A bicategorical approach to Morita equivalence for rings and von Neumann algebras

46 Pages·0.45 MB·English

by  R. M. Brouwer

A Bicategorical Approach to Morita Equivalence
for Rings and von Neumann Algebras∗
3 R.M. Brouwer†‡
0
0
February 8, 2008
2

n
a
J

0
Abstract
3

] Rings form a bicategory[Rings], with classes of bimodules as horizontalarrows,
A
and bimodule maps as vertical arrows. The notion of Morita equivalence for
O rings can be translatedin terms of bicategoriesin the following way. Two rings
. areMoritaequivalentifandonlyiftheyareisomorphicobjectsinthebicategory.
h
t We repeatthisconstructionforvonNeumannalgebras. VonNeumannalgebras
a form a bicategory [W∗], with classes of correspondences as horizontal arrows,
m
and intertwiners as vertical arrows. Two von Neumann algebras are Morita
[ equivalent if and only if they are isomorphic objects in the bicategory [W∗].

1
v
Introduction
3
5
3 This paper is concerned with the Morita theory of rings and von Neumann
1
algebras. Abstract ring theory was initiated around 1920, by, amongst others,
0
Fraenkel, Brauer, Artin, Hasse, and Emmy Noether. Following from abstract
3
0 field theory, ring theory has found its use in many areas of mathematics. Von
/ Neumann algebras, first introduced by von Neumann in 1930, are now widely
h
used in analysis and mathematical physics.
t
a At first sight, the rather abstract field of ring and module theory, and the
m
more physical von Neumann algebra theory are wide apart. For example, ring
: theoryisusedinnumbertheoryandalgebraicgeometry. VonNeumannalgebras
v
i find their use in quantum field theory and statistical mechanics, as well as in
X
representation theory and ergodic theory. In this paper, the author has tried
r to show analogies between the two, using Morita equivalence and bicategories.
a
However, bicategories, though interesting objects as such, are nevertheless just
a tool for handling Morita equivalence.
KiitiMoritaintroducedthetraditionalnotionofMoritaequivalenceforrings,
where it is said that two rings R,S are Morita equivalent if their categories of
∗Thispaperisthemaster’sthesisIwroteundersupervisionofN.P.Landsman,University
ofAmsterdam. Anabbreviatedversionwillappear intheJournalofMathematicalPhysics.
†Universiteit van Amsterdam, Korteweg-de Vries Instituut, Faculteit Natuurwetenschap-
pen,WiskundeenInformatica,PlantageMuidergracht24,1018TVAmsterdam
‡Current address: CWI, Postbus 94079, 1090 GB, Amsterdam. Email:
[email protected]
1
right modules, M and M , are equivalent. In the case of von Neumann al-
R S
gebras, M,N are said to be Morita equivalent if there exists a correspondence
M→H←N for which the representation of M on H is faithful and
M≃ (Nop)′ holds. However, we may choose a definition of Morita equivalence
for von Neumann algebras, similar to the definition of Morita equivalence for
rings (in terms of representation categories), which is equivalent to our defini-
tion.
The main result of this paper is the following: For both rings and von Neu-
mann algebras, it is possible to prove that Morita equivalence is nothing but
isomorphism in their respective bicategories. Despite the fact that rings and
von Neumann algebras have their use in different areas of mathematics, they
have the same underlying structure as far as Morita equivalence is concerned.
This paper is organized as follows. Section 1 contains the basic definition of
a bicategory. A few examples will be discussed.
Section2handlesthecaseofrings. First,thetensorproductoftwomodules
willbeexplainedinSubsection2.1. Next,wewillconsiderthetraditionalMorita
theory, which handles progenerators and categories of modules in Subsection
2.2. The notion of a bicategory emerges in Subsection 2.3, where we will show
that rings form a bicategory, and we will state the Morita theory in terms
of bicategories. To justify the use of the terminology “Morita theory” in the
bicategorycase,we willshow thatthe results ofSubsections 2.2 and2.3leadto
equivalent theories. This is done in Subsection 2.4.
Section 3 handles von Neumann algebras. The goal of this section is to ex-
plainthenotionofMoritaequivalence,andtoexhibitcorrespondingstatements
in terms of bicategories. It turns out to be the case that for the construction
of the bicategory of von Neumann algebras, one needs the standard form and
the identity correspondence of a von Neumann algebra, as well as the concept
of Connes fusion. These are discussed in Subsections 3.1,3.2 and 3.3. Finally,
one will find the Morita theory of von Neumann algebras in Subsection 3.4.
2
1 Bicategories
1.1 Definitions
This section will explain the notion of a bicategory that we will use in later
sections. It is assumed that the reader is familiar with categories,functors and
natural equivalences or natural isomorphisms. In this paper, we will use the
convention that the class of objects of a category C is denoted by C ; the class
0
of morphisms of C is denoted by C . The notation (A,B) is used for all arrows
1
B →A, which allows us to write the composition of arrowsconveniently. For a
standard text on categories the reader is referred to [16]. See also [2] and [16]
for an overview of bicategories.
InseveralsituationswherewehaveabifunctorB×B →B(foracategoryB),
this bifunctor is notassociative. Ifitis, andhas aunit element,ourcategoryB
becomesaso-calledstrictmonoidalcategory. Fora(relaxed)monoidalcategory,
there exist natural equivalences such that the bifunctor is associative up to
isomorphism. Further,sometimeswewouldliketodefinebifunctorsB×C →D
for categories B,C,D. Such bifunctors or composition functors give rise to a
2-category if they are associative. Again, composition functors are generally
not associative. An example is a bicategory, where the composition functor is
merely associative up to isomorphism.
Definition 1.1.1. A bicategory B consists of the following ingredients:
1. A set B of objects.
0
2. For all pairs (A,B) of objects, a category. If there is no confusion pos-
sible, this category will also be denoted by (A,B). The class of all such
categories will be denoted by B ; it contains all morphisms or horizontal
1
arrows of B. The morphisms (arrows) (A,B) of the category (A,B) are
1
called vertical arrows.
3. For each triple (A,B,C) of objects in B , a composition functor.
0
C(A,B,C):(A,B)×(B,C)−→(A,C). (1.1)
If (P,Q) is an element of (A,B)×(B,C), we will write P ∗Q for
C(A,B,C)(P,Q). The same notation will be used on the arrows.
4. For each object B of B , an object I of (B,B). I is called the identity
0 B B
arrow of B.
5. For each quadruple (A,B,C,D) of objects in B , a natural isomorphism
0
β between the functors
F = C(A,B,D) ◦ Id ×C(B,C,D) , (1.2)
(A,B)
(cid:2) (cid:3) (cid:2) (cid:3)
and
G= C(A,C,D) ◦ C(A,B,C)×Id , (1.3)
(C,D)
(cid:2) (cid:3) (cid:2) (cid:3)
where
F,G:(A,B)×(B,C)×(C,D)−→(A,D). (1.4)
3
This means there exists a natural equivalence β which assigns to every
object (E ,E ,E ) in (A,B)×(B,C)×(C,D) an arrow
1 2 3
β(E ,E ,E ):F (E ,E ,E )→G (E ,E ,E ),
1 2 3 0 1 2 3 0 1 2 3
such that for every arrow
h:(E ,E ,E )→(E ′,E ′,E ′)∈(A,B)×(B,C)×(C,D),
1 2 3 1 2 3
the diagram
F (E ,E ,E ) β(E1,E2,E3) //G (E ,E ,E ) (1.5)
0 1 2 3 0 1 2 3
F1(h) G1(h)
F (E ′,E(cid:15)(cid:15) ′,E ′) β(E1′,E2′,E3′)//G (E ′,E(cid:15)(cid:15) ′,E ′)
0 1 2 3 0 1 2 3
commutes. Note that β depends onthe quadruple(A,B,C,D), but when
no confusion is possible, we will omit this.
6. Foreachpairofobjects(A,B),twonaturalequivalencesR andL
(A,B) (A,B)
called left and right identities. Here L is a natural equivalence be-
(A,B)
tween
C(A,A,B) ◦ I ×Id , (1.6)
A (A,B)
(cid:2) (cid:3) (cid:2) (cid:3)
and the canonical functor from 1×(A,B) to (A,B). R is a natural
(A,B)
equivalence between
C(A,B,B) ◦ Id ×I , (1.7)
(A,B) B
(cid:2) (cid:3) (cid:2) (cid:3)
and the canonical functor from (A,B)×1 to (A,B). When no confusion
is possible, the subscript of the right and left identities will be omitted.
7. Thenaturalisomorphismsβ,L andR arerequiredtosatisfythe
(A,B) (A,B)
so-called coherence axioms.
• Associativity coherence:
If (P,Q,R,S) is an object in (A,B)×(B,C)×(C,D)×(D,E), the
following diagram commutes:
β(P,Q,R)∗Id
((P ∗Q)∗R)∗S // (P ∗(Q∗R))∗S
β(P∗Q,R,S) β(P,Q∗R,S)
(cid:15)(cid:15) (cid:15)(cid:15)
(P ∗Q)∗(R∗S) P ∗((Q∗R)∗S)
β(SPS,SQS,SRS∗SSS)SSSSSS)) uukkkkkkIkdk∗kβk(kQk,kRk,S)
P ∗(Q∗(R∗S)).
(1.8)
4
• Identity coherence:
If (P,Q) is an object in (A,B)×(B,C) the following diagram com-
mutes:
(P ∗I )∗Q β(P,IB,Q) //P ∗(I ∗Q) (1.9)
BR(PMM)∗MIMdMMMMMM&& xxqqqqqqIqdq∗qLq(Q) B
P ∗Q.
One would like that all diagrams concerning β,L and R are commutative.
In fact, if the coherence axioms are satisfied, all such diagrams commute. ([16]
Section XI.7; c.f. the coherenceaxioms in the case of a monoidalcategory,[16],
Section VII.2, where commutativity of all diagrams is proven.)
Considering a bicategory and taking isomorphism classes in the categories
(A,B)asarrows,wegetacategory. ThenaturalisomorphismsLandRprovide
left and right identities, the natural isomorphism β provides associativity. The
coherence axioms even make this category monoidal. A 2-category, where the
composition is associative, is a special case of a bicategory: In this case the
naturalisomorphisms β,L and R are all identities, so the coherenceaxioms are
satisfied immediately.
For later use, we need a notion of isomorphism in a bicategory, which is
broader than the usual notion of isomorphism in category theory..
Definition 1.1.2. Two objects A,B in a bicategory B are isomorphic in the
b
bicategory, A≃B, if aninvertible (horizontal)arrowP exists, P ∈(A,B), i.e.
P ∗P−1 ∼=I , (1.10)
A
and
P−1∗P ∼=I , (1.11)
B
where the symbol ∼= denotes isomorphism of objects in (A,A) and in (B,B)
respectively, in the usual sense of categories.
Notethateveryobjectofthebicategoryisisomorphictoitselfviaitsidentity
arrow, which is its own inverse. Further, two objects A,B that are isomorphic
in the usual sense, are also isomorphic in the bicategory, since in that case the
categories (A,A) and (A,B) are equivalent. Hence we have isomorphism on
objects through the natural equivalence. The invertible arrow in this case is
given by the image of I in (A,B).
A
1.2 Examples
• All 2-categoriesare bicategories. For example:
– The class of all categories as objects, functors as horizontal arrows,
and natural transformations as vertical arrows.
– Theclassofalltopologicalspacesasobjects,continuousmapsashor-
izontal arrows,and homotopy classes of continuous maps as vertical
arrows.
5
• Each(relaxed)monoidalcategoryM formsa bicategory,whichin general
isnota2-category. Thebicategoryconsistsofoneobject(namelyM),the
objectsofthecategoryM formthehorizontalarrowsofthebicategory. A
compositionfunctorM×M →M whichis associativeupto isomorphism
exists,sinceM ismonoidal. ThearrowsM formtheverticalarrowsofthe
1
bicategory. Thenaturalisomorphismsthatareassociatedtothemonoidal
category make that the coherence axioms are satisfied.
• More instructive examples of bicategories are the bicategory [Rings] and
the bicategory [W∗]. The bicategory [Rings] consists of rings as objects,
categories of bimodules as horizontal arrows, and linear maps as vertical
arrows. Thebicategory[W∗]consistsofvonNeumannalgebrasasobjects,
categoriesofcorrespondencesashorizontalarrows,andintertwinersasver-
tical arrows. However, it is not easy to show that [Rings] and [W∗] are
indeed bicategories. Especially the definition of the composition functor
is nottrivial. Thereforea proofcanbe found inlater sections(see Propo-
sitions 2.3.1 and 3.5.3). However, these examples are the main reason to
discuss bicategories,since we use bicategories to show that the notions of
Morita equivalence for rings and von Neumann have the same underlying
structure.
6
2 Morita theory for rings
It is assumed that the reader is familiar with the notion of modules and bi-
modules of rings. If not, basic ring and module theory may be found in [6].
Throughoutthis section, allrings will have a unit. Let R, S be rings. A left R-
module M will be denoted by M. Right modules will be denoted analogously
R
by M . A R-S bimodule N will be denoted by N or R → N ← S. Both
R R S
notations will be used in the following.
In this section, we will present the ”traditional” Morita theory. After that,
this theory will be reformulated in terms of bicategories. Finally, it will be
shown that these theories are equivalent. In both approaches, the notion of a
tensor product of two (bi)modules is needed, so first of all, we will discuss the
tensor product. See [5] for an extensive discussion of the tensor product.
2.1 The tensor product of bimodules
To define the tensor product of two modules, we need the following definitions.
LetM,N andLbeabeliangroups,R,S andT rings. Amapψ :M×N →Lis
calledbilinear ifitsatisfiesψ(m+m′,n)=ψ(m,n)+ψ(m′,n)andψ(m,n+n′)=
ψ(m,n)+ψ(m,n′),form,m′ ∈M,n,n′ ∈N. IfM isaleftR-module,N aright
T-module and L a R-T bimodule, a linear map ψ : M ×N → L is called R-T
linear if the map ψ intertwines the R and T actions. Further, if M is a right
S-module andN is a left S-module,a mapψ :M×N →L is calledS-balanced
if it satisfies ψ(m,sn)=ψ(ms,n), for m∈M,n∈N,s∈S.
Proposition2.1.1. GiventhreeringsR,S andT,andtwobimodules M and
R N
N ,thereexistsanR-T bimodule (M⊗ N) andanS-balancedR-T bilinear
S T R S T
map τ : M ×N → (M ⊗ N) with the following universal property: For
R S T
every R-T bimodule L and every S-balanced R-T bilinear map φ:M ×N →L
there exists a unique R-T bilinear map α : M ⊗ N → L such that φ = α◦τ.
S
In a commutative diagram:
φ
M ×N // L (2.1)
MMMMMMτMMMM&& vvv!αvvvvvvv::
M ⊗ N.
S
Proof. Existence of M⊗ N followsdirectly by construction. Consider Y, the
S
free Z-module on M ×N with embedding i : M ×N → Y. Now quotient to
I, which is generated by elements of the form (m+m′n)−(m,n)−(m′,n) or
(m,n+n′)−(m,n)−(m,n′)or(ms,n)−(m,sn),form,m′ ∈M, n,n′ ∈N, s∈
S. Let π : Y → Y/I be the canonical surjection. The remaining quotient Y/I
formsthebimoduletensorproductM⊗ N of M and N . Onehastoshow
S R S S T
thatthis tensorproductisanR-T bimodule. The leftR-actiononM isdefined
on M ⊗ N by r(m⊗ n) := (rm⊗ n), which is defined because of the left
S S S
R-action on M. Note that I is closed under the left R-action, so the R-action
passes to the quotient. The right T-action descends to the quotient likewise.
Define τ =π◦i. The structure of M ⊗ N causes τ to be S-balancedand R-T
S
bilinear.
7
Let L be a R-T bimodule and let φ : M ×N → L be an S-balanced, R-T
linear map. We now obtain the following diagram:
φ
M ×N //88LOO (2.2)
A
i α
(cid:15)(cid:15)
π
Y //M ⊗S N.
The universal property of a free module provides the decomposition φ=A◦i,
where A is a linear map. Further, A vanishes on the elements of I, since φ is
bilinear and S-balanced, so we have a decomposition A = α◦π, where α is a
linear map. Trivially, α is bilinear and S-balanced because of the definition of
M ⊗ N. We need to show that α is an R-T bilinear map:
S
α(r(m⊗ n)t) = α(rm⊗ nt)=α(τ(rm,nt))
S S
= φ(rm,nt)=rφ(m,n)t
= r(α(m⊗ n))t, (2.3)
S
for m∈M,n∈N,r ∈R,t∈T.
Uniqueness of α follows from taking a second R-T bilinear S-balanced map
α˜ such that φ = α˜◦τ. Defining A˜ = α˜◦π, a similar diagram as above can be
formed,withα˜insteadofαandA˜insteadofA. However,theuniversalproperty
of a free module guaranteesthat A is unique, so A˜=A. Now α˜◦π =α◦π and
surjectivity of π shows α˜ =α.
Finally, we show uniqueness of the pair (M ⊗ N,τ). Suppose we have a
S
second pair (M^⊗ N,τ˜) that satisfies the properties stated in the proposition
S
above. Applying the universalpropertyto both (M⊗ N,τ) and(M^⊗ N,τ˜),
S S
we obtain the following commutative diagram, where both α and α˜ are unique:
M ⊗ N (2.4)
S
qqqqτqqqqqq88 OOOO
M ×NLLLLLLτ˜LLLL !α˜ !α
%% (cid:15)(cid:15)(cid:15)(cid:15)
M^⊗ N.
S
We obtain
τ =α˜◦τ˜ τ =α˜◦α◦τ;
⇒ (2.5)
τ˜=α◦τ(cid:27) τ˜=α◦α˜◦τ˜.
Byconstruction,τ issurjective,soequation(2.5)impliesα˜◦α=idonM⊗ N.
S
Hence α◦α˜ = id on M^⊗ N. Now α˜ = α−1 so M ⊗ N ∼= M^⊗ N and
S S S
τ =α−1◦τ˜.
Remark 2.1.2. Note that, giventhe aboveproposition,we areable to construct
the tensor product M ⊗ N between a left R-module N and a right R-module
R
M,for anyringR. By consideringthe left R-moduleN as aR-Zbimodule and
the right R-module M as a Z-R bimodule, the proof of the above proposition
applies.
8
Let R,S,T be rings. As a preparation for the categorical statements of
Morita theory, we will show that the tensor product ⊗ defines a functor. Let
S
(R,S) denote the following category: The class of objects (R,S) consists of
0
R-S bimodules, the class of arrows (R,S) consists of R-S linear maps. The
1
categories (S,T) and (R,T) are defined likewise.
On objects, ⊗ is defined by:
S
⊗ :(R,S)×(S,T) −→ (R,T)
S
M × N 7−→ (M ⊗ N) , (2.6)
R S S T R S T
for M ∈(R,S), N ∈(S,T).
R S S T
Onarrows,⊗ acts as follows: Letφ: M → K be anarrowin (R,S), and
S R S R S
ψ : N → L be an arrow in (S,T). Then, for (m ⊗ n ) ∈M ⊗ N:
S T S T i i S i S
P
(φ⊗ ψ): (M ⊗ N) −→ (K⊗ L)
S R S T R S T
(m ⊗ n ) 7−→ (φ(m )⊗ ψ(n )). (2.7)
i S i i S i
Xi Xi
Since both φ and ψ intertwine the S-action, φ⊗ ψ is well-defined.
S
The construction of the tensor product (Proposition 2.1) shows that on ob-
jects, the image of⊗ lies in(R,T) . Onarrows,one has to show thatfor each
S 0
pairof arrows(φ×ψ)∈ (R,S)×(S,T) , the image(φ⊗ ψ)is anR-T linear
1 S
map. For r ∈R, and i(cid:0)(mi⊗S ni)∈M(cid:1)⊗S N, one has
P
r(φ⊗ ψ) (m ⊗ n ) = r φ(m )⊗ ψ(n )
S i S i i S i
Xi Xi (cid:0) (cid:1)
= rφ(m )⊗ ψ(n )
i S i
Xi (cid:0) (cid:1)
= φ(rm )⊗ ψ(n )
i S i
Xi (cid:0) (cid:1)
= (φ⊗ ψ) (rm ⊗ n )
S i S i
Xi
= (φ⊗ ψ) r (m ⊗ n ) . (2.8)
S i S i
(cid:16) Xi (cid:17)
A similar computation shows that (φ⊗ ψ) preserves the right T-action.
S
It is left to show that ⊗ is a functor. By definition, a functor F from a
S
category C →D is a map which assigns to each object in C an object in D and
to each arrow f :c→c′ in C an arrowF(f):F(c)→F(c′) in C such that
F =id , (2.9)
idc F(c)
and
F(f)◦F(g)=F(f ◦g), (2.10)
for all objects c in C and all arrows f,g in C , whenever the composition of
0 1
9
arrows f ◦g is defined in C . Let M ∈(R,S) and N ∈(S,T). Then
1 R S S T
⊗ ◦id M,N = ⊗ M,N
S (R,S)×(S,T) S
(cid:0) (cid:1) = (M(cid:0) ⊗ N(cid:1))
R S T
= id (M ⊗ N)
(R,T) S
= id M,N ,
⊗S (R,S)×(S,T)
(cid:0) (cid:1)
(cid:0) (cid:1) (2.11)
and
⊗ ((f ×f )◦(g ×g )) (m ⊗ n ) = ((f ◦g )⊗ (f ◦g )) (m ⊗ n )
S 1 2 1 2 i S i 1 1 S 2 2 i S i
Xi Xi
= ((f ◦g )(m )⊗ (f ◦g )(n ))
1 1 i S 2 2 i
Xi
= (f ⊗ f ) (g (m )⊗ g (n ))
1 S 2 1 i S 2 i
Xi
= (f ⊗ f )◦(g ⊗ g ) (m ⊗ n )
1 S 2 1 S 2 i S i
Xi
= ⊗ (f)◦⊗ (g) (m ⊗ n );
S S i S i
Xi
(2.12)
wherethelastequationholdswheneverf◦g isdefinedin(R,S)×(S,T). Hence
⊗ is a functor. However, we will see later that ⊗ is not associative.
S S
2.2 Traditional Morita theory
Following[14], we will discuss the traditionalMorita theory. Our approachem-
phasizes the algebraic side of the theory,starting with modules and generators.
Later on, categories and functors will appear. However, see [7] for a review of
Morita theory which stresses the functoriality. All theory below concerns right
modules. Ofcourse,anequivalenttheoryforleftmodules exists. First,weneed
some general notions and definitions.
Definition 2.2.1. Let R be a ring. Then M denotes the category of right
R
R-modules, the arrows of M being given by R-module maps.
R
Definition 2.2.2. LetR,S betworings. RandS arecalledMorita equivalent,
M
denoted by R ∼ S, if there exists a categorical equivalence between M and
R
M , i.e. a functor F : M → M and a functor G : M → M such that
S R S S R
(F ◦G)≃id and (G◦F)≃id .
MS MR
Definition 2.2.3. A right R-module P is a generator for M if Hom (P,−)
R R
is a faithful functor from M to the category of abelian groups. A finitely
R
generated projective generator is called a progenerator. An (R,S)-progenerator
P isafaithfullybalancedR-Sbimodule(i.e. abimoduleforwhichR∼=End(P )
S
and S ∼=End( P)) that is a progenerator for M .
R S
Recall that a module P is finitely generated when for all families of sub-
modules {Ni}i∈I with i∈INi =P there exists a finite subset J ⊆I suchthat
P
10