department of mathematics and computer science university...

DANIELE MUNDICI

Department of Mathematics and Computer Science

University of Florence, Florence, Italy

[email protected]

A quotation (and a research program)

Von Neumann algebras are blessed with an excess of

structure—algebraic, geometric, topological—so much,

that one can easily obscure through proof by overkill,

what makes a particular theorem work.

...

If all the functional analysis is stripped away (by hands more brutal than mine), what remains should stand firmly as a substantial piece of algebra, completely accessible through algebraic avenues

S.K.Berberian, “Baer *-rings”, Springer 1972 (From the Introduction)

C*-algebras, AF-algebras, MV-algebras

(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)

x⊕0 = x

¬¬x = x

x⊕¬0 = ¬0

y⊕¬(y ⊕ ¬x) = x⊕¬(x ⊕ ¬y)

An algebra A over C is a *-algebra iff *:AA is a linear map such that for all y,z∈A and µ∈ C, z**=z, (zy)*=y*z* and (µz)*=µ’z*.

A C*-algebra is a *-algebra A with a norm making A into a Banach space, and such that ||1A||=1, ||yz||≤||y|| ||z|| and ||zz*|| = ||z||2.

An AF-algebra is the norm closure of an ascending sequence of finite-dimensional C*-algebras.

An MV-algebra is a structure (B,0,¬,⊕) satisfying the equations

Throughout this talk, all groups will be abelian

Stipulation

The material developed in the first part will find use in the final part, dealing with the Lawson-Scott coordinatization of

countable MV-algebras by suitable inverse semigroups. This material can also be applied to the Wehrung

coordinatization of all MV-algebras.

This preliminary material is of independent interest.

Projections p and q in an AF-algebra A are equivalent if p=wqw* for some w ∈ A satisfying 1 = w*w = ww*.

L(A) = set of equivalence classes [p] of projections p∈A.

The Murray-von Neumann order on L(A) is given by: [p]≤[q] iff p is

equivalent to a sub-projection of q.

Elliott partial addition + is the operation on L(A) given by adding two

projections whenever they are orthogonal.

This operation is commutative, associative, monotone and has the

residuation property: For any projection p∈A, the equivalence class

¬[p]=[1–p] is the smallest class [z] satisfying [z] + [p] = [1].

Elliott partial monoid

from AF-algebras to MV-algebras THEOREM (G. Panti, D.M., J. Funct. Analysis, 117, 1993)

Let A be an AF-algebra.

Elliott partial addition + has at most one extension to an

associative, commutative, monotone operation ⊕ over L(A) having the residuation property, stating that [1–p] (denoted ¬[p]) is the smallest class [z] with [z] ⊕ [p] = 1.





The extension ⊕ exists if and only if L(A) is a lattice.





The extension ⊕ exists if and only if L(A) is a lattice.

The map A(L(A), 0, ¬, ⊕) is a one-one correspondence between AF-algebras with lattice ordered L(A) and countable MV-algebras.

the one-one correspondence

the five defining equations of MV-algebras DEFINITION An MV-algebra is a structure (A,0,¬,⊕) satisfying:

(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) x⊕y = y⊕x x⊕0 = x ¬¬x = x x⊕¬0 = ¬0 y⊕¬(y ⊕ ¬x) ⊕ = x⊕¬(x ⊕ ¬y)

DEFINITION An MV-algebra is a structure (A,0,¬,⊕) satisfying:


Writing x–y instead of x⊕¬y, the last axiom y–(y–x)=x–(x–y) generalizes a well known property of set-theoretic difference.

Indeed, for sets, y–(y–x) = x∩y = y∩x=x–(x–y).

the five defining equations of MV-algebras

DEFINITION An MV-algebra is a structure (A,0,¬,⊕) satisfying:

Commutativity follows from the remaining 5 axioms (Kolarìk,

Math. Slovaca 63, 2013)


a boolean algebra is an MV-algebra satisfying x ⊕ x = x

the five defining equations of MV-algebras

the categorical equivalence Γ and its adjoint Ξ

THEOREM (D.M., J. Functional Analysis 65, 1986) For any MV-algebra A, let Γ(A) = [0,1] be the unit interval of A equipped with truncated addition and involution ¬ x = 1–x. Further, for any homomorphism h : AB let Γ(h) be the restriction of h to A. Then Γ is a categorical equivalence between unital l-groups and MV-algebras.

the categorical equivalence Γ and its adjoint Ξ

THEOREM (D.M., J. Functional Analysis 65, 1986) For any MV-algebra A, let Γ(A) = [0,1] be the unit interval of A equipped with truncated addition and involution ¬ x = 1–x. Further, for any homomorphism h : AB let Γ(h) be the restriction of h to A. Then Γ is a categorical equivalence between unital l-groups and MV-algebras.

A good sequence in A is a sequence of elements ai where ai+1 ⊕ ai = ai and all elements are zero, up to finitely many exceptions.

Good sequences form a cancellative lattice ordered monoid Q with unit 1, whose Grothendieck group G = Q—Q is the unital lattice ordered group corresponding to A. Ξ(A) = (G,1)

Error-correcting communication with feedback (Dobrushin-Berlekamp)

94B60

Ulam-Rènyi game of Twenty Questions 91A46 Łukasiewicz logic 03B50 C.C.Chang MV-algebras 06D35 Lattice-ordered groups with strong unit 06F20 AF-algebras with Murray-von Neumann order of projections. Inverse semigroups

46L80 20M18

Rational polyhedra and the fans of nonsingular toric varieties

14M25

Piecewise linear functions with integer coefficients, unimodular triangulations

57Q05

Classification and recognition of orbits of the affine group over the integers acting on euclidean n-space

37C85

MV-algebras and some of their friends

unital dimension groups

Dimension group = partially ordered group G which is

directed (in the sense that G=G+–G+)

unperforated (in the sense that nx≥0, n>0 x≥0) and

has Riesz decomposition (i.e., sums of intervals are intervals).

G is unital if it has a unit u (i.e., any g∈G is dominated by some nu).

A simplicial group is an order-isomorphic copy of the free abelian group Zk equipped with the product ordering.

THEOREM (Effros Handelman Shen, and Grillet, op.cit.) (G,u) is a unital dimension group iff it is the (direct) limit of a direct system of simplicial unital groups with monotone unit-preserving homomorphisms.

K0 of a unital AF-algebra A

Mn(A) denotes the C*-algebra of n x n matrices with values in A.

We have the “North-West” embedding Mn(A) Mn+1(A).

M∞(A) denotes the limit C*-algebra lim Mn(A).

P(A) denotes the set of projections of M∞(A).

P(A)/≈ denotes the set of Murray von-Neumann equivalence

classes [p] of projections p in M∞(A).

K0(A)+ denotes the cancellative monoid obtained by equipping

P(A)/≈ with the operation [p]+[q]=[p⊕q].

K0(A) denotes the Grothendieck group of K0(A)+ .

K0 of a unital AF-algebra A

Define x≤y iff x–y ∈ K0(A)+. Then K0(A) becomes a countable partially ordered abelian group with [1] as the distinguished unit. For any two unital AF-algebras A, B we have:

THEOREM (Effros-Handelman-Shen, Grillet, + Elliott, op.cit.)

A ≅ B if and only if (K0(A), K0(A)+, [1A]) ≅ (K0(B), K0(B)+, [1B]).

A unital partially ordered group (G, G+, u) is isomorphic to (K0(A), K0(A)+, [1A]) for some unital AF-algebra A iff it is a unital countable dimension group.

combining Γ and K0 THEOREM (D.M. J. Functional Analysis 1986) The composite functor ΓK0 maps (isomorphism classes of) AF-algebras whose Murray-von Neumann order of projections is a lattice, one-one onto (isomorphism classes of) countable MV-algebras.

THEOREM (G.A. Elliott, D.M, Math. Zeit. 213, 1993) Suppose G has the Riesz decomposition property and the intersection of two compact open subsets in the prime spectrum of G is compact. Suppose whenever Q is a nonzero quotient of G having no nonzero ideals with 0 intersection then Q is totally ordered. Then G is a lattice.

the one-one correspondence is now a functor

the one-one correspondence is now a functor

we will be interested in these algebras

Geometry of Bratteli diagrams via Schauder bases

The free AF-algebra M1 corresponding to the free one

generator MV-algebra F1

ΓK0(M1)=F1 i.e., K0(M1)=Ξ(F1)

(includes Effros-Shen and Behnke-Leptin algebras)

background in MV-algebras

Basic material, 2000 Advanced topics, 2011

http://www.matematica.uns.edu.ar/IXCongresoMonteiro/Comunicaciones/Mundici_tutorial.pdf

Basic material, 2000 Advanced topics, 2011

C.C.Chang 1959 completeness theorem: MV=HSP[0,1]

Let [0,1] be the unit real interval with the distinguished constants 0 and 1, and the operations: ¬ x = 1 – x, called negation, and x ⊕ y = min(1, x + y), called truncated addition.

THEOREM. The equational class of MV-algebras is generated by the MV-algebra [0,1]. In symbols, MV = HSP[0,1].

COROLLARY The free n-generator MV-algebra Fn is the MV-algebra of functions f : [0,1]n [0,1] obtained from the coordinate functions x1, x2,..., xn by pointwise application of negation and truncated addition.

C.C.Chang 1959 completeness theorem: MV=HSP[0,1]

we will first consider the all-instructive case of the one generator free MV-algebra F1

Let [0,1] be the unit real interval with the distinguished constants 0 and 1, and the operations: ¬ x = 1 – x, called negation, and x ⊕ y = min(1, x + y), called truncated addition.

THEOREM. The equational class of MV-algebras is generated by the MV-algebra [0,1]. In symbols, MV = HSP[0,1].

COROLLARY The free n-generator MV-algebra Fn is the MV-algebra of functions f : [0,1]n [0,1] obtained from the coordinate functions x1, x2,..., xn by pointwise application of negation and truncated addition.

Since MV=HSP[0,1] then any f ∈F1 is continuous, piecewise linear, and each linear piece has integer coefficients

x

¬f

f

f⊕g f

g

1

1

McNaughton theorem (1951) These conditions are sufficient for f to belong to the free algebra F1

x

¬f

f

f⊕g f

g

1

1

Since MV=HSP[0,1] then any f ∈F1 is continuous, piecewise linear, and each linear piece has integer coefficients

the first four bases S0,S1,S2,S3 : start from 0/1 and 1/1 and insert Farey mediants via the freshman sum of fractions

Sn has 2n +1 hats; the hat of Sn at p/q has height 1/q and is nonzero over the open interval between the two adjacent points of p/q in the nth Stern-Brocot sequence

S0

S1

S2

S3

the two points 0/1 and 1/1

the Farey mediant ½ = (0+1)/(1+1)

adding the Farey mediants 1/3 and 2/3

adding the Farey mediants of the above four intervals

Every Schauder hat is a McNaughton function.

This follows from the unimodularity of each interval [p/q,r/s], i.e., rq–ps=1.

So each basis Sk is contained in the free MV-algebra F1

S0

S1

S2

S3


the constant map 1 equals 1 copy of the red map + 1 copy of the blue

= 1 red map + 2 copies of the green map + 1 of the blue map

= 1 red + 3 violet + 2 green + 3 brown + 1 blue

= 1 red + 4 orange + 3 violet +5 black +2 green +5 blue + 3 yellow + 4 pink +1 brown

S0

S1

S2

S3






S0

S1

S2

S3

the hats of each basis Sn are linearly

independent in the vector space of real

valued functions defined over [0,1].

So each Sk is a basis of the free abelian

group Ξ(F1)

Each hat of Sn is a linear combination of

the hats of Sn+1 with positive integer

coefficients.


the simplicial groups freely generated by the bases S1,S2,S3,... in Ξ(F1) define a “Bratteli diagram”

1 2 1

1 1 3 3 2

1 4 3 5 2 5 3 4 1





S0

S1

S2

S3

1 2 1

1 1 3 3 2

1 4 3 5 2 5 3 4 1

this is the (Farey-Stern-Brocot) Bratteli diagram of the AF-algebra M1 corresponding via ΓK0 to the

free MV-algebra F1 (D.M., Advances in Math. vol 68, 1988)

F1=ΓK0(M1)

an equivalent presentation of the Farey AF C*-algebra M1 corresponding via ΓK0 to the free MV-algebra F1 (D.M.,

Advances in Mathematics, vol. 68, 1988, page 25)

These matrices embed the simplicial group Free(Sk) freely generated by the basis Sk into the simplicial Free(Sk+1).

The limit dimension group is K0(M1) = Ξ(F1). The free MV-algebra F1 coincides with ΓK0(M1).

twenty years later, M1 is rediscovered F. Boca, Can. J. Math., 60.5, 2008, page 977

this Bratteli diagram defines exactly the AF-algebra M1 (D.M., Rendiconti Accademia Lincei, 20, 2009)

The Effros Shen algebras are precisely the infinite-dimensional simple quotients of M1 (D.M., Advances in Math. 68, 1988)

Each irrational rotation C*-algebra is embeddable into a simple quotient of M1 (D.M., ibid.)

All primitive ideals of M1 are essential.(D.M.,Milan J.Math.79, 2011)

The automorphism group of M1 has exactly two connected

components. (ibid.)

M1 has a faithful invariant tracial state. (D.M., Rend.Lincei 20, 2009)

M1 is Hopfian, i.e., is not isomorphic to a proper quotient of itself

(D.M. Forum Math., to appear)

M1 is locally finite-dimensional (D.M. Forum Math., to appear)

some properties of the AF-algebra M1

The center of M1 is the C*-algebra C of continuous complex valued functions on [0,1], (F. Boca, op.cit.)

The state space of C is affinely weak* homeomorphic to the space of tracial states on M1 (F. Boca, op.cit.)

Every state on the center C of M1 has a unique tracial extension to M1 (Eckhardt, Math. Scand. 108 (2011), 233–250)

further properties of M1

Eckhardt (op.cit.) generalizes the Gauss map, which is a Bernoulli shift for continued fractions, in the noncommutative setting of M1.

the most interesting property of M1

each simplicial group Free(Sk) in K0(M1) is freely generated

by the Schauder basis Sk of the free MV-algebra F1=ΓK0(M1). Further, K0(M1) = ∪kFree(Sk). This is the tip of an iceberg:


THEOREM (V.Marra, J. Algebra 225, 2000) Every lattice ordered group (G,u) is ultrasimplicial, i.e., is the directed union of the simplicial groups Fi contained in G, whose positive cone is contained in G+, all with the same unit u.

each simplicial group Free(Sk) in K0(M1) is freely generated by the Schauder basis Sk of the free MV-algebra F1=ΓK0(M1). Further, K0(M1) = ∪kFree(Sk). This is the tip of an iceberg:


THEOREM (V.Marra, J. Algebra 225, 2000) Every lattice ordered group (G,u) is ultrasimplicial, i.e., is the directed union of the simplicial groups Fi contained in G, whose positive cone is contained in G+, all with the same unit u.

Elliott proved the special case when (G,u) is totally ordered.

Handelman asked if all lattice ordered groups are ultrasimplicial

The simplicials Free(Sk) are freely generated by Schauder bases over all Farey-Stern-Brocot sequences and their unimodular generalizations

each simplicial group Free(Sk) in K0(M1) is freely generated by the Schauder basis Sk of the free MV-algebra F1=ΓK0(M1). Further, K0(M1) = ∪kFree(Sk). This is the tip of an iceberg:

an example: f,g,h ∈ F1

f

g

h

THEOREM (D.M. Adv.Math., vol 68, 1988) Every finite set of elements of F1 is a linear combination of finitely many hats of some

basis, with positive integer coefficients

a basis of 13 Schauder hats, over a suitable triangulation of [0,1] satisfying Farey unimodularity



each one of f, g, h is a linear combination of the 13 basic hats, with integer coefficients ≥ 0.

these 13 hats are free generators of some simplicial group sitting inside (G,u) = Ξ(F1)



all simplicials inside (G,u) arise from bases in this way

the coordinatization program (Lawson-Scott, arXiv: 1408.1231)

Lawson and Scott define a class of inverse monoids having the property that their lattices of principal ideals naturally

form an MV-algebra. An arbitrary MV-algebra is said to be coordinatized if it is

isomorphic to one constructed in this way from such a monoid. They prove that any countable MV-algebra can

be so coordinatized. The particular inverse monoids needed to establish this result are constructed from Bratteli diagrams of unital dimension groups, and arise naturally as limits of finite

direct products of finite symmetric inverse monoids.

Lawson-Scott coordinatization DEFINITION For any set X the symmetric inverse monoid

I(X) is the monoid of all partial bijections of X. THEOREM 0 Every finite MV-algebra can be coordinatized

by some finite direct product of finite symmetric inverse monoids.

THEOREM 1 Every Bratteli diagram B uniquely determines the inverse monoid I(B) constructed as the limit of a system of finite direct products of finite symmetric inverse monoids and standard morphisms.

COORDINATIZATION THEOREM Let E be a countable MV-algebra. Then there is a limit S of finite direct products of finite symmetric inverse monoids such that the poset of principal ideals of S is isomorphic to E as an MV-algebra.

main steps in the proof of the coordinatization theorem

Let E be a countable MV-algebra.

Then, the Γ functor theorem shows that E is isomorphic to the

MV-algebra [0, u] for some countable unital dimension group (G,u).

By Effros-Handelman-Shen, (G,u) has some Bratteli diagram B.

Let I(B) be the inverse monoid of B, as given by Theorem 1.

Then the poset of principal ideals of I(B) is isomorphic to E.

For short, E is coordinatized by the inverse monoid I(B)

We must revisit the above proof and replace every occurrence of “there is some” by “there is exactly one”. To do this we will need the sophisticated methods given by the following theorems: Chang completeness, the McNaughton representation of free MV-algebras, the Γ functor, and Schauder bases.

From ∃ to ∃! using the adjoint Ξ of Γ


So let us assume we are given a countable MV-algebra E. “There is a unital dimension group (G,u) such that E is the unit interval [0,u]”



So let us assume we are given a countable MV-algebra E. “There is a unital dimension group (G,u) such that E is the unit interval [0,u]”

The Ξ functor gives precisely one unital lattice ordered group (G*,u*)=Ξ(F1) such that E is the unit interval [0,u*] = [0,1]. (G*,u*) is the Grothendieck group of the semigroup of good sequences of F1


“There is some Bratteli diagram B for (G,u), arising from some sequence of simplicial groups converging to (G,u)”

From ∃ to ∃! via the largest Bratteli diagram


THEOREM (V.Marra, for unitals) Every unital lattice ordered group (G,u) is ultrasimplicial, i.e., it is the union of the unital simplicial groups (Fi,u) ⊆ (G,u), whose positive cone is contained in G+.

Any two simplicials Fi, Fj are contained in another simplicial Fk.



Thus there is precisely one largest direct system of simplicial groups whose limit is (G*,u*).






Direct system of simplicial groups ≈ the simplicials (Fi*,u*) ⊆ (G*,u*) limit of the direct system ≈ ((∪Fi*), u*)

Geometric bonus for free MV-algebras: simplicial group ≈ Schauder basis



“There is some Bratteli diagram B for (G,u)”

There is precisely one largest Bratteli diagram B*(G*,u*) for the unital lattice ordered group (G*,u*) corresponding to the free MV-algebra F1 via the Γ functor: It is obtained from the Schauder hats of F1.

the largest diagram of (G*,u*) = Ξ(F1)

Let Q=F1/J be a quotient of F1 for some ideal J of F1.

Q is any possible one-generator MV-algebra.

E.g., the Effros-Shen and the Behnke-Leptin MV-algebras

the diagram of a quotient Q of F1/J

the diagram of a quotient Q of F1/J

Let Q=F1/J be a quotient of F1 for some ideal J of F1.

Q is any possible one-generator MV-algebra.

E.g., the Effros-Shen and the Behnke-Leptin MV-algebras.

Then the simplicials sitting inside the unital lattice ordered group Ξ(F1/J) determine the largest diagram for F1/J.

THEOREM (D.M., J. Functional Analysis 65,1986) Every AF-algebra is a subalgebra of a quotient of M. These quotients are all AF-algebras whose Murray-von Neumann order of projections is a lattice.

Let Fω be the free countably generated MV-algebra.

The universal AF-algebra M is defined by Fω = ΓK0(M).

countable and uncountable MV-algebras

THEOREM (D.M., J. Functional Analysis 65,1986) Every AF-algebra is a subalgebra of a quotient of M. These quotients are all AF-algebras whose Murray-von Neumann order of projections is a lattice.

Let Fω be the free countably generated MV-algebra.

The universal AF-algebra M is defined by Fω = ΓK0(M).

M , just like M1 , has a largest Bratteli diagram, given by the direct system of all Schauder bases of the free MV-algebra Fω, i.e., free generating sets of all unital simplicials sitting inside Ξ(Fω). Any countable MV-algebra inherits from M a largest Bratteli diagram.

countable and uncountable MV-algebras

Any (possibly uncountable) MV-algebra A has a largest Bratteli diagram, given by the unital simplicials sitting inside Ξ(A).

The Schauder bases of any free MV-algebra F are free generating sets of the simplicial groups inside Ξ(F)

to see this, we more generally use Schauder bases over the

cube [0,1]3, then over

[0,1]4, [0,1]5,..., [0,1]ω,..., [0,1]κ,...,

for any cardinal

hats over [0,1]

hats over [0,1]2

thank you

department of mathematics and computer science university...

Documents