6
$\begingroup$

Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \varphi_L : P \to Q $ of lattices. My question is, what can we say about the relation between $ M $ and $ N $, knowing only that their projection lattice structures are isomorphic?

For example,

  1. Suppose $ M $ is a factor, and $ P $ is isomorphic to $ Q $ as lattices, is $ N $ a factor as well? If so, do they have the same type?

  2. Are there any examples where $ P $ and $ Q $ are isomorphic as lattices, but $ M $ and $ N $ are not isomorphic as involutive algebras?

Here is some of my thoughts about question 1.

I believe that we can conclude $ N $ is also a factor in this case. If I remember correctly, we can identify factors as von Neumann algebras which can not be further decomposed as the direct sum of two (smaller) von Neumann algebras (can anyone give a reference for a proof of this fact, or in the case I am wrong, a counterexample?). Such a direct sum decomposition of a von Neumann algebra would be reflected on its projection lattice structure.

As for the types of $M$ and $N$. By the fundamental theorem of projective geometry, it is easy to see that if $ M $ is a type $ I_n $ factor, $ N $ a type $ I_m $ factor, and $ P $ is isomorphic to $ Q $ as lattices, then $ m = n $. This naturally raises the question of whether the same can be said for factors of other types.

Question 2 seems more intractable to me. But it sure is interesting in its own right. I think maybe some easy examples can be constructed, as it seems too wild to conjecture that $M$ is isomorphic to $N$ if $P$ is isomorphic to $Q$. Yet, I can not produce such an example after some effort.

Perhaps answers to these questions are well-known among experts, as they seem rather basic to me. In this case, please kindly point out the relevant references.

$\endgroup$
4
  • 1
    $\begingroup$ This is close to a duplicate of: mathoverflow.net/questions/256062/… In particular, counterexamples to 2 are given by Connes' examples of von Neumann algebras not *-isomorphic to their opposites: jstor.org/stable/1970940 $\endgroup$ Oct 18, 2018 at 11:50
  • $\begingroup$ @RobertFurber Thanks for the link. Am I right in assuming question 1 is still unanswered? The type of the factor depends not only on the projection lattice, but also on the von Neumann-Murray comparaison theory. I don't see exactly how the Jordan $\ast$-homomorphism can be used to "detect" the comparaison of projections. $\endgroup$ Oct 18, 2018 at 12:35
  • $\begingroup$ Question 1 is answered here (plus the fact that type I factors are characterized by having minimal projections). $\endgroup$
    – Nik Weaver
    Oct 18, 2018 at 13:59
  • $\begingroup$ This paper seems directly relevant to your second question; it characterizes lattice isomorphisms between projection lattices of von Neumann algebras in some cases, e.g. type III factors: arxiv.org/abs/2006.08959 $\endgroup$ Apr 7, 2021 at 16:26

1 Answer 1

2
$\begingroup$

Related to Which complete orthomodular lattices arise from von Neumann algebras? but I post only one answer here.

A improved version of the question is: extend von Neumann equivalence for finite factors to an equivalence (again in absence of type I$_1$ and I$_2$ components) between a real vNa $A$ (as ring or $*$-ring), its ($*$-)ring of classical quotients $Q(A)$, given by locally measurable possibly unbounded operators, and its (ortho)lattice of projections $L(A)$. This is answered here explicitly in presence of orthocomplementation and only implicitly in absence. Here I make the involutionless case explicit, after a recall of some things from that post.

[Usual disclaimer: in 1992 - 1994 my brain was still half working, and I am confident about things that I knew at that time. In 2014 much less so, and today nothing at all. Luckily these are things that I mostly studied back then so unless I completely mess up my rememberings the great picture is correct]

[Disclaimer$^2$. My point of view wants to maximize interaction between lattice and rings, with language translations of equivalent conditions. It searches to minimize computations (leaved to luckily already known results). Not everyone likes this (see Kadison's memoir of a Kaplansky - Segal interaction about operator algebras). Moreover, a complete picture of equivalences was liked by von Neumann much more than by others, and is usually considered bad didactics.]

Recall from the type III post, for a real or complex, here better seen as real polarizable, vNa $A$ with no type I$_1$ or type I$_2$ components:

  • The lattice $L(A)$ (without involution) is equivalent to the classical ring $Q(A)$ of (right, and left by involution) quotients of $A$ realized as locally measurable operators. following Saito (1971)

[When $A$ is finite i.e. $L$ is modular, then the all affiliated operators are (locally) measurable and $Q$ is also the classical and maximal ring of (right) quotients of $A$, with unique extension of the involution. If not, then the involution does not extend to the maximal ring of maximal right quotients: it is always regular and right self-injective, so a involution can be there only when its coordinatized lattice is a continuous geometry.]

The equivalence of $*$-rings between $A$ and $Q$ is given by the explicit Saito - Berberian construction of $Q$ from $A$ (Saito, theorem 4.2), with the reconstruction of $A$ from $Q$ given by the "subring of bounded elements". As seen by Saito and Berberian, $Q$ is a Baer$^*$-ring with the same projections as $A$, and the classical ring of quotients (every $q\in Q$ has the form $a/b$ with $a,b\in A$ and $b$ invertible in $Q$).

This implies that $Q$ as ring depends only on the ring $A$, and as Baer / Rickart ring its lattice is the same as that of $A$: even if $Q$ has more idempotents than $A$, the order associated to the divisibility preorder (is the associated lattice $L$ and) is the same thanks to projections giving a unique representative in each equivalence class.

The idempotents for one side generate the ring and for the other side they are identified with particular complementary pairs in $L$. In detail:

when matrix units of order at least 3 exist (and the decomposition and dimension theory, which depends only on $L$, gives a reduction to such a case), the ring is generated by its idempotents $e,f,...$ with the relations $e\oplus f=e+f-fe$ when $ef=0$, and these relations depends only upon the lattice: a $e$ "is" a complementary modular pair in $L$ (thanks to O-symmetry, all usual modularity conditions on a pair are equivalent; "independent" pairs) "kernel and image of $e$"; $ef=0$ is "image of the first in the kernel of the second" and then the partial operation $\oplus$ on idempotents is "independent join of the images, dually for kernels".

Here the $Q$ is the largest possible ring over $L$ to be realized by right ideals generated by a idempotent i.e. with algebraic complement: all "independent" complementary pairs in $L$ give a idempotent in $Q$ (and conversely only "independent" complements in $L$ can come from a Rickart ring). The algebraic sum of corresponding subspaces of $H$ (kernel and image of the idempotent as linear operator) could be only dense (and so one has a densely defined linear operator); the important point is the validity of the four modularity conditions for a pair in $L(A)$ (not in the larger $L(H)$); this permits the construction of the ring from the generators and relations, realized by concrete linear algebra computations with closed subspaces of $H$.

The purely algebraic setting of this is common in the general theory of rings of quotients (see for example Rowen's book about ring theory); one ends up with only densely defined operators, and it is decisive the identification of maps which are defined and coincide on a "dense" common part. Here one has endomaps and a modular lattice setting (abelian subobjects); the "inspiring" cases are instead with distributive lattices and functions towards a fixed external structure, like the real numbers: maps which are continuous and coincident on a dense open subset (a dense G$_\delta$ set in a Baire setting; outside a set in a ideal of zero measure set is another common setting; rational maps and Zariski topology in algebraic geometry).

While $A$ can be reconstructed from $L$ following the "linear algebra proof" of von Neumann coordinatization using linear endomorphisms of $H$, for $Q$ one must use also tricks (identify operators equal on a "large/dense" common domain, as usual in the theory of ring of quotients); however, for any countable subring of $Q$ a common dense domain of definition exists (like intersection of dense G$_\delta$ sets in Baire spaces, or co-negligible sets in measure spaces) that permits to realize the subring with linear operators in a dense subspace of $H$.

Note that what matters here is not what kind of ring of quotients form the locally measurable operators, or the other specific properties above (like the fact that the idempotents of $Q$ are exactly the "independent" complementary pairs in $L$, and not other kinds of complementary pairs). Even if having an equivalence also with $Q$ is a nice complement to the global picture, the above is only used as model for the reconstruction of $A$ from $L$, using another kind of complementary pairs.

  • The ring $A$ is instead equivalent to the lattice $L$ with the additional structure of S. Maeda semiorthogonality: "ideals generated by idempotents with product 0". The reconstruction of $A$ from $L$ is as above, except that one now uses only the semiorthogonal pairs (which are more restricted than the independent ones).

[As seen below, these are exactly the complementary pairs such that the associated algebraic direct sum decomposition of the (Hilbert) vector space $H$ on which $A$ acts as "ring of operators" is also a topological decomposition, i.e. the skew-projections are continuous (and open, thanks to the Banach setting). $A$ is realized as the ring of bounded linear operators associated to $L$ when $L$ is realized in a $H$. $Q$ instead includes also unbounded ones.]

(The semiorthogonality relation is also studied by L. Herman, who relates it with Topping nonasymptoticity, and implicitly by Bures "modular separation" in 1984, with definitive improvements by S. Maeda in 1986 with the paper "Modular pairs in the lattice of projections of a von Neumann algebra").

  • So to complete the equivalence of $L$ with $A$ it is enough to see that the semiorthogonality can be recovered (in the class of vNa) from the lattice alone $L$. This recovering (the only point not explicit in the Type III post) is given below. Note that once $L$ and $A$ are known to be equivalent one immediately has (by general categorical arguments already used by von Neumann, even before categories were defined), that $L$ equipped with a antiautomorphism, involution, orthocomplementation is equivalent to $A$ equipped with a antiautomorphism, involution, proper (a.k.a. positive definite: $xx^*=0\Rightarrow x=0$) involution.

Lattice reconstruction of semiorthogonality:

first for type I factors, then in general.

For $A$ a type I factor i.e. for the lattice $L$ of closed subspaces of a Hilbert space $H$ one has:

a disjoint pair $X,Y$ is dually modular iff the algebraic sum $X+Y$ is again closed iff the natural map from $X\oplus Y$ [as external direct sum of topological vector spaces] to $X+Y$ (as topological vector subspace of $H$) is a linear and topological isomorphism (Mackey, 1943 - 1945, in Banach spaces).

Dually modular is then equivalent to all other modularity conditions (in Hilbert spaces, one has O-symmerty of the lattice, hence cross-symmetry, M-symmetry and dually. See F. Maeda, S. Maeda, theory of symmetric lattices).

Then following L. Herman and S. Maeda one identifies semiorthogonal pairs in type I factors with these "independent" pairs.

Another equivalent condition: there exists a new orthocomplementation in $L$ i.e. (see again Maeda Maeda for the equivalence) a scalar product in $H$ with the same topology as the original one that makes the pair $X,Y$ orthogonal. (A orthogonal pair is semiorthogonal. Conversely, one can take $X,Y$ as Hilbert subspaces of $H$ and use the Hilbert space direct sum structure $X\oplus Y\oplus (X+Y)^\perp$ which is linearly and topologically isomorphic to $H$ with the natural map. Note: the new scalar product is given by a linear change of variables, hence the new involution in $A$ is obtained from the old with a internal automorphism).

This last equivalent condition shows how the "compatibility" (simultaneous experimentability) relation for quantum logic propositions (defined in orthomodular structures as "to be contained in a boolean subalgebra") can be defined in the lattice $L(H)$ without ortocomplementation: $X,Y$ "commute" (another name for that relation, since for projections in $A$ it coincides with commutativity in the ring) iff there are $X',Z,Y'$ in $L$ which are independent (pairwise disjoint modular pairs, and also the sum of two independent from the third) such that $X'\oplus Z=X$, $Y'\oplus Z=Y$. [The usual definition of compatibility uses existence of such orthogonal decompositions, but the above shows that existence of semiorthogonal ones is equivalent].

Once one has defined "commutativity" in purely lattice terms in $L(H)$, one can define "faithful vNa representations" for a general $L$ (without involution) in a type I factor: poset (hence complete lattice) isomorphism between $L$ and a subset of $L(H)$ (poset of closed subspaces, topology is fixed but not the scalar product) which is its own double-commutant. [One obtains exactly the $L(A)$ for a spatial vNa in $H$, and one sees that all scalar products that give the same complete topology give the same $L(A)$ as lattices, even if the induced orthocomplementation is not the same]. Then one has, following S. Maeda 1986: a disjoint pair is semi-orthogonal iff in (one hence) all such representations of $L$ it goes to a independent (modular, semiorthogonal, nonasymptotic) pair in $L(H)$. This is a purely lattice reconstruction (without a chosen orthogonality).

Note that this can only work for vNa. For representing JBW algebras, one must use also octonionic orthoplanes together with Hilbert lattices $L(H)$.

Note that a "finite" $A$ gives a modular $L(A)$ but not all among its pairs are modular pairs in the larger $L(H)$; semiorthogonality is stronger than modular disjoint. (It is weaker than orthogonality, being "orthogonality for a compatible orthocomplementation" which can be different from the originally given one)

Another equivalent definition of semiorthogonality of a disjoint pair is "absolute modularity": for all faithful representations of $L$ as complete sub ortholattice in a vNa, the pair goes to a (dually, cross, etc.) modular pair.

For AW$^*$ (and its Jordan analogue) the last version (absolute modularity, using all AW$^*$-embeddings) might work or not: since the basic relations among idempotents which are used here survive enlarging the ring, semiorthogonality is "absolute for (complete otholattice / AW$^*$) embeddings", but here we lack a class of cases (like the type I factors for spatial embeddings of vNa) with two properties together: (a) semiorthogonality is characterized as modular disjoint in these cases; (b) every AW$^*$ embeds as double commutant in one such case. Perhaps (probably?) type III AW$^*$ factors (including the wild ones) might work for this class. They should work using the more general AW$^*$-embeddings, but this is unpleasant: to be independent from orhocomplementation, one must say "for each orthocomplementation ..."

Complementary notes:

  • The present case is analogous to the case of "elementary geometry". Really it is a generalization, since purely real $A$ are included here together with those with a $i=-i^*$, $ix=xi$, $i^2=-1$. The real type I$_n$ case is "elementary geometry". The affine structure (over a euclidean hence uniquely orderable field) can be defined by pure incidence (or equivalently by betweenness); it does not uniquely define a orthogonality structure (polar reciprocity on the hyperplane at infinity, or metric on the vector space of translations), but all such additional structures are isomorphic (classification of quadratic forms, only one case is anisotropic).

  • In a sense, von Neumann calling his algebras "rings of operators" is almost justified also for modern readers: the ring structure uniquely defines the real algebra structure, but not the complex one (which exists iff a $i$ exists, and existence of $i$ can be important, for example for some applications to physics, but the specific choice of $i$ is not, even for physics; unnaturality of the complex structure is emphasized also by the existence of factors with a $i$ where all ring i.e. real algebra anti-automorphisms must change sign to $i$). On the other side, a "compatible" involution structure is also not unique, and again it is unique up to isomorphisms; but in this case it is important not only the existence, but also the specific choice.

It is almost like Veblen that in 1904 pretended that (real) euclidean geometry can be axiomatized as affine (betweenness without orthogonality) because a compatible metric structure is unique up to (non-unique) isomorphism. It took Tarski to clearly spell out that no, unique up to a unique isomorphism is needed, and so Tarski axiomatized the various elementary geometries with more primitive concepts, even a redundant system of primitive concepts so that axioms are easier to state (directly using primitive concepts) and understand. And the same did von Neumann with quantum logic; even if the name he choose for the "analytic" equivalent of his "synthetic" axioms would suggest a Veblen-like oversight ("$*$-ring of operators" would have been the name), he was conceptually unacceptable like the two best logicians (of his time, and not only).

  • Notable non-counterexample: for complex type I$_n$ factors, there exist orthocomplementations in $L$ (the Grassmannian of a finite dimensional projective geometry) i.e. proper involutions in $A$ which do not come from a Hilbert space structure on the vector space $H$: one takes a discontinuous involution of the complex numbers (and any vector basis of $H$, then declared orthonormal for a hermitian form for the discontinuous involution. By Birkhoff and von Neumann, 1936, all orthocomplementations on $L$ i.e. proper involutions on $A$ come in this way). These are not counter examples; here one asks for unicity of the $A$ in the special class of vNa that corresponds to $L$, and one has only shown coordinatization by another $A$ outside that class: the ring is the same, but the involution makes $A$ not $*$-ring isomorphic to any vNa unless the complex involution gives a fixed subfield isomorphic to the reals; this is possible ($z^*=f^{-1}(\bar{f(z)})$ with $f$ discontinuous automorphism), but when it happens the lattice with involution is again isomorphic to the standard one, so that unicity (up to non-unique isomorphism) in the special class is not disproved.

Already Mackey and Kakutany in 1944 and 1946 knew and proved that this complex type I$_n$ case is the only "almost counterexample" possible for type I factors; in all other cases, all possible orthocomplementations of the lattice and proper involutions of the ring give a vNa (and always a isomorphic one, over a given lattice). The delicate point is to show that in the complex infinite dimensional case (the other cases being easier by absence of exotic involutions on the skew field) the fact that one uses only closed linear subspaces for a Banach topology excludes discontinuous involutions.

  • There is a way to represent a generic ring isomorphism between $A$ and $A'$ (equivalently, rings automorphisms of $A$) in terms of $*$-rings isomorphisms (i.e. real vNa isomorphism) and spatial, internal automorphisms of $B(H)$ for a suitable Hilbert space representation.

First a theorem of Kaplansky 1953 for complex semisimple Banach algebras:

if $\phi$ is a ring isomorphism from one semi-simple Banach algebra $A$ onto another, then $A$ is a direct sum $A_1\oplus A_2\oplus A_3$ with $A_1$ finite dimensional, $\phi$ linear on $A_2$, and $\phi$ conjugate linear on $A_3$.

In our case the part $A_1$ is uninteresting (see the non-counterexample above), so one can assume a real linear $\phi$.

Yood 1958 explicitly covers the real case (among other generalizations). Still in 2010 generalizations were produced.

Then one has a result of Gardner, 1964, where C$^*$-algebras are complex and isomorphisms $\psi$ are linear, not only ring ones; preservation of $*$ is not requested and continuity is part of the thesis: "In the proof, we make use of the fact that $\psi$ is necessarily bounded [((Dixmier 1957)), p. 15, Exercise 5]"):

Even for W$^*$-algebras, the question has remained open: if $A$ and $A'$ are algebraically isomorphic, are they necessarily $*$-isomorphic? See, e.g. [((Sakai 1962)), p. 1.53, Problem (i)]. In this note, the above question is answered affirmatively for the more inclusive class of C$^*$-algebras [Theorem 3]. Theorem 2 gives the structure of isomorphisms of C$^*$-algebras, showing that each is, in a certain canonical sense, spatial in nature.

Spatial: there is a Banach space isomorphism between the direct sum of all GNS representations relative to pure states of $A$ and $A'$ which extends the given isomorphism between $A,A'$. So, when both "atomic universal" representations are given in the same Hilbert space $H$, a inner automorphism of $B(H)$ implements the given isomorphism between $A,A'$.

Another description is the "polar decomposition" of such a isomorphism: Okayasu, 1974: "any isomorphism of von Neumann algebras is decomposed uniquely as the product of a $*$-isomorphism and an automorphism implemented by an invertible positive element." "Any isomorphism of C$^*$-algebras is decomposed uniquely as the product of a $*$-isomorphism and a positive automorphism, and this decomposition is norm-continuous."

[One can use the complexification $A+iA$ of a real or complex $A$ to obtain a complex-linear ring isomorphism between the complexifications and apply the theorem].

There are also modern proofs pag. 1761.

One can also use unicity of the topology on a real or complex semisimple Banach algebra, Johnson 1967, to have continuity of real-linear ring isomorphism among (real or complex) C$^*$-algebras.

Another proof, Aupetit 1982 using the spectral radius and linearity. "The main idea in the proof is to remark that the spectral radius is independent of the Banach algebra norm."

  • Recent related papers (the authors might or might be not fully aware of the relevant old literature on the subject, I cannot say):

https://arxiv.org/abs/2006.08959 https://arxiv.org/abs/2010.01627 https://arxiv.org/abs/2010.03176

(Note also https://arxiv.org/abs/2107.05806 which considers $*$-regular subrings of a vNa i.e. of $B(H)$; the author might be unaware of the important work of C. Herrmann and coauthors around 2000 - 2015 in strictly related areas).

The different points of view give not only different proofs, but also theorems with nonzero boolean difference in applicability, for example: ortholattice complete hom. among AW$^*$-algebras 2014.

  • Note older results concerning Jordan maps, example but here are relevant really much older results about Jordan maps.

This is related to the present question, as shown by the equivalence of the Jordan structure on self-adjoint elements (or also the structure of "effects" with spectrum in $[0,1]$) with $A$ and $L$ including the involution.

  • Is the "general linear" group (invertible elements of $A$) or its unitary subgroup ($uu^*=1=u^*u$) another equivalent structure? This is considered (also for other rings known to be equivalent to their associated lattice) in old papers by Ehrlich, [[with modern followes and old ones like Whitesitt who considered, before Maeda, the generalization of regular rings where idempotent generated ideals form a sublattice of all right ideals; then Maeda considered the corresponding generalization of Rickart rings using annihilators of pairs of idempotents; equivalently, of the products of pairs of idempotents (in place of arbitrary elements)]]. Above all the (Feldman -) Dye theorem [Ann. of Math. 61 and 63, (1956)] shows (sometimes only in the finite $A$ i.e. modular $L$ case) a "almost equivalence" of the above structures ($L$ or $A$ or $Q$) with or without involution with their associated group (invertible element of the ring, or unitary elements of the $*$-ring). However, note "almost" (up to a finite, order two, quotient of the group of automorphisms: like the relation between a algebraically closed field with involution and its polarized version, i.e. the field of pairs in a real closed field): the linear group (with its group of automorphism and anti-automorphisms) can only determine $L$ i.e. $A$ with their group of automorphisms and antiautomorphisms (an "unoriented", slightly weaker structure than $L$ i.e. $A$, like betweenness on a totally ordered set is weaker than the total order, and separation of pairs of points is weaker than cyclic order on a circle / projective line). (For the structures with involution and unitary group the case is even more delicate, Dye notes I$_{2n}$ complex exceptions.) This is a kind of "Klein Erlanger program" but with the group alone, without a externally given set for its action; in this, it is like Tits buildings. After this one expects also a study of the Lie structure on $A$, and how much it can determine $A$ i.e. $L$; also this exists but I never looked at it.

  • More details for the case with involution excluded might appear in future here; some other complements are already there.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.