . for example , how awesome would it be if the following result , which is a fairly simple consequence of alaoglu s theorem , had a c-analogue ?  [ subthm : pointultraweak ]let be a banach space , be a von neumann algebra and be a bounded net of linear maps .then has a cluster point in the point - ultraweak topology .do nt see the point ?well , imagine the theorems you could prove if , for example , every sequence of asymptotically multiplicative u.c.p .maps had a point - norm convergent subsequence i.e. , gave rise to a -homomorphism !( for example , you could prove that all c-algebras are type i , a fantastically false result . )the point i m trying to make is that in a von neumann algebra you can do things we c-fundamentalists only dream of doing .passing to a weak closure and gettin crazy can be a lot of fun not to mention fruitful .indeed , when trying to prove a c-theorem my first thought is usually this : how do i translate this problem into a w-question ? andhow will i come back ? " of course , certain c-questions are more amenable to this approach than others , but in these notes i ll try to explain some of the ways that von neumann algebras have been used to prove c-theorems , and vice versa .the results are impressive , by any standard . andbeautiful !i hope you enjoy them as much as i do .   hereare some general tools that facilitate the passage between norm - closed and weakly - closed algebras . the first result , one of the oldest in the subject , is still used daily .let be a c-algebra acting nondegenerately .the weak - operator - topology closure of is equal to the double commutant .  let be a c-algebra acting nondegenerately .then the unit ball of is weakly dense in the unit ball of .let be a c-algebra acting nondegenerately on a separable hilbert space .for each self - adjoint , there exists a decreasing sequence of self - adjoints in such that  1 . in the strong operator topology , and 2 .  for each , there exists an increasing sequence of self - adjoint in , such that ( as ) in the strong operator topology .let be a c-algebra acting nondegenerately .for every finite set of vectors , , projection and self - adjoint , there exist a self - adjoint and a projection such that , for all , and .the ( banach space ) double dual of a c-algebra is a von neumann algebra .moreover , the ultraweak topology on ( coming from its von neumann algebra structure ) agrees with the weak- topology ( coming from ) , and hence restricts to the weak topology on ( coming from ) .  from a c-algebraist s point of view , the double dual theorem is probably the most important as it allows one to come back from the world of von neumann algebras .that is , suppose one wants to prove a c-theorem , exploiting the enormous w-toolkit .well , in any representation of the given c-algebra one could take the weak closure and bang away .but how to come back to the c-algebra of interest ?answer : the hahn - banach theorem .that is , rather than work in any old weak closure , one should work in the double dual von neumann algebra , where the hahn - banach theorem implies that convex sets have the same norm and weak closures .that probably makes little sense , so let s look at an illustrative example .let s show that if is semidiscrete , then is nuclear .recall that a linear map is _ completely positive _if the matrix - level map , defined by ) = [ { \varphi}(a_{i , j})],\ ] ] is positive ( i.e. , maps positive matrices to positive matrices ) for every .special attention will be paid to the cases where is unital ( u.c.p . ) or , more generally , contractive ( c.c.p . ) .[ defn : nuclear ] a map is _ nuclear _ if there exist c.c.p .maps and such that in norm , for all .a c-algebra is _ nuclear _ if the identity map is nuclear , i.e. , if there exist c.c.p .maps and such that in norm , for all .a w-algebra is _ semidiscrete _ if there exist c.c.p .maps and such that ultraweakly , for all .  to show semidiscrete nuclear , " we need an important matrix - fact .[ prop : cpmapsfrommatrix ] let be a -algebra and be matrix units of .a map is c.p . if and only if {i , j} ] , it follows that .hence the flner condition is equivalent to requiring , which is often how it gets used in our context . ]a sequence of finite sets such that for every is called a _flner sequence_.  in the context of discrete groups , the following result illustrates the connections between nuclearity and von neumann algebras .[ thm : amenablegroup ] let be a discrete group .the following are equivalent :  1 .[ thm : grpam1 ] is amenable ; 2 .[ thm : grpam3 ] satisfies the flner condition ; 3 .[ thm : grpam9 ] is nuclear ; 4 .[ thm : grpam9.5 ] the canonical -homomorphism is -continuous ( i.e. , extends to ) ; 5 .[ thm : grpam10 ] is semidiscrete ; 6 .[ thm : grpam11 ] is injective .the equivalence of ( 1 ) and ( 2 ) is classical ( see ( * ? ? ?* theorem 2.6.8 ) for details ) . : let be a sequence of flner sets . for each  be the orthogonal projection onto the finite - dimensional subspace spanned by .identify with the matrix algebra and let be the canonical matrix units of .one can check that for each we have unless , and if . since ,we have let be the u.c.p .map defined by . now define a map by sending evidently this map is unital ; it is also completely positive , as one can check .the s and s do the trick .since the linear span of is norm dense in , it suffices to check that for all .this follows from the definition of flner sets together with the following computation : hence the reduced group c-algebra is nuclear . : this follows immediately from proposition [ prop : tensornuc ] , since is always -continuous . : applying the trick to we get a u.c.p .map such that for all .hence , lemma [ lem : invmean ] implies is amenable . is similar to the proof of . : let and be as in the definition of semidiscreteness . by arvesons extension theorem , we may assume the s are defined on all of . by theorem [ subthm : pointultraweak ] , we can find a point - ultraweak cluster point of the maps and this is evidently a conditional expectation . is an immediate consequence of lemma [ lem : invmean ] .as mentioned earlier , one of the reasons i ve taken the time to prove the theorem above is that it is nt all that hard . andit suggests there might be a more general result lurking in the bushes .   in the proof of theorem [ thm : amenablegroup ] we used the group in every implication except when proving .this implication followed from the general fact that if is nuclear , then there is a unique c-norm on for every ( proposition [ prop : tensornuc ] ) .however , we ve seen that nuclear c-algebras have injective commutants in any representation ( proposition [ prop : comminj ] ) .thanks to haagerup standard form ( ) , we can take another step .[ prop : nucinj ] if is nuclear , then is injective .first we represent in standard form .then , there is a conjugate - linear isometry such that and . by proposition [ prop :comminj ] , there is a conditional expectation .now check that is a conditional expectation onto , and we re done .here s the big theorem we ve been after .[ thm : nucchar ] for a c-algebra , the following are equivalent :  1 .  for every is a unique c-norm on ; 2 . is nuclear ; 3 . is semidiscrete ; 4 . is injective .we ve already seen ( proposition [ prop : seminuc ] ) , ( proposition [ prop : tensornuc ] ) and ( proposition [ prop : nucinj ] ) .the remaining implication , mainly due to connes and choi - effros , is very hard work too hard for these notes .it depends on some deep results in von neumann algebra theory ( e.g. , existence of modular automorphisms ) and , in my opinion , is one of the great w-achievements. see ( * ? ? ?* section 9.3 ) for more on the proof .  at presentthere is no proof of the equivalence of conditions ( 1 ) and ( 2 ) above a purely c-algebraic statement ! that avoids von neumann algebras .  if is a closed 2-sided ideal , then .hence the next corollary is easily deduced from the previous theorem .[ cor : nucquot ] let be a short exact sequence . then is nuclear if and only if both and are nuclear .  to the c-algebraist that does nt yet appreciate von neumann algebras, i have a challenge : find a c-proof of the fact that nuclearity passes to quotients .good luck ...  here s another nice c-application ( due to blackadar ) .it requires the fact that nuclearity passes to quotients , hence von neumann algebras deserve much of the credit .[ cor : nuctypei ] a separable -algebra is type _i _ if and only if every subalgebra is nuclear .recall glimm s theorem : a separable c-algebra is _ not _ type i if and only if every uhf algebra arises as a subquotient of ( see ( * ? ? ?* section 6.8 ) ) .this implies , first of all , that subalgebras of type i are again type i. indeed , if a subalgebra were not type i , then it would have uhf subquotients and hence the larger algebra would too . since typei c-algebras are nuclear ( ( * ? ? ?* proposition 2.7.4 ) ) , this evidently implies the only if " direction .  for the opposite direction ,assume is not type i. by glimm s theorem , we can find a subalgebra with a prescribed uhf quotient .it is a fact that is a subquotient of every uhf algebra ( cf .* corollary 8.2.5 ) ) , hence we can find which has as a quotient .since is not nuclear ( cf .theorem [ thm : amenablegroup ] ) , is not nuclear .now that we ve seen a few c-theorems that require w-proofs , i want to turn the tables and show you a few c-contributions to von neumann algebra theory .here s a horribly short summary of some c-facts we ll need .a c-algebra is _ exact _ if there exists a faithful , nuclear ( see definition [ defn : nuclear ] ) -representation .a discrete group is exact if is exact .it turns out that exact c-algebras enjoy an important approximation property that is nt as well known as it should be .[ defn : locref ] a -algebra is _ locally reflexive _ if for every finite - dimensional operator system is a self - adjoint linear subspace containing the unit of . ] , there exists a net of c.c.p .maps which converges to in the point - ultraweak topology .  in a brutal tour de force, kirchberg proved the following remarkable result ( see ( * ? ? ?* chapter 9 ) ) .[ cor : exactlocref ] exact -algebras are locally reflexive .. as such , it is another example of a c-theorem that requires von neumann algebras .but we re about to come full circle and use this result to prove ozawa s solidity theorem for von neumann algebras , so it s a bit misleading to say that ozawa s work is an application of c-theory to w-algebras .it s really a brilliant reminder of the unity of operator algebras ! ]   in hopes of making ozawa s work easier to digest , i ll cut it into bite - size chunks .the first bite is a tasty application of local reflexivity .[ lem : lem ] let be a von neumann algebra which contains a weakly dense exact c-algebra .assume is a von neumann subalgebra with a weakly continuous conditional expectation , such that there exists a u.c.p .map with the property that .then is injective .let be a finite - dimensional operator system and be contractive c.p .maps converging to in the point - ultraweak topology . by arvesons extension theorem , we may assume each is defined on all of ( and now takes values in ) .then one readily checks that are u.c.p . maps with the property that ultraweakly for all ( since for all ) . taking a cluster point in the point - ultraweak topology we get a u.c.p. map which restricts to the identity on . taking another cluster point of the maps ( over all finite - dimensional operator systems ) we get the desired conditional expectation .( i love theorem [ subthm : pointultraweak ] ! )now , let s specialize to the case of group von neumann algebras .another spectacular feature of finite von neumann algebras is the existence of conditional expectations ; that is , if we had assumed to be finite in the previous proposition , then the existence of a conditional expectation would have been automatic ( see ( * ? ? ?* lemma 1.5.11 ) ) .[ lem : thelem ] assume is exact and let be a von neumann sublagebra with trace - preserving conditional expectation . if is -continuous , then is injective .  since , the trick applied to yields a u.c.p .map such that .hence lemma [ lem : lem ] applies .a von neuman algebra is called _ solid _ if the relative commutant of every diffuse von neumann subalgebra is injective .when i first saw this definition , i thought , huh ? what s that all about ? "well , it turns out to be a very strong structural statement .for example , solidity passes to subalgebras and implies that nontrivial tensor product decompositions are impossible ( for non - injective algebras ) .[ prop : prime ] if is finite and solid , and is a non - injective subfactor , then is prime ( i.e. , is not isomorphic to the tensor product of ii-factors ) .assume is not prime .then we can write where is diffuse and is not injective . by definition ,the relative commutant of ( in ) is injective . butthis commutant contains , which is a contradiction ( since there is a conditional expectation , hence injectivity would pass to ) .here s ozawa s celebrated solidity theorem .[ thm : taka ] assume is exact and the canonical map is -continuous .then is solid .let be diffuse and be a masa ( in ) . since and there is a conditional expectation , it suffices to prove is injective . since is diffuse , is non - atomic i.e. , .hence we can find a generating unitary such that ultraweakly .let and be the unique trace - preserving conditional expectation . as is well known , we can define a conditional expectation by taking a cluster point of the maps note that contains in its kernel , since ultraweakly .evidently is a trace - preserving conditional expectation of onto ; hence , by uniqueness , .moreover , since we have that lies in the multiplicative domain of .thus , for all and . in particular , for all and .  by lemma [ lem : thelem ] ,to prove is injective , it suffices to show that is -continuous .but we assumed that is -continuous , and the map factors through the calkin algebra ( since is in its kernel ) , so -continuity follows from the fact that for all and .recall that is amenable if and only if the canonical -homomorphism is -continuous ( theorem [ thm : amenablegroup ] ) .since the hypotheses of ozawa s result looks frighteningly close to this , one should worry about the existence of nonamenable groups to which the result applies .however , it turns out that many such groups exist .even better , proving that they exist is a beautiful synthesis of geometric group theory , nuclearity and c-crossed products .hence our next section is a quick summary of the theory of amenable actions and their crossed products .after that we ll come to examples .the crossed product construction is fundamental in operator algebras and appears in several texts .hence we wo nt recall the details see ( * ? ? ?* section 4.1 ) but we will review the set - up . if is a discrete group and is a group homomorphism into the group of automorphisms of a c-algebra , then one can construct a full " ( or universal " ) crossed product  satisfying the universal property that any covariant representation of extends to  and a reduced " crossed product that is defined via a left - representation - like covariant representation .when is amenable , the two constructions yield the same algebra ( see ( * ? ? ?* theorem 4.6.2 ) ) .however , we ll be interested mainly in nonamenable groups that _ act _ amenably .we ve seen that amenable groups are defined via the canonical action of on . to define an amenable action we consider another canonical action . is the set of probability measures on  which we identify with the set of positive , norm - one elements in and topologize by restricting the weak- topology ( coming from ) .note that acts by left translation on ( just restrict the action on to absolutely summable sequences ) and this leaves invariant .hence we have a canonical action of on which we ll denote by for all and .( as before , . )though is never compact ( when is infinite ) , it is the prototype of an amenable action .we call a compact hausdorff space a _ -space _ if it is equipped with an action of ( by homeomorphisms ) .we let denote the action of on .[ defn : amenableactionii ] an action of on a compact space is called ( topologically ) _ amenable _ ( or , equivalently , is an _amenable -space _ ) if there exists a net of continuous maps , such that for each , where is the probability measure that associates to the point . is continuous if and only if for each convergent net we have for all . ]  in other words , an action of on is amenable if it can be asymptotically intertwined with the canonical action of on :   it is worthwhile to check that every action of an amenable group is amenable in the sense defined above .( if is a flner sequence , let map every point to the normalized characteristic function over .) however , it is a remarkable fact that virtually all groups one normally encounters act amenably on some compact space .( for example , all linear groups see . )[ rem : borel ] definition [ defn : amenableactionii ] requires the maps to be continuous .however , it is sufficient for the s to be borel , meaning that for every the function , is borel .see ( * ? ? ?* proposition 5.2.1 ) for a proof of this useful fact .the following result connects exactness with amenable actions .it is due to ozawa and , independently , anantharaman - delaroche , following an important contribution by guentner and kaminker .see ( * ? ? ?* chapter 5 ) for details .[ thm : hmm ] for a discrete group , the following are equivalent :  1 .[ thm : actsc1 ] is exact ; 2 .[ thm : actsc3 ] acts amenably on some compact topological space .  for our purposes ,the most important fact about amenable actions is that the associated crossed products are nuclear .the proof is somewhat technical , but is close in spirit to the proof of in theorem [ thm : amenablegroup ] , in that one can construct approximating maps by hand .see theorem 4.3.4 and lemma 4.3.7 in for a proof of the following fact .[ thm : crossedproductamenableaction ]if acts amenably on , then the associated ( reduced ) crossed product is nuclear .there is much more that could , and probably should , be said about amenable actions , but i do nt want to distract us from our goal of w-applications . hence i ll close this subsection with an exercise a very important exercise that we ll need later .[ exer : biexact ] assume acts on and there exist two -invariant subalgebras such that ( i ) is amenable while is trivial and ( ii ) is trivial while is amenable . prove that the action of on is amenable .since our goal is applications to group von neumann algebras , it will be necessary to stay on the hilbert space .crossed products do nt naturally act on this hilbert space ; at least , it is nt obvious that they do .the _ uniform roe algebra _ of is the c-subalgebra of generated by and .it turns out that the uniform roe algebra is a crossed product a very important one , too .the following result is a simple computation , but since we did nt define reduced crossed products , we refer the reader to ( * ? ? ?* proposition 5.1.3 ) for a proof .[ prop : roecross ] let be the left translation action .then there is an isomorphism such that and for all and .note that is universal in the following sense : if is a compact -space , then there is a covariant homomorphism and , thus , a -homomorphism . to see thissimply pick a point and consider the orbit .the ( automatically ) continuous map defined by extends to a continuous map from the stone - ech compactification to .moreover , it is equivariant when is equipped with the ( extension of the ) left translation action of on . in other words , picking a point determines a commutative diagram   from this observation it is easy to see that if acts amenably on , then the left translation action of on is also amenable .it follows that admits an amenable action on some compact space if and only if the left translation action of on is amenable .this remark is essentially useless if one hopes to determine whether or not a given group acts amenably on some space . but it s useful for other things ; combining the results above , we have   admits an amenable action on some compact hausdorff space if and only if is nuclear .the problem of constructing amenable actions is a hard one , at least for an operator algebraist .we ll soon see that our friends in geometric group theory are far better equipped to handle this problem , but before getting concrete let s set the stage with some general preparations .[ defn : grpcompactification ] a _ ( covariant ) compactification _ of a group is a compact topological space containing as an open dense subset and with the property that the left translation action of on extends to a continuous action of on .for such a compact space , there is a canonical embedding determined by the neutral element .namely , a function is identified with while is identified with .( injectivity follows from the density of in . ) _ for the remainder of these notes , we will make this identification without reference ._  [ defn : small ] a compactification is said to be _ small at infinity _ if for every net converging to a boundary point and every , one has that . is a quotient of the higson corona of . ]  to understand this definition , recall that also acts on by right translation ; i.e. , the mapping , where , and for all , defines a right action of on .it is worth pointing out that this action , like its leftist counterpart , is spatially implemented . butthis time the right regular representation does the job : , for all and .  to see what being small at infinity has to do with the right translation action , we suggest proving the following lemma .let be a group and be a compactification .the following are equivalent :  1 .the compactification is small at infinity ; 2 .the right translation action extends to a continuous action on in such a way that it is trivial on ; 3 .one has for every and .having wandered into the trees a bit , let s step back and take a look at the forest .recall that our goal is to give examples of groups satisfying the hypotheses of theorem [ thm : taka ] .our next result says that it suffices to find groups with sufficiently nice compactifications .[ prop : ao ] assume admits a compactification which is small at infinity and such that the left action of on is amenable .then the canonical map -continuous .hence , is solid .-algebra is exact , the existence of an amenable action implies exactness of the group ( since the associated crossed product is nuclear and contains ; cf .theorem [ thm : hmm ] ) .thus the hypotheses of this proposition imply that theorem [ thm : taka ] is applicable . ]it suffices to show that there exists a nuclear c-algebra such that and commutes with , where is the quotient map .indeed , if such exists , then we have an inclusion and a natural -homomorphism .so , let be a left - amenable compactification which is small at infinity .according to theorem [ thm : crossedproductamenableaction ] , the crossed product is nuclear and contains .so we only have to check that commutes with , after passing to the calkin algebra .  since is generated by and  and commutes with  it suffices to check that \in { { \mathbb k}}(\ell^2({\gamma})) ] , as desired .akemann and ostrand first proved that if is a free group , then is -continuous ( cf .hence , we refer to this as the _ akemann - ostrand property_.  a theorem of choi implies that free groups are exact ( cf .together with the work of akemann and ostrand , this implies that free groups satisfy the hypotheses of theorem [ thm : taka ] .hence , modulo citation , we have shown that free group factors are solid .- proof of solidity of free group factors ( cf . ) , but it does nt extend to the hyperbolic groups and other examples that c-techniques can reach . ] in particular , we deduce a celebrated result that was first proved using free probability theory .free group factors are prime .though there is nothing wrong with relying on the work of choi and akemann - ostrand , it turns out that more can be said much more if we instead prove that free groups satisfy the hypotheses of proposition [ prop : ao ] .that will be the topic of the next section .in this section we exploit gromov s view of discrete groups as geometric objects to show that the free group on two generators satisfies the hypotheses of proposition [ prop : ao ] .it turns out that this approach works equally well for general hyperbolic groups , but the ideas are particularly transparent if we first look at .recall that the cayley graph of a group has vertices labeled by the group elements and an edge between two vertices whenever the corresponding group elements differ by a generator ( in particular , it depends on the choice of a generating set ) .if we choose the canonical generating set of , denoted by the letters , then the center " of the cayley graph looks like this :    to construct the right compactification of , we have to consider geodesics in the cayley graph i.e. , paths that never backtrack .note that since the cayley graph of is a tree , there is a _unique _ geodesic connecting any pair of vertices .however , we must consider infinite geodesics , too .let denote the set of geodesics ( both finite and infinite ) in the cayley graph of which start at the neutral element .we can view this compactification as follows :    the circle around the outside represents the infinite geodesics and we think of each such path as converging to a boundary point .let me first describe the topology on in words . if is a finite path , then a net of paths converges to if and only if for all large .however , if is infinite , then means that the s agree with more and more and more ; a bit more precisely , for every there exists a such that the first -steps along the path agree with the first -steps along , for all .however , the topology on is best understood pictorially ; means :    for those uncomfortable with loose definitions ( like me ) , let s be more rigorous .first we identify a path with a ( finite or infinite ) reduced word in the obvious way :    then we identify reduced words with the sequences in determined by the following rule : either for all ( infinite words ) or , if , then for all ( finite words ) .it is a good exercise to check that restricting the product topology on to this copy of yields the topology described above .moreover , the complement of in is open ( another simple exercise ) , so is compact .now we must show that is a compactification in the sense of definition [ defn : grpcompactification ] ; that it is small at infinity ; and that the left action of on is amenable .first , we identify with the set of finite paths in .it is easy to check that is dense and open . to describe the left action of on , we use the reduced - word pictureindeed , if is a reduced word and then acts on by left concatenation and cancellation of any inverses , i.e. , if , then ( and otherwise a bit of cancellation may occur ) .clearly this extends the left action of on itself , and it s easily seen to be continuous . thus is a compactification .next we check the small - at - infinity condition which is my favorite part , because pictorially it s completely trivial .indeed , the right action of on itself is also by concatenation so if converge to an infinite path and is arbitrary , then evidently as well .finally , we must see why the left action of on is amenable .that is , we must construct continuous maps , such that for each , given , we write it as a reduced word and define , where .( if is a finite word , then we put for all . ) now , for we define by where is the point mass concentrated at . in the casethat is an infinite word , is just the normalized characteristic function over the first steps along the path determined by ; when is a finite word , is converging to the point mass at ( as ) .one checks that each is continuous , and for every we have where denotes the word length of .one can verify this inequality by hand , but it s easier to see geometrically .indeed , if is an infinite word and , then after cancellation we have for some .a computation shows that is just the normalized characteristic function over the first steps along the geodesic that starts at , goes back to , and then proceeds along the remainder of .thus , in pictures , we have :    it is clear from this picture that is zero over the intersection of the two paths , and the number of points where the paths do nt overlap is bounded above by . the case that is a finite path is similar , so we conclude that .letting completes the proof .it turns out that everything we ve done in this section extends to hyperbolic groups .the details are significantly more annoying , but the geometric intuition is virtually identical .hence , i ll sketch the ideas , but refer the reader to ( * ? ? ? * section 5.3 ) for details .let be a connected graph with the graph metric ( the distance between vertices is the length of the shortest path connecting them ) .for every pair of vertices , there exists a ( not necessarily unique ) geodesic path connecting to and we let  ] connecting them .quick question : if is a tree , what does a geodesic triangle look like ?that s right , it s awfully skinny .[ defn : hyperbolicgraph ] for , we say a geodesic triangle is _ -slim _ if each of its sides is contained in the open -tubular neighborhood of the union of the other two i.e. , \subset n_\delta([y , z]\cup[z , x]) ] and hence is generated by a single unitary element ( e.g. , ) .fixing such a generator , we define a c.p .map from into by where the limit is taken along some fixed ultrafilter .it is not hard to see that is a ( nonunital ) conditional expectation onto and that is a trace - preserving conditional expectation from onto . by uniqueness , one has for every .it follows that for and .   are nt the closing remarks of the preceding subsection reminiscent of the proof of theorem [ thm : taka ] ?well , we ll soon use them to prove an analogue of that result in the context of bi - exactness . butfirst , we have to recall an important theorem ( of popa ) that is also needed in the proof ( see ( * ? ? ?* appendix f ) for details ) .[ thm : popafinbimod ] let be finite von neumann algebras with separable predual and let be a nonzero projection .then , for a von neumann subalgebra , the following are equivalent :  1 .[ thm : popafinbimod1 ] there is no sequence of unitary elements in is a partial isometry in such that . ] in such that for every ; 2 .[ thm : popafinbimod2 ] there exists a positive element with such that the ultraweakly closed convex hull of does not contain ; 3 .[ thm : popafinbimod3 ] there exists a --submodule of with ; 4 .  [ thm : popafinbimod4 ] there exist nonzero projections and , a unital normal -homomorphism and a nonzero partial isometry such that and such that and .note that if there is a nonzero projection such that embeds in inside , then embeds in inside ( as condition ( [ thm : popafinbimod4 ] ) in theorem [ thm : popafinbimod ] evidently implies ) .[ cor : popafinbimod ] let be a finite von neumann algebra with separable predual and be a sequence of von neumann subalgebras .let be a von neumann subalgebra such that does not embed in inside for any .then , there exists a diffuse abelian von neumann subalgebra such that does not embed in inside for any .[ thm : gvnacn ] assume is bi - exact relative to and let be a projection and be a von neumann subalgebra .if the relative commutant is noninjective , then there exists such that embeds in inside .  by contradiction , suppose that the conclusion of the theorem is not true .then , by corollary [ cor : popafinbimod ] , there is a diffuse abelian von neumann subalgebra such that does not embed in inside for any . to get our contradiction, it suffices to show is injective ( cf .the proof of theorem [ thm : taka ] ) .  to do this , we ll use proposition [ prop : injcric ] ( and the remarks and notation which follow it ) with , , and .thus , it suffices to show is -continuous on .but to do this , it would be enough to know that , where is the map given to us by lemma [ lem : clrmid ] . since we already observed that for and , and we know , our task is further reduced to proving that  if we can do this , the proof is complete .  with , observe that is the jones projection onto and hence with .therefore , it follows from the definition of that is a positive element in such that .since we re assuming does not embed in inside , the ultraweakly closed convex hull of contains for every element of finite trace ; since for all unitaries , it follows that .finally , since is in the multiplicative domain of , this implies that for every and , or equivalently , .  to the von neumann algebraist that does nt yet appreciate c-algebras, i have a challenge : find a proof of the previous theorem that does nt depend on c-theory .- algebraists has the weight of history on its side unlike this w-challenge .hence , i ll be less surprised ( but still quite surprised ! ) if someone finds a w-proof of theorem [ thm : gvnacn ] someday .indeed , similar results have already been proved without appealing to c-algebras ( cf . , , ) . ]the next two pages demonstrate this theorem s remarkable power .let be diffuse and assume is not injective .then , by theorem [ thm : gvnacn ] , embeds in inside . by condition( 4 ) in theorem [ thm : popafinbimod ] , we can find a projection and a unital -homomorphism .thus is a direct summand of , which implies has a minimal projection .this contradicts our assumption that is diffuse .we ve already seen that hyperbolic groups satisfy the hypothesis of the previous result , but there are others .for example , ( * ? ? ? * corollary 15.3.9 ) states that the wreath product of an amenable group by a group which is bi - exact relative to is again bi - exact relative to .[ lem : dpvnac ] let be groups that are bi - exact relative to and be the direct product . define a family of subgroups of by then is bi - exact relative to .let be groups that are bi - exact relative to ( e.g. , hyperbolic groups ) and be noninjective -factors . if there exists an embedding for some projection , then .the proof is by induction on .if , we re simply claiming that noninjective subfactors of are prime which follows from corollary [ cor : bisolid ] .so assume the result holds for all products of length and we have an embedding .since we see that this relative commutant is noninjective .thus , theorem [ thm : gvnacn ] and lemma [ lem : dpvnac ] imply that , after permuting indices , one has for some nonzero projections and . by the induction hypothesis , , so we re done .  in particular , and ( for noninjective ii-factors ) ca nt be isomorphic , which extends ge s theorem that free group factors are prime .however , free entropy degenerates in tensor products , so ge s techniques ca nt be adapted to handle the case of higher tensor powers .inspired by the isomorphism problem for free group factors , we end this paper with one more application of theorem [ thm : gvnacn ] .but first we ll need two more of popa s w-theorems . heres an older result on normalizers in free products ( cf . ) .[ thm : freenormalizer ] if and are finite von neumann algebras and is diffuse , then there is no unitary such that .moreover , if , then .[ thm : finbimodii ] assume and are ii-factors , is diffuse and is a factor .if embeds in inside , then there exists a unitary such that .  to apply theorem [ thm : gvnacn ] to free products , we need to know that is bi - exact relative to the family of subgroups , whenever each is exactthis is nt trivial , but it is true see ( * ? ? ? * proposition 15.3.12 ) .[ lem : thelem2 ] assume , with each exact and i.c.c ., is infinite for all , and it ensures that is a factor . ] and has the following properties : each is a ii-factor , and is not injective .then , there exists and a unitary such that .  by theorem [ thm : gvnacn ] , there is an such that embeds in inside .since is diffuse and is a factor , theorem [ thm : finbimodii ] provides a unitary such that . from theorem[ thm : freenormalizer ] we have that .since , the proof is complete .first write where is nonamenable . then, by theorem [ thm : freenormalizer ] , is a noninjective ii-factor .thus , applying the previous lemma to each , we can find unitaries and indices such that .similarly , we can find unitaries and indices such that .hence ,    this result can be improved a bit ( cf .* corollary 15.3.15 ) ) , but it s good enough to imply the following striking result : if denotes the injective ii-factor , then without the s , this is the isomorphism problem for free group factors ( cf . , ) .nathanial p.brown and narutaka ozawa , _-algebras and finite - dimensional approximations _ , graduate studies in mathematics , * 88*. american mathematical society , providence , ri , 2008 .xvi+509 pp .

Abstract: these days it is common for young operator algebraists to know a lot about c-algebras , or a lot about von neumann algebras but not both . though a natural consequence of the breadth and depth of each subject , this is unfortunate as the interplay between the two theories has deep historical roots and has led to many beautiful results . we review some of these connections , in the context of amenability , with the hope of convincing ( younger ) readers that tribalism impedes progress .


ultra - high energy cosmic - ray ( uhecr ) particles are the highest energy particles observed by humankind , well beyond what is accessible at the large hadron collider .current experiments are recording large amounts of high quality data .recent observations do not draw a simple nor consistent picture of the nature of uhecr particles .there are hints of anisotropy in the arrival directions  , favouring the presence of light primary particles .using the current high energy hadronic interaction models , typical air - shower observables do not generally support the light component hypothesis : the pierre auger collaboration observes the depth of the electromagnetic shower maximum , muon production maximum , rise - time of the shower front , etc . that are all pointing in the direction of a primary composition dominated by heavy particles at ultra - high energies  . on the other hand ,the hires collaboration claims that their measurement of the depth of the shower maximum is compatible with protons up to the highest energies  , which is currently supported by the first ta data  .it is important to notice that whereas auger publishes in the atmosphere by using a fiducial volume selection to minimize acceptance biases , hires and ta presently do not apply such corrections and present at the detector level .also the possible difference of the energy scale between auger and hires / ta ( % )  contribute to the difficulties to explain the data  .the current situation could point to exciting physics aspects that are about to emerge with higher accumulated event statistics , as for instance , systematic differences of the cosmic - ray flux on the northern and southern hemisphere .the pierre auger observatory observes the southern , while hires / ta the norther sky , with very different astrophysical objects in their direct field - of - view .it is clear , that the understanding of the primary mass composition of uhecr is one of the major steps towards the final solution of the uhecr puzzle .depending on a reliable measurement of the mass composition very different scenarios of the nature of uhecr will finally emerge .  in this articlewe discuss the concept of _ observed _ versus _ true _-distributions to finally demonstrate that the _ observed _ average of depth of shower maxima , , can result in a non - linear relation with the average logarithm of the mass number , , of the uhecr .these effects are very specific for a given experimental setup and have to be accounted for in order to allow a comparison or interpretation of these data .fluorescence telescopes measure the ultraviolet light , which is proportional to the energy deposited by the passage of the charged particles of the air - shower cascade through the atmosphere .this makes possible to reconstruct the longitudinal profile of the electromagnetic shower  .the atmospheric depth at which the energy deposit is maximal is the shower maximum , , and is related to the nature of the primary particle . at the same primary energy , primary nuclei with mass produce air - showers with shower maxima at different atmospheric depths .this can be approximated as  where is the cross - section of cosmic - ray primaries with air and {g / cm^2} ] , which corresponds to h=1.5 km , as explained below , and the opening angle of the field - of - view is 20 .we consider air - showers of fixed primary energy , with random arrival directions , sampled from up to 60 zenith angle , as well as distances from the telescope , sampled from .the requirement of a particular shower to be observed is that at least photons reach the telescope from the location of the shower maximum on the shower axis .the number of photons is computed as where is the number of fluorescence photons emitted at the shower maximum , is the aperture of the telescope , is the geometric distance , the integrated depth distance from the telescope to the location of , and {g / cm^2} ] , of the telescope of {g / cm^2} ] , {g / cm^2} ] for the light component as well as {g / cm^2} ] and {g / cm^2} ] and an rms of {g / cm^2} ] and an rms of {g / cm^2} ] for the light and {g / cm^2} ] biased with respect to the input distributions .   from the energy and thus the distance of air - showers from the telescope .the upper axis indicates the resulting average distance of showers from the telescope depending on the primary energy.[fig : energy ] ]   as a function of the light cosmic - ray fraction , .the _ observed _ values for pure light and pure heavy compositions are shifted with respect to the _ true _ .the relation between and , and therefore with by eq .[ eq : lnaxmax ] , is linear for the true distributions , whereas it is non - linear for the case of the observed .the curve labeled `` observation a '' indicates the effect on a pure composition of intermediate mass . ]the bias , defined as the difference between the observed and true value , changes with distance to the telescope and thus also with the primary energy of the cosmic - ray particles according to eqs .( [ eq : ndet ] ) and ( [ eq : nph ] ) . in fig .[ fig : energy ] this is shown for the cases of pure light , pure heavy and an equal mixture of light and heavy primaries .for this study we consider a change of with energy of {g / cm^2}$ ] with respect to the default values used otherwise in this paper .it is interesting to note that the resulting bias depends on the underlying mass composition . in general , for smaller energies the showers are located closer to the telescope and the effect becomes stronger .for example , the low energy fluorescence telescope extensions heat  at the pierre auger observatory and tale  at the telescope array are important to limit these biases by providing a much wider field - of - view for showers at close distances .furthermore , the bias can be positive as well as negative , depending on the overlap of the true distribution with the telescope efficiency function , c.f .[ fig : eff ] .however , the impact will qualitatively be always as shown in fig .[ fig : energy ] : at short distances the field - of - view cuts shallow showers and thus is overestimated , at larger distances this effects becomes smaller and eventually might even reverse leading to an underestimation of the true value .it is important to realize that the primary effect is due to a different distance of showers , and that any parameter that affects the typical observation distance will have a similar impact .such parameters are typically related to the detector setup , for example the diaphragm opening , the sampling time , but also the atmospheric density profile and optical absorption characteristics .   on the non - linearity .]  if we consider a mixture of light and heavy particles at fixed primary energy with a given fraction of light and of heavy primary particles , the average observed shower maximum becomes where .this relation is non - linear in and thus not in . in figure[ f : fxmax ] we show the results of this study .the true as well as the observed are shown , together with the linear interpolation from eq .( [ eq : linear ] ) and the non - linear version eq .( [ eq : nonlinear ] ) .the latter describes perfectly well the results of our monte carlo study .we also study how a different detector setup affects our result .in general , by changing parameters of the telescope detector setup , very different values of can be obtained . in fig .[ fig : scan ] we show how the predictions of eq .( [ eq : nonlinear ] ) change for a wide range of different values of .it summarizes the effect described in this paper , and the miss - interpretation it can induce on the inference of the primary mass composition from .for instance , would imply that a real fraction 50% proton and 50% iron , that is f=0.5 and , would be interpreted as 75% proton and 25% iron , , if just the linear relation eq .( [ eq : linear ] ) is assumed .the data collected by a fluorescence telescope are affected by acceptance effects , which can have an important impact on the detailed interpretation of the data .this is problematic for researchers not from within a particular experimental collaboration who , for example , have no access to a detailed monte carlo simulation of the detector setup .the average efficiency , which describes the response to a flat distribution in , is one of the crucial properties of a fluorescence telescope . however , in addition to the acceptance effects also the detector resolution and reconstruction biases have to be known for a full data analysis .  we have demonstrated that the bias in the average shower maximum introduced by the acceptance of fluorescence telescopes induces a non linearity between and .we would like to thank a. olinto for the initial discussions that inspired this work during the taup2011 meeting at munich .j. bellido , d. harari , m. bueno , m. unger , r. conceio and m. pimenta for careful reading of this manuscript and their comments .finally we thank our colleagues from the pierre auger collaboration for helping us to get the necessary insight on the fluorescence detection techniques .l.c . wants to thank fundings by fundao para a cincia e tecnologia ( cern / fp11633/2010 ) , and fundings of mctes through poph - qren tipologia 4.2 , portugal , and european social fund .j. abraham _ et al . _[ pierre auger collaboration ] , phys .lett .  * 104 * , 091101 ( 2010 ) .facal , d. garcia - pinto , d. garcia - gamez for the pierre auger collaboration , 32nd international cosmic ray conference , beijing , china , august 2011 , arxiv:1107.4804 .matthews [ ta collaboration ] , nucl .* 212 - 213 * , 79 ( 2011 ) .d. ikeda to appear in the proceedings of the 12th internation conference of topics in astroparticle and underground physics , taup2011 , munich , 2011 .r. engel [ pierre auger collaboration ] , aip conf .proc .  * 1367 * , 50 ( 2011 ) .a. olinto , to appear in the proceedings of the 12th international conference of topics in astroparticle and underground physics , taup2011 , munich ( 2011 ) .

Abstract: the measurement of the average depth of the shower maximum is the most commonly used observable for the possible inference of the primary cosmic - ray mass composition . currently , different experimental collaborations process and present their data not in the same way , leading to problems in the comparability and interpretation of the results . whereas is expected to be proportional to in ideal conditions , we demonstrate that the finite field - of - view of fluorescence telescopes plus the attenuation in the atmosphere introduce a non - linearity into this relation , which is specific for each particular detector setup .   cosmic - rays , extensive air - showers , shower maximum , telescope , field - of - view


the portfolio selection problem is one of the most interesting and important topics in investment theory which plays an important role in financial research nowadays .a huge number of papers are dealing with a mean - variance portfolio and its characteristics ( e.g. , markowitz ( 1952 ) , tobin ( 1958 ) , merton ( 1972 ) , gibbons et al .( 1989 ) , jobson and korkie ( 1989 ) , white ( 1998 ) , korkie and turtle ( 2002 ) , okhrin and schmid ( 2006 ) , basak et al .( 2009 ) ) .currently , the mean - variance analysis derived by markowitz ( 1952 ) is of great importance for both researchers and practitioners on the financial sector ( cf . litterman ( 2003 ) , brandt ( 2010 ) ) .it is the first systematic treatment of an investor s conflicting objectives , a high return versus a low risk .the mean - variance analysis is based on a quadratic parametric optimization model for the single - period portfolio selection problem .an explicit analytical solution is obtained for an investor trying to maximize his expected wealth without exceeding a predetermined risk level and an investor trying to minimize his risk ensuring a predetermined wealth respectively .alternatively , two other quadratic optimization procedures have been recently considered in literature ( see , e.g. , ingersoll ( 1987 ) , brandt and santa - clara ( 2006 ) , bodnar and schmid ( 2008b , 2009 ) , elikyurt and zekici ( 2007 ) , fu et al . , ( 2010 ) , cesarone et al.,(2011 ) ) .note that a quadratic optimization problem satisfies the bernoulli principle without imposing the assumption of normality on the distribution of the asset returns ( see tobin ( 1958 ) ) .  in thefollowing we consider an investor who holds a portfolio consisting of assets .the -dimensional vector of the asset returns is denoted by .let be the mean vector of , i.e. , and be its covariance matrix .we assume that is positive definite .the vector and the vector denote the -dimensional vector of portfolio weights and the -dimensional vector of ones , respectively .the first problem considered in the paper is the so - called mean - variance optimization problem suggested in the seminal paper of markowitz ( 1952 ) .kroll et al .( 1984 ) showed that the mean - variance optimal portfolio also possesses a maximum expected utility or is very close to this value for utility functions commonly used in financial literature . in the optimization problem of markowitzan investor is considered who minimizes the portfolio risk for a given level of the expected return .it is formulated as throughout the paper we refer to ( [ problem1 ] ) as the markowitz mean - variance ( m ) optimization problem .secondly , we consider an investor who maximizes the mean - variance utility function .the corresponding optimization procedure , to which we refer as the mean - variance utility ( mvu ) problem , is given by the quantity is the slope parameter of the quadratic utility function . in generalit is not equal to the risk aversion coefficient as defined by pratt ( 1964 ) .moreover , the quantity determines the risk of the investor under the assumption of a normal distribution .note that the mean - variance utility function assuming normally distributed asset returns coincides with the exponential ( or cara ) utility function where denotes the investor s wealth ( see , e.g. , merton ( 1969 ) , elikyurt and zekici ( 2009 ) ) .the comparison of different utility functions can be found in yu et al .( 2009 ) .the portfolio that maximizes ( [ problem2 ] ) is called the expected utility ( eu ) portfolio .it has been heavily discussed in financial literature recently ( see , e.g. , ingersoll ( 1987 ) , okhrin and schmid ( 2006 ) , bodnar and schmid ( 2008b , 2009 ) ) . while okhrin and schmid ( 2006 ) obtained the first two moments of the estimated weights of the eu portfolio , bodnar and schmid ( 2008b , 2009 ) derived the sample distributions of its estimated expected return and variance .  finally , our third optimization procedure is , usually , considered in the multi - stage portfolio selection theory ( see , e.g. , brandt and santa - clara ( 2006 ) ) .it is based on the following quadratic utility function , where denotes the investor s wealth and determines the investor attitude towards risk .it is formulated as where with and where is the initial value of the wealth . without loss of generalitywe put .we refer to ( [ problem3 ] ) as the quadratic utility ( qu ) optimization problem .  in table 1we summarize the optimization problems ( [ problem1])-([problem3 ] ) and present their solutions .although these three quadratic optimization problems , i.e , the markowitz mean - variance problem ( m ) as well as the problems based on the mean - variance utility function ( mvu ) and the quadratic utility ( qu ) , are intensively discussed in financial theory and financial practice we have not found papers which compare the three considered quadratic optimization problems with each other .the comparison of different existing models is always meaningful to understand the theory more deeply .the only papers , where a similar problem has been briefly discussed , are the papers written by li and ng ( 2000 ) and leippold et al .( 2004 ) . in these papersit was pointed out that the three quadratic optimization problems are mathematically equivalent but not economically .although their solutions are lying on the same set in the mean - variance space , we show that they are not obviously the same. solutions of the three considered optimization problems lie on a parabola in the mean - variance space .however , only using the ( mvu ) optimization problem , the investor always gets a mean - variance efficient portfolio because only in this case all solutions lie on the upper part of the parabola .this is not the case for the optimization problem ( m ) and ( qu ) since in these cases we can get portfolios from the lower part of the parabola .nevertheless , they can be made equivalent by adding a corresponding constraint in markowitz s problem as well as in the problem based on the quadratic utility function .however , the constraints themselves depend on the unknown quantities and , hence , they can not be checked in practice . as a result , the three problems are mathematically equivalent but not from a stochastic point of view . ]   &  ' '' ''  ' '' ''   + &  ' '' ''  ' '' ''   + &  ' '' ''  ' '' ''   +  in this paper these results are extended in several directions .first , we derive sufficient conditions for the mathematical equivalence which are new and have not been previously discussed in financial literature to the best of our knowledge .these conditions only depend on the parameters of the asset returns .the investor only has to change his risk aversion parameter to obtain an equivalent solution for another optimization problem .second , the influence of parameter uncertainty on the three optimization problems is investigated .it holds that the parameters of the asset returns , namely the mean vector and the covariance matrix , are usually unknown in practice . as a result, the sufficient conditions for the equivalence can not be directly verified in practice since they are functions of these two parameters .the question arises whether the solutions of markowitz s problem and of the quadratic utility problem are always mean - variance efficient . if not then we would like to know how likely are inefficient portfolios obtained . in order to take the parameter uncertainty into accountwe replace the parameters in the equivalence conditions by their sample estimators .we derive the distributions of these estimated quantities assuming that the asset returns are independently and identically normally distributed .although the assumption of normality is heavily criticized in financial literature , it can be applied in the case of the mean - variance investor ( see tu and zhou ( 2004 ) ) .furthermore , by the detailed analysis of the probability that the solutions of the optimization problems coincide , we derive a test whether a given solution is mean - variance efficient .contrary to the classical testing theory for the mean - variance efficiency of a portfolio ( see e.g. gibbons et al .( 1989 ) , britten - jones ( 1999 ) , bodnar and schmid ( 2009 ) ) , the suggested test on efficiency is constructed under the alternative hypothesis and , consequently , it can be accepted . in order to obtain the finite sample distributions of the estimators and exact tests on the corresponding population quantities ,the results of bodnar and schmid ( 2008b , 2009 ) are used .moreover , using real data we show that if these parameters are estimated and replaced by the corresponding estimators then the probability of getting an inefficient portfolio by using markowitz s optimization problem and the optimization problem based on the quadratic utility function can reach .the rest of the paper is organized as follows . in section 2we discuss the case with known parameters . in theorem 1conditions are derived under which the solution of ( [ problem1 ] ) coincides with the solution of ( [ problem2 ] ) and as the solution of ( [ problem3 ] ) is the same as the solution of ( [ problem2 ] ) , respectively .it appears that these conditions depend on the unknown parameters and . in section 3 the problem of parameter uncertaintyis taken into account .we replace the parameters and in ( [ problem1])-([problem3 ] ) by their sample estimators . by doing thiswe get estimators of the corresponding optimal portfolio weights . in theorem 2, we derive the probability that the estimated portfolio based on ( [ problem1 ] ) ( as well as the estimated solution based on ( [ problem3 ] ) ) are not mean - variance efficient . in section 3.2 a test theory for the mean - variance efficiency is developed . an empirical illustration is provided in section 4 .section 5 presents final remarks .the proof of theorem 1 is given in the appendix ( section 6 ) .throughout this section we assume that both parameters of the asset return distribution , i.e. and , are known quantities . under this assumption merton ( 1972 )showed that all solutions of ( [ problem1 ] ) are lying on a parabola in the mean - variance space .the upper part of this parabola is called the efficient frontier , which is the set of all optimal portfolios .the parabola is fully defined by the three parameters , which are known as the efficient set constants ( see , e.g. pennacchi ( 2008 ) ) .  using ( [ problem2 ] ) bodnar and schmid ( 2008b ) suggested an equivalent representation of the efficient frontier given by where are the expected return and the variance of the global minimum variance portfolio ( gmv ) which is the portfolio with the smallest variance .the quantity denotes the slope coefficient of the parabola .the set of characteristics is easier tractable than the set of the constants introduced by merton ( 1972 ) .moreover , bodnar and schmid ( 2008b ) showed that the solution of ( [ problem2 ] ) for coincides with the efficient frontier , i.e. all optimal portfolios obtained by maximizing given are mean - variance efficient .it is important to note that in general the solution of ( [ problem1 ] ) as well as the solution of ( [ problem3 ] ) do not coincide with the efficient frontier .they both lie on the parabola in the mean - variance space , but these sets can be larger or smaller than the efficient frontier .the aim of this section is to provide a comprehensive analysis of this point .first , we present a short illustration that fully motivates the topic . for this reason ,we make use of which are considered as the true parameters of the efficient frontier in this example . in figure 1we plot the solutions of the m problem and of the qu problem for and .we observe that the graphs of the solutions of ( [ problem1 ] ) as well as ( [ problem3 ] ) include that of ( [ problem2 ] ) . while the graph of the solutions of ( [ problem2 ] ) is equal to the upper part of the parabola , the efficient frontier , the solutions of ( [ problem1 ] ) and ( [ problem3 ] ) may lie on the lower part , respectively .it has to be noted that the lower part of the parabola does not consist of optimal portfolios any longer , since for the same value of the variance we are able to specify a portfolio from the upper part of the parabola which has a larger expected return .moreover , the lower parts of the corresponding solutions are quite large in both cases , especially in the case of the qu optimization procedure .  in theorem 1 we state the conditions under which the optimization problems ( [ problem1 ] ) , ( [ problem2 ] ) , and ( [ problem3 ] ) are equivalent ,i.e. they all generate a solution that lies on the efficient frontier .let be a random return vector with mean vector and positive definite covariance matrix .a.  let be given .then a solution of ( [ problem1 ] ) is a solution of ( [ problem2 ] ) and it is obtained by choosing equal to if , however , then the solution of ( [ problem1 ] ) does not lie on the efficient frontier and thus it is not a solution of ( [ problem2 ] ) .b.  let be given .then a solution of ( [ problem3 ] ) is a solution of ( [ problem2 ] ) and it is obtained by choosing equal to if , however , then the solution of ( [ problem3 ] ) does not lie on the efficient frontier and thus it is not a solution of ( [ problem2 ] ) .the proof of theorem 1 is given in the appendix .for ( ) the solution is the global minimum variance portfolio .similar findings are also given in leippold et al .( 2004 ) who pointed out that the optimization problems are mathematically equivalent but not economically .however , no condition of the equivalence of ( [ problem1 ] ) , ( [ problem2 ] ) , and ( [ problem3 ] ) is presented in that paper . if is a known quantity then the conditions of mean - variance efficiency in theorem 1a can be directly checked .however , the quantities and are unknown in practice and have to be replaced by the suitable estimators . in this casewe are unable to guarantee that the solution of ( [ problem1 ] ) or / and ( [ problem3 ] ) lies always on the estimated efficient frontier .the problem is much more difficult in this case and it will be treated in section 3 .  in the empirical illustration considered in figure 1 , it holds that for and for . for these values of and the corresponding solutions of ( [ problem1 ] ) and ( [ problem3 ] ) are no longer mean - variance optimal portfolios .[ cols= " < , < " , ]   the above discussed testing procedures possess several nice interpretation .they present us an intuition about how large the values of and have to be chosen in order to guarantee investors using ( [ problem1 ] ) and ( [ problem3 ] ) to get an efficient portfolio and , consequently , avoid unexpected losses .moreover , the importance of these tests seems to increase in the multi - period portfolio context where the above mentioned optimization problems are commonly used and the unexpected loss can be accumulated from one period to another .thus , for investors using ( [ problem1 ] ) and ( [ problem3 ] ) it is reasonable to correct the parameters and before constructing the portfolio .no correction is necessary for investor using the optimization problem ( [ problem2 ] ) .the assumption of normality and independence of the asset returns is , of course , very restrictive .however , assuming more complicated distributions or processes it is very difficult to get such expressions as derived in section 3 .the real asset returns are heavily tailed with a high peak in the center so that the probability of getting an inefficient portfolio can be even larger .that is why we consider the above mentioned procedure as a benchmark how the probability of mean - variance inefficiency can be determined .in 1952 g. markowitz suggested a novel approach how a optimal portfolio can be constructed .the idea of markowitz was to minimize the portfolio variance for a given level of return .this optimization problem is known in literature as the mean - variance analysis .recently , other optimization procedures have been suggested ( see , e.g. , ingersoll ( 1987 ) , brandt and santa - clara ( 2006 ) , bodnar and schmid ( 2008b , 2009 ) ) which are based on quadratic utility functions .  in the present paperwe compare three approaches with each other .it turns out that under some conditions they are mathematically equivalent but not always mean - variance efficient .it holds that only all optimal portfolios obtained by maximizing the mean - variance utility function are mean - variance efficient .conditions are derived under which the solutions of the other considered optimizations problems lay on the efficient frontier .these conditions , however , can not be checked in a practical situation because they all depend on the unknown parameters of the asset returns .we provide a comprehensive treatment of this problem by deriving the exact expressions of the probabilities that the estimated solutions of the markowitz problem and the one obtained by maximizing the quadratic utility are mean - variance efficient . because the derived probabilities deviate from onewe conclude that the three optimization problems are not stochastically equivalent .finally , exact tests for the mean - variance efficiency of the obtained solutions are developed .in this section the proof of theorem 1 is given .* proof of theorem 1 : *  a.  from merton ( 1972 ) the solution of ( [ problem1 ] ) is given by where the constants , , , , , and are defined at the beginning of section 2 . with some mathematical calculations ,we get + the last expression is the solution of ( [ problem2 ] ) with ( see , e.g. , bodnar and schmid ( 2008b , p. 1003 ) ) .the statement of the theorem is proved .b.  the solution of ( [ problem3 ] ) is given by + the application of the sherman - morrison formula ( harville ( 1997 , theorem 18.2.8 ) ) leads to + because and we get where the last equality follows from .the theorem is proved .the authors are thankful to the referees and the editor for their suggestions which have improved the presentation in the paper .bodnar , t. and w. schmid , ( 2008b ) , estimation of optimal portfolio compositions for gaussian returns ._ statistics decisions _ 26 , 179 - 201 .bodnar , t. and w. schmid , ( 2009 ) , econometrical analysis of the sample efficient frontier . _ the european journal of finance _ 15 , 317 - 335 .jobson , j.d . and b. korkie , ( 1989 ) , a performance interpretation of multivariate tests of asset set intersection , spanning , and mean - variance efficiency ._ journal of financial and quantitative analysis _ 24 , 185 - 204 .yu , b.w.t ., pang , w.k . , troutt , m.d . and s.h .hou , ( 2009 ) , objective comparisons of the optimal portfolios corresponding to different utility functions ._ european journal of operational research _ 199 , 604 - 610 .

Abstract: in the paper , we consider three quadratic optimization problems which are frequently applied in portfolio theory , i.e , the markowitz mean - variance problem as well as the problems based on the mean - variance utility function and the quadratic utility . conditions are derived under which the solutions of these three optimization procedures coincide and are lying on the efficient frontier , the set of mean - variance optimal portfolios . it is shown that the solutions of the markowitz optimization problem and the quadratic utility problem are not always mean - variance efficient .   the conditions for the mean - variance efficiency of the solutions depend on the unknown parameters of the asset returns . we deal with the problem of parameter uncertainty in detail and derive the probabilities that the estimated solutions of the markowitz problem and the quadratic utility problem are mean - variance efficient . because these probabilities deviate from one the above mentioned quadratic optimization problems are not stochastically equivalent . the obtained results are illustrated by an empirical study .   * on the equivalence of quadratic optimization problems commonly used in portfolio theory * + taras bodnar , nestor parolya and wolfgang schmid corresponding author . e - mail address : schmid-frankfurt-o.de ]  +  + _ department of statistics , european university viadrina , po box 1786 , 15207 frankfurt ( oder ) , germany _ +  jel classification : g11 , c18 , c44 , c54 + _ keywords _ : investment analysis , mean - variance analysis , parameter uncertainty , interval estimation , test theory .


it is as well theoretically proven as experimentally tested that the exploitation of entangled states improves our ability to manage information in several ways .the development of quantum information theory is definitely due to this fundamental result .however , it has been recently discovered that even multipartite separable states can play a relevant role in performing better - than - classical communication and information protocols . in general , then , the usefulness of a particular state of a quantum system for such tasks can be traced back to the presence of internal correlations among parts of the system , i.e. , to the knowledge that an observer alice , probing a subsystem , gains about the state of another subsystem , controlled by another observer bob , and vice versa .  the quantitative and qualitative evaluation of such correlations and the proper discrimination of their nature classical versus quantum stand as open problems .several quantifiers of non - classicality of correlations have been introduced in literature ( not to be confused with the non - classicality of quantum optical states ) , but there are still neither clear criteria of faithfulness or _ bona - fide_-ness for them , nor a well established hierarchy of reliability . in spite of that, they are heavily used in current research .the most popular one is by far the _ quantum discord _ , a measure of non classical correlations which goes beyond entanglement and whose definition has an immediate interpretation in information theory : discord equals the difference of the total correlations between two subsystems and before and after a local measurement process is performed on one of them .the quantum discord admits at least three other operational interpretations , in contexts ranging from thermodynamics to communication protocols , such as the quantum state merging .the evaluation of the quantum discord is a hard task from a computational point of view , implying an optimization of the conditional entropy between alice and bob , , over all local ( generalized ) measurements on one party , which is often obtainable by numerical methods only .a closed analytical solution is known in the case of arbitrary two mode gaussian states , under the restriction of gaussian local measurements . narrowing our overview to two qubit states , an analytical expression of discord has been derived in particular for the subclass of so - called  , and a successful attempt to generalize this procedure has not been advanced yet ( to the best of our knowledge ) , apart from an upper bound quantity for the discord defined in .the difficulty in calculating quantum discord motivated the introduction of alternative measures of non - classical correlations .in particular , the _ geometric discord _ is one such a measure , which quantifies the amount of non - classical correlations of a state in terms of its minimal distance from the set of genuinely classical states .the geometric discord involves a simpler optimization and is easily computable analytically for general two qubit states .however , its relationship with the original quantum discord is not entirely clear at the present stage .  in this work, we present an algorithm to calculate quantum discord for general two qubit states .first , we obtain an explicit and simplified expression for the conditional entropy , exploiting the bloch representation of the density matrix ; then , we employ new variables that allow us to set the optimization conditions in a closed form . finally , we associate them to constraints over the eigenvalues of the statistical ensemble obtained after the measurement process .our approach qualifies as the most efficient and reliable way to evaluate quantum discord for arbitrary states of two qubits .  exploiting our algorithm, we perform a detailed numerical exploration of the hilbert space of two qubit states to compare quantum discord and the geometric discord as quantifiers of non - classical correlations .we shed light on the relationship between these two quantities by identifying the states that extremize geometric discord at fixed quantum discord ( and vice versa ) .we are motivated by the aim of establishing a reliable hierarchy of non - classical states based on physically and mathematically consistent criteria .we find that , interestingly , the quantum discord of a two qubit state can never exceed its ( normalized ) geometric discord .in analogy with the study of maximal entanglement , we find that the notion of maximal non - classical correlations is measure dependent : therefore , the feasibility in a specific experimental realization will determine which , among the various classes of maximally quantumly correlated states , are the most suitable ones to be employed for applications .the paper is organized as follows .section [ sec1 ] provides an introduction to quantum discord and geometric discord ( hereafter denoted by and , respectively ) , presenting definitions , main properties and a summary of the results obtained in the literature . in section [ sec2 ] , we pursue an analytical approach to the calculation of quantum discord for general two qubit states , eventually recasting the optimization problem into a system of two transcendental equations , whose solution specifies the local measurement that minimizes the conditional entropy .section [ sec3 ] concerns the comparison between quantum discord and geometric discord by using the results of the previous sections ; we identify the classes of states with maximal and minimal geometric discord at fixed quantum discord .finally , section [ sec4 ] recalls the main results of our work and suggests further issues worthy of investigation .one of the lessons we can apprehend from quantum mechanics is that the measurement process disturbs the state in which a physical system is set .this differs from what happens in the classical scenario , hence it is possible to conclude that the disturbance induced by a measurement on a state is a good evidence of its `` quantumness '' .now , let us suppose to have a bipartite system in a certain state and make a measurement on one of its subsystems .we can analyze the nature of the internal correlations of the system in such a state by studying in which way they are affected by the measurement .information theory provides the tools to accomplish this task .  as a starting point ,let us consider the case of information stored in two classical probability distributions .the quantity expressing the total amount of correlations between two random variables and , assuming values with probability , is the _ mutual information _ , defined as where is the shannon entropy associated to the random variable ; consequently is the entropy for while is the joint entropy of and .following bayesian rules , we can retrieve equivalent formulations , , where here the conditional entropy is straightforwardly defined as , and represents how much uncertainty ( ignorance ) we have on given the value of ( and vice versa for ) .  in the quantum scenario, we consider a bipartite system described by a density matrix , and subsystems with marginal density matrices .the mutual information can be used once more to quantify the total correlations between and .the quantum analogue of the expression ( [ ixy ] ) straightforwardly reads where  ] ,  ] is the square of the hilbert schmidt norm of a matrix , the geometric discord has been introduced as   it is possible to obtain an explicit closed expression of for two qubit states .first , one needs to express the density matrix of a two qubit state in the so called bloch basis : where  ] given by the solution to .the geometric discord of these states is simply , while their quantum discord is calculated in .we highlight the presence of the `` pimple '' at the joint between branches ( i ) and ( ii ) , a recurring feature in the profile of extremal states involving quantum discord .3 .  ( red online )this branch accommodates asymmetric of the form with \,,\ ] ] solution to . for these states , and $ ] .4 .  ( black online )the top - right - most branch accommodates just pure states , for which the discord equals the marginal von neumann entropy , , and the geometric discord equals the marginal linear entropy , , where we have denoted the eigenvalues of the reduced density matrix by .  on the other hand , the upper boundary ( v ) in fig .[ zurvsgeo ] , despite being single branched ,is more involved and we are unable to provide a tractable parametrization of the states that saturate it . they can be sought among symmetric of full rank , but with the two biggest eigenvalues dominating the other two . the extremal curve has been obtained as the result of extensive numerical optimization , in which the parameter space has been finely sliced in discrete intervals of nearly constant discord , and for each interval the datapoint corresponding to the random state with the maximum geometric discord has been selected . joining all such extremal stateswe have obtained the smooth ( magenta online ) line of fig .[ zurvsgeo ] .the two measures and correctly coincide on classical quantum states eq .( [ cq ] ) , where both vanish , and on maximally entangled bell states , where both reach unity .we have presented a reliable and effective algorithm for the evaluation of the quantum discord of general two qubit states .we have simplified the optimization involved in calculating the conditional entropy , by removing the redundant degrees of freedom that can be set to zero by means of local unitaries in the first place , and by properly taking into account the symmetries of the problem . the optimization problem for the conditional entropy , and equivalently for the discord ,is recast into a compact form that implies an elegant relationship among the eigenvalues of the ensemble obtained after the local measurement process on one qubit .the derived transcendental constraints are amenable to direct numerical solution .we have then compared quantum discord with an alternative but affine quantity , the geometric discord , identifying the classes of states with extremal values of geometric discord at fixed quantum discord . for a fixed geometric discord ,maximal quantum discord is attained by different families of states depending on the degree of non - classical correlations , encompassing pure as well as mixed , symmetric and nonsymmetric states . in general , the hierarchical bound holds for all two qubit states .we hope that our results can provide further insight into the fascinating but still not well - understood paradigm of non - classical correlations beyond entanglement in composite quantum systems .the methods presented here can be generalized to higher dimensional and continuous variable systems .a. datta , s. t. flammia , and c. m. caves , phys .a * 72 * 042316 ( 2005 ) ; 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Abstract: we present a reliable algorithm to evaluate quantum discord for general two qubit states , amending and extending an approach recently put forward for the subclass of . a closed expression for the discord of arbitrary states of two qubits can not be obtained , as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters . we apply our algorithm to run a numerical comparison between quantum discord and an alternative , computable measure of non - classical correlations , namely the geometric discord . we identify the extremally non - classically correlated two  qubit states according to the ( normalized ) geometric discord , at fixed value of the conventional quantum discord . the latter can not exceed the square root of the former for systems of two qubits .