“Operator Algebras Associated to some Mathematical Structures”

Operator Algebras Associated to

Some Mathematical Structures

Marius M˘antoiu

Universidad de Chile

August 2016

1 _{Some basic facts about C}∗_-algebras.

2 C∗-algebras associated to topological spaces and groups.

3 _C∗_{-algebras associated to dynamical systems.}

Introduction

Many spacesCappearing in mathematics and their applications (physics,

etc) composed of (a) functionsf : Ω→C, (b) matrices, (c) operators

T :H → Hin some Hilbert space (d) others share a commun algebraical and topological structure:

can be composed by addition and multiplication and can be multiplied by scalars:

(αf +βg)(ω) :=αf(ω) +βg(ω), (fg)(ω) =f(ω)g(ω),

(αS+βT)u:=αSu+βTu, (ST)u=S(Tu),

they often have natural (complete) norms

kfk∞:= sup

ω∈Ω

|f(ω)|, kTk:= sup kuk=1

kTuk

and natural involutions

Introduction

Some extra compatibility relations (natural and satisfied in most of the cases) leads to very rich structures

general enough to include a lot of examples,

particular enough to allow classification, precise and far reaching results,

interesting combinations of algebra and analysis (nicer than Banach spaces),

with potential applications to other fields,

very often based on a simpler structure (topological space, group, groupoid, graph, dynamical system)

Abstract Banach and

C

-algebras

Definition

1 A∗-algebra is an algebraBwith an involution∗:B→Bsatisfying

fora,b∈B, α, β ∈C:

(αa+βb)∗=αa∗+βb∗, (ab)∗=b∗a∗,

(a∗)∗=a.

2 B isunitalif ∃1∈Bwitha1=a=a1, ∀a∈B. 3 _{B iscommutativeif ab=ba,} ∀_a,b∈_B.

4 _ABanach ∗_{-algebrais a Banach space and a}∗_{-algebra such that}

kabk ≤ kak kbk. ka∗k=kak.

5 _AC∗_{-algebrais a Banach}∗_{-algebraBsuch that}

for everya∈Bone has ka∗ak=kak2_.

Substructures and morphisms

Definition

Obvious notions in a∗-algebra:

subalgebra,∗-subalgebra, ideal (of all kinds),∗-ideal (of all kinds)

A closed∗-subalgebra of a Banach∗-subalgebra is

a Banach∗-algebra.

A closed∗-subalgebra of aC∗-subalgebra is aC∗-algebra.

(Bad) Convention

In a Banach∗-algebra; IDEAL = closed bi-sided self-adjoint ideal.

Definition

Morphism between two Banach∗-algebras:

Some examples (the red ones are

C

-algebras)

IfHis a (complex separable) Hilbert space, then

B(H):={T :H → H |T is linear and continuous(bounded)},

FR(H):={T ∈B(H)|THhas finite dimension}, K(H):={T :H → H |T is compact},

B2(H):= Hilbert-Schmidt operators .

If Ω is a (Hausdorff) locally compact space, then

C(Ω):={f : Ω→C|f is continuous},

BC(Ω):={f : Ω→C|f is continuous and bounded},

BCu(Ω):={f : Ω→C|f is uniformly continuous and bounded}, C0(Ω):={f : Ω→C|f is continuous and limω→∞f(ω) = 0},

Cc(Ω):={f : Ω→C|f is continuous and supp(f)is compact},

They all coincide if Ω is compact.

Cper(_R):={f :_R→_C|f cont. ,f(x+ 3,14) =f(x), ∀x∈_R},

Concrete

C

-algebras

Definition

Aconcrete C∗-algebrais a closed subspace CofB(H) which is

Stable under multiplication : S,T ∈C ⇒ST ∈C.

Stable under involution: T ∈C ⇒T∗∈C.

Examples

1 {0},_C≡_C1_H and_B(H)areC∗-algebras.

2 _K(H)is a closed two-sided∗-ideal, hence aC∗-algebra.

3 _C0(Rn_),BCu(Rn_),BC(Rn_),→_BL2_(Rn_{)are AbelianC}∗_-algebras.

4 TheC∗-algebra C∗(T)generated byT ∈_B(H) is Abelian

(commutative) iffT is normal (T∗T =TT∗) .

5 _{Many, many, manyothers.}

Avon Neumann algebrais a concreteC∗-algebra C⊂B(H)

Representations of

C

-algebras

Definition

1 Arepresentationof theC∗-algebraCin a Hilbert space His a

morphismr :C→_B(H)(it respects the algebraic operations) :

r(αa+βb) =αr(a) +βr(b), r(ab) =r(a)r(b), r(a∗) =r(a)∗

2 Representations can be reducible, irreducible, direct sums, unitarily

equivalent, etc.

3 The representation r is non-degenerateif ∩_a∈Cker[r(a)] ={0},

which is equivalent to the fact that r(C)HspansH.

Theorem

1 The kernel of a representation is a closed bi-sided∗-ideal.

2 The ranger(C) of a representation is a (concrete)C∗-algebra.

3 _{A representation is automatically contractive (thus continuous).}

Some properties

1 IfD is aC∗-subalgebra ofCanda∈D, then sp_D(a) =sp_C(a)

(stablility under inversion).

2 _{The norm} k_ak_{of a normal elementa}∈_{Ccoincides with its spectral}

radius ρ(a) := sup{|λ| |λ∈sp(a)}= lim

n→∞ka

n_k1/n_.

3 _{Close connections between representations and states.}

The GNS construction.

4 Every abstractC∗-algebraadmits a faitful representation in a Hilbert

space (so itis isomorphic to a concreteC∗-algebra).

5 _{Many others.}

Envelopping

C

-algebras

Given a Banach∗-algebraB, we try to embed it into aC∗-algebra

B,→j C:=Env(B)(nicer properties). This one should satisfy

The universal property (definition)

If (B→µ D=C∗-algebra) is a morphism, then there exists a unique

morphismEnv(B)Env−→(µ)Dsuch that Env(µ)◦j=µ.

Theorem

1 _{Such an object exists and is unique up to a canonical isomorphism.}

2 If ν:B1→B2 is a morphism of Banach∗-algebras, then

canonically Env(ν) :End(B1)→Env(B2) .

Essentially,Env(B) can be the completion ofB in the norm

Special elements in (unital)

C

-algebras

Definition

LetCbe aC∗-algebra.

a∈Cisself-adjointifa=a∗(⇔ sp(a)⊂_R) .

b∈Cisnormalifbb∗=b∗b(C∗(b) is Abelian ) .

p∈Cisa projection ifp2_=p=p∗ _{(⇔ sp(a)⊂ {0,1}) .}

u∈Cisunitary ifu∗_u=1=uu∗ _{(⇔ sp(u)⊂}

S1) . v ∈Cis an isometryifv∗v=1.

If π:C→B(H) is a representation, then π(c)∈B(H) has the same

Concrete Abelian

C

-algebras

LetΩbe a (Hausdorff) locally compact space, meaning that eachω∈Ω

has a basis of compact neighborhoods.

C0(Ω):={f : Ω→C|f is continuous,limω→∞f(ω) = 0}

is an AbelianC∗-algebra with

(f +g)(ω) :=f(ω) +g(ω), (αf)(ω) :=αf(ω),

(fg)(ω) :=f(ω)g(ω), f∗(ω) :=f(ω),

kfk≡kfk∞:= sup

Topological to algebraical properties

Ω composed of npoints ⇔ C0(Ω)∼=Cnis n-dimensional.

Ω compact ⇔ C0(Ω) unital.

minimal unitization of C0(Ω) isC(Ω•) (Alexandrov).

unitizations ofC0(Ω) ↔ compactification of Ω .

IDEAL ↔ closed subset.

maximal IDEAL ↔ point.

maximal unitization ofC0(Ω) isomorphic toC(βΩ)

f ∈C0(Ω) self-adjoint ⇔ f(ω)∈R, ∀ω∈Ω .

f ∈C(Ω) unitary ⇔ |f(ω)|= 1, ∀ ω∈Ω .

f ∈C0(Ω) projection ⇔ Ω = Ω0tΩ1, f|Ω0= 0, f|Ω1 = 1 .

Ω (compact) is connected ⇔ C0(Ω) has no non-trivial projections.

conected component ↔ mimimal projection6= 0 (define!)

Ω second countable ↔ C0(Ω) separable.

Ω∼Ω0 _{(homeomorphic) ⇔ C}

Abelian

C

-algebras

Definition

LetAan abelianC∗-algebra. Itsspectrum (character space)is

SP(A)≡Ab :={ϕ:A→C|ϕis a ∗−morphism}.

It is a locally compact space with the topology of the uniform convergence on compact sets.

b

Aidentified with the familyMID(A)of maximal IDEALS of A.

Ifa∈Cis a normal element andA:=C∗(a) , thenSP(A)∼sp(a).

Schur’s Lemma

The representation ϕ: G→_B(H) is irreducible iff

only the constant operators λid commute with allϕ(a) .

Consequence

The irreducible representations of an AbelianC∗-algebra are

Gelfand’s Theorem

Theorem (Gelfand)

G:A→C0(Ab), [G(a)](ϕ) :=ϕ(a), ∀a∈A, ∀ϕ∈Ab

is an isomorphism ofC∗-algebras.

IfA=C0(Ω) , thenAb and Ω are homoemorphic.

Functor

Gelfand’s transformation definesa contravariant functor

from the category of AbelianC∗-algebras

Locally compact groups

Not all the groups are finite!!!

Theorem

Topological group (G,·,T), where

(G,·) is a group, with unit e,

T is a topology onG (always Hausdorff),

G×G3(x,y)→xy∈G, G3x→x−1∈G are continuous.

It isa locally compact groupifehas a basis on compact neighborhoods.

Examples

Discrete(in particular finite) groups.

All kind of p-adicgroups.

Lie groups(Rn, orthogonal, unitary, simplectic, Heisenberg, Galilei,

Lorentz, Poincar´e, etc)

Infinite-dimensional normed spaces are topological groups,

Integration on locally compact groups

Algebra and analysis on_Zn _or

Rnare ”nice and easy” because

sums and Lebesgue integrals are invariant under translations.

Theorem

On any locally compact group G there is a(left) Haar measure, i.e.

a positive Borel measure m:Bor(G)→[0,∞)such that it is

left invariant: m(xA) =m(A) if x∈G andA∈Bor(G),

finite on compact sets: m(K)<∞ifK ⊂G is compact,

regular in a certain sense.

Ifm0 is a second Haar measure, thenm0=cm withc>0 .

Consequence

Z

G

f(ax)dm(x) =

Z

G

f(x)dm(x), ∀a∈G.

Group

C

-algebras

We fix a unimodular (for simplicity) locally compact groupGwith unite

and Haar measurem. Then(L1_{(G),k · k}

1)is a Banach∗-algebra with:

(f ∗g)(x) := Z

G

f(y)g(y−1x)dm(y), f∗(x) :=f(x−1_).

The envelopingC∗-algebraC∗(G):=Env

L1_(G)

is called

the (universal) groupC∗-algebra of G .

Theorem

1 _{It is unital iff G is discrete. The unit isδe.}

In this case δx ∗δy =δxy andδx∗=δx−1.

2 C∗(G) is commutative iff G is commutative.

By Gelfand’s Theorem, it should be isomorphic to someC0(Ω) ,

where Ω is a locally compact space.

Unitary group representations

Definition

(Strongly continuous unitary) representationof the group G in the

Hilbert spaceH: U : G→B(H) such that

U(x)∗U(x) = Id =U(x)U(x)∗, ∀x ∈G,

U(x)U(y) =U(xy), ∀x,y ∈G,

The map G3x→U(x)v ∈ His continuous for every v∈ H.

The representations can be (ir-)reducible, unitarily equivalent, etc.

Schur’s Lemma ⇒ the single irreducible representations of an Abelian

Group

C

-algebras and representations

Theorem

One-to-one correspondence between

strongly continuous unitary representationsU : G→U(H) ,

non-degenerate representations r :C∗(G)→_B(H) ,

respecting unitary equivalence, (ir)reducibility, direct sums, etc.

1 _{IfU is given, setrU:L}1_(G)→_B(H_{) by (integrated form)}

rU(f) :=

Z

G

f(x)U(x)dm(x).

2 _{Ifr is given and G is discrete, setUr(x) :=r(δx)for everyx} ∈_{G . In}

The Pontryagin group dual

Definition

Let G be a locally compact Abelian group. Its(Pontryagin) dual:

b

G: ={χ:A→C|χ(xy) =χ(x)χ(y), χcontinuous}

=the irreducible representations of G =characters.

Theorem

b

G is a locally compact Abelian group with

The product (χξ)(x) :=χ(x)ξ(x), ∀x∈G.

The topology of uniform convergence on compact sets.

b

G is discrete iff G is compact andG is compact iff G is discrete.b

Theorem (Pontryagin duality)

Isomorphism of topological groups: _Gb_b ∼_{= Gby}

Some Fourier transformations

Particular cases: Rcn∼=Rn, Tcn∼=Zn, Zcn∼=Tn. For example

Rn3y→χy ∈_Rcn, χy(x) :=e−ix·y.

Recall some Fourier transformations of the formFG:L1(G)→C0(bG)

F_Tn(f)(n) :=

Z

Tn

e−in·τf(τ)dτ ,

F_Zn(g)(τ) :=

X

n∈Zn

e−iτ·ng(n),

F_Tn(h)(y) :=

Z

Rn

The Fourier transform for Abelian locally compact group

Definition

Ifg is a ”reasonable” complex function on the Abelian locally compact

group G , itsFourier transform is a function onG :b

FG(g)

(χ) :=

Z

G

χ(x)g(x)dm(x)

Theorem

Bounded linear FG:Lp(G)→Lq(G) ifb p∈[1,2] and 1/p+ 1/q= 1 .

Unitary operator FG :L2(G)→L2(G) (Plancherel).b

Contractive linearFG:L1(G)→C0(bG) (Riemann-Lebesgue).

Inversion formula: F−1

G =inv◦ FbG, i.e.

b g(χ) :=

Z

G

χ(x)g(x)dmG(x) ⇔ g(x) := Z

G

χ(x)_bg(χ)dm

b

The Gelfand transform for Abelian locally compact groups

Recall thatL1_{(G) is a (convolution) Banach}∗_{-algebra whileC}

0(G) is ab

(point-wise)C∗-algebra.

Theorem

The Fourier transformFG:L1(G)→C0(G) is a (contractive) morphism:b

It is linear, k FG(f)k∞≤ kfk1,

FG(f ∗g) =FG(f)FG(g) , FG(f∗_{) =FG(f) .}

It extends to an isomorphism ofC∗-algebras FG:L1(G)→C0(bG) .

Theorem

If G is an Abelian locally compact group, then

SP[C∗_{(G)] is homeomorphic tothe dual group}

b G,

the Gelfand transform is essentially the Fourier transform and

Twisted group

C

-algebras

Definition

1 Two-cocycle with values in the torus (or multiplier) of G is a

continuous function γ: G×G→_T s.t. for allx,y,z∈G :

γ(x,y)γ(xy,z) =γ(y,z)γ(x,yz), γ(x,e) = 1 =γ(e,x).

2 Projective representation with 2-cocycleγ:

strongly continuous function U : G→U(H) satisfying

U(x)U(y) =γ(x,y)U(xy), ∀x,y∈G.

The (non-commut.) twisted group algebraC_γ∗(G)defined as before, but

(f ∗γg)(x) := Z

G

γ(y,y−1x)f(y)g(y−1x)dm(y),

f∗γ_{(x) :=γ(x,x}−1_)f(x−1_).

One-to-one correspondence between projective representations ofG and

Commutative and non-commutative tori are twisted group

C

-algebras (just the case

n

= 2)

For the group G =Z2andθ∈Rlet the 2-cocycleγθ:Z2×Z2→T

γθ (n1,n2),(m1,m2)

:=e−iθ(n1m2−n2m1)_.

Result

If you write down the twisted group algebraC∗

γθ(Z

2_{) you find out that it}

is isomorphic to the rotation algebra (non-comm. torus)

Aθ≡C∗(u,v;θ)generated by two unitary elementsu,v satisfying the

commutation relationuv =eiθ_vu.

This happens basically because the groupZ2is generated by two elements

(1,0) and (0,1) . Settingu:=δ(1,0),v:=δ(0,1)∈`1(Z2)⊂Cγ∗θ(Z

2_{) :}

u∗∗u=δe=u∗u∗, v∗∗v =δe=v∗v∗, u∗v=eiθv∗u.

Ifθ∈2π_Zthen γθis trivial andCγ∗θ(Z

2_{) =C}∗₍

Z2)∼=C(Tn).

C

-dynamical systems

Definition

AC∗-dynamical systemis a triple (A, α,G)where

G is a locally compact group, Ais aC∗-algebra,

α: G→Aut(A) is a strongly continuous action by automorphisms.

Example

LetA:=C0(Y) be an AbelianC∗-algebra. Any automorphism

α:A → Acomes from an homeorphismβ :Y →Y by

[α(ϕ)](y) :=ϕ[β−1(y)], ϕ∈C0(Y),y∈X.

So in this case theC∗-dynamical system is given by

a topological dynamic system(Y, τ,G)

Crossed products

Definition

To a aC∗-dynamical system (A, α,G) we associate

the crossed productC∗-algebra A_oαG:=EnvL1(G;A),

where the Banach∗-algebra structure onL1_{(G,A) is}

kFk(1):= Z

G

kF(x)kAdm(x),

(FG)(x) := Z

G F(y)αy

G(y−1x) dm(y),

F(x) :=αx

F(x−1)∗ .

Examples

1 IfA=_Cthenα_x =idand_{C o}G =C∗(G) .

2 _{If G =}{e} _thenA_oG≡ A_.

3 In general A_oαG is a sort of twisted tensor product A

α

Covariant representations

Definition

Let (A, α,G) be aC∗-dynamical system.

A covariant representationis a triple(r,U,H)where

His a Hilbert space,

U : G→U(H) is a (strongly continuous) unitary representation,

r :A →B(H) is a representation (∗-morphism),

U(x)r(ϕ)U(x)∗=r[αx(ϕ)], for everyx ∈G .

Theorem

One-to-one correspondence between

covariant representations of theC∗-dynamical system (A, α,G) ,

non-degenerate representations of the crossed productAoαG .

Integrated formof (r,U,H) : firstroU :L1(G;A)→B(H)

(roU)(F) :=

Z

G

A particular case

Assume thatAis aC∗-subalgebra ofBCu(G) invariant under the left

translationsand set [αy(ϕ)(x)] :=ϕ(y−1x). One gets theC∗-dynamical

system (A, α,G) and the crossed productAoαG .

On the dense subsetL1(G;A)⊂ AoθG the composition law

(FG)(x) :=

Z

G

F(y)αyG(y−1x)dm(y)

becomes more explicit

(FG)(x,q) = Z

G

F(y,q)G y−1x,y−1q dm(y),

andF_{(x) :=α}

x

F(x−1₎∗

The Schr¨

odinger representation

Definition

1 The Schr¨odinger representation r,U,L2(G)is

[U(y)v] (x) :=v y−1x,

r(ϕ)v :=ϕv.

2 Its integrated form is given forF ∈L1(G;A) andv ∈L2(G) by

[(roU)(F)v] (x)=

Z

G

F(z,x)v(z−1x)dm(z)

= Z

G

F(xy−1,x)v(y)dm(y).

Easy to see thatroU:L2(G×G)→B2