.

LISBON LECTURES 1

Subalgebras of graph C*-algebras

Stephen C. Power,

Lancaster University

Summer School Course for the ICM 2006 satellite workshop on

Operator Algebras, Operator Theory and Applications, Lisbon, 1-5

September 2006.

Introduction

Part I : The Cuntz algebras, intuitively

Part II : Toeplitz algebras for directed graphs

Part III : Subalgebras of On and graph C∗-algebras

1draft July 31, 20071

2

Introduction

I will consider two novel classes of nonselfadjoint operator algebras,

(i) generalised analytic Toeplitz algebras LG associated with a di-

rected graph G,

(ii) subalgebras of graph C*-algebras (mainly On).

These topics are largely independent but in both cases I focus on

classifying isomorphism types and the recovery of underlying

”foundational structures” such as graphs or groupoids.

operator algebra : a complex algebra of bounded linear operators

on a separable complex Hilbert space.

spatial setting : the Hilbert space is in place at the outset and

(often) the operator algebra is generated by generators related to the

presentation of the Hilbert space..

space-free setting : eg. an operator algebra, within some huge

general class of operator algebras, satisfying a universal property

for a set of generators and relations.

Question: When are two ”given” operator algebras isomorphic ?

3

.

Gelfand Naimark Theorem

LetA be a C*-algebra (an involutive complete normed algebra with

‖ab‖ ≤ ‖a‖‖b‖ and ‖a∗a‖ = ‖a‖2 for all a, b ∈ A). Then there is

a Hilbert space H (a separable one is possible if A is separable) and

an isometric star homomorphism A → B(H).

4

PART I

The Cuntz Algebras, intuitively.

On is a C*-algebra generated by n isometries, S1, . . . , Sn satisfying

S1S∗1 + · · · + SnS

∗n = 1. The range projections SiS

∗i are pairwise

orthogonal and sum to the identity operator.

A remarkable uniqueness property:

Proposition. If n ≥ 2 and s1, . . . , sn is any family of n isometries

in a unital C*-algebra with s1s∗1+· · ·+sns

∗n = I , then C∗({s1, . . . , sn})

is naturally isometrically isomorphic to On.

The ”interval picture” for On:

Isometric operators S1, . . . , Sn on L2[0, 1] whose ranges are

L2[i− 1

n,i

n].

Define Sif (x) =√

nf (i−1+xn ).

5

Figure 1. Interval picture for the operator S1S2 in O2.

Sµ = Si1Si2 . . . Sik has range L2[Ek] where |Ek| = n−k.

µ is the word i1i2 . . . ik.

Distinct ranges if |µ| = |ν| = k so S∗µSν = 0 if µ 6= ν. In general

S∗µSν = S∗µ′ where µ = νµ′, or S∗µSν = Sν′ where ν = µν ′.

For |µ| = |ν| = k, SµS∗ν : L2[Eν] → L2[Eµ]

The set of operators

{SµS∗ν : |µ| = |ν| = k}

form a matrix unit system with span = Fnk = Mnk(C).

Hence the tower

Fn1 ⊆ Fn

2 ⊆ Fn3 . . .

Thus the generators of On give a subalgebra Fn∞ = ∪∞k=1Fn

k : as a

matricial star algebra of type n∞.

6

Fn is the closure.

7

Generalised Fourier Series

Proposition. (i) Each operator a in the star algebra generated

by S1, . . . , Sn has a representation

a =

N∑i=1

(S∗1)ia−i + a0 +

N∑i=1

aiSi1

where ai ∈ Fn∞ for each i. This representation is unique if for each

i ≥ 0 we have ai = aiPi and a−i = Pia−i where Pi is the final

projection of Si1.

(ii) The linear maps Ei, defined by Ei(a) = ai, extend to continu-

ous, contractive, linear maps from On to Fn.

(iii) The generalised Cesaro sums

Σk(a) =

N∑k=1

(1− |k|

N

)(S∗1)kE−k(a) +

N∑k=0

(1− |k|

N

)Ek(a)sk

1

converge to a as N →∞.

Key to uniqueness - the ”recovery formula”

a0 =

∫Tγz(a)dz.

A Riemann integral of a norm continuous function.

8

The ”Cantorised interval picture” for On:

X is the set of infinite paths. This is equal to∏∞

k=1{0, 1}.

Typical path/point: x = x1x2 . . . .

Each vertex word w ”suspends” an ”interval” Ew in X of points x

which start with w.

Product topology yields a Cantor space. Intervals form a base of

closed-open sets.

For vertex words w1, w2 there is a partial homeomorphism

αw2,w1 from Ew1 to Ew2 ”matching tails” :

if x = w1xpxp+1 . . . then αw2,w1(x) = w2xpxp+1 . . .

This p.h. has an action on X like Sw2S∗w1

and its interval picture.

9

(NB Generators now relabeled S0 and S1.)

Put the product probability measure on X and present On on

L2(X) in terms S0 and S1 defined by the partial homeomorphisms

α∅,0 : x → 0x, α∅,1 : x → 1x.

That is,

S0f (x) =√

2f (α∅,0(x)), S1f (x) =√

2f (α∅,1(x)).

Beauty of Cantorised interval picture :

(a) It provides a useful representation of On on a Hilbert space.

(b) The graphs of the partial homeomorphisms provide a means

for invariants for subalgebras of Fn (and On) in the category of

binary relations on X (and groupoids).

10

Normalising partial isometries

Definition. A partial isometry in Fn (or, more generally, in On)

is C-normalising if vCv∗ ⊆ C and v∗Cv ⊆ C.

eg the matrix units SµS∗ν , |µ| = |ν| = k, and certain sums Also,

may multiply v by a unitary diagonal element d in C.

Support which can be indicated pictorially, shown as a ”continuous

matrix” (but no d information, only ”action information”).

Figure 2. The support the partial isometry S2S∗1 + S11S

∗22.

Theorem Let v be an element of Fn. Then the following as-

sertions are equivalent :

(i) v is a C-normalizing partial isometry.

(ii) v is an orthogonal sum of a finite number of partial isome-

tries of the form dSµS∗ν , where |µ| = |ν| and d ∈ C.

(iii) For all projections p, q ∈ C, the norm ‖qvp‖ = 0 or 1.

11

Key Fact : How operators b in Fn may be ”constructively ap-

proximated” by elements ∆m(b) in ”small explicit subalgebras” :

For b ∈ Fn the ”diagonal part” ∆(b) ∈ C may be defined as the

limit as k →∞ of the block diagonal operators

bk := Σiekiibe

kii

where ekii are the diagonal matrix units of Fn

k .

∆ : Fn → C is a projection (and faithful ∆(b∗b) = 0 =⇒ b = 0).

Likewise one can use block diagonal maps (via matrix unit projec-

tions taken from the commutants of Fnm in Fn

k , k = m, m + 1, . . . )

to define explicit maps

∆m : Fn → F̃nm

where F̃nm = C∗(C,Fn

m). (This algebra = Fnm ⊗ (em

11Cem11.) We have

∆m(b) → b as m →∞ for all b ∈ F n.

12

Sketch Proof of Theorem 2.4 : Assume v satisfies the zero one

norm condition (iii). Choose m large so that

v = ∆m(v) + v′, ‖v′‖ < 1.

Since v satisfies (iii) so too does ∆m(v) and v − ∆m(v)(= v′). (use

the def of ∆m(v).)

The implication (iii) implies (ii) is straightforward for elements of

F̃nk . Thus it remains to show that if v′ has norm less than one and

satisfies the zero one norm condition then v′ = 0. This follows from

approximation.

13

Subalgebras of On ?

First Mn !

Let E ⊆ {1, ..., n}×{1, ..., n} be a binary relation, symmetric and

transitive. Then

A(G) := span{eij : (i, j) ∈ E}

is a subalgebra of Mn, known as a digraph algebra (or incidence

algebra). It contains the diagonal algebra:

Cn ⊆ A(G) ⊆ Mn(C)

Theorem. (Recovery theorem.) If A(G1) and A(G2) are isomet-

rically isomorphic algebras then the graphs G1, G2 are isomorphic.

14

Binary relation invariants.

Define R = R(Fn) ⊆ X ×X to be the union of the graphs of the

”nondilating” partial homeomorphisms :

R = {(α(x), x) : x ∈ dom(α), α = αµ,ν, |µ| = |ν| = k, k = 1, 2, . . . }

Write Eki,j for the subset of R that is the graph of the partial home-

omorphism for the matrix unit e(k)ij in Fn

k . (A ”support set” for this

matrix unit.) The diagonal matrix units e(k)ij provide closed-open sets

Ekii in the diagonal ∆ = {(x, x) : x ∈ X} and these give a base for

a natural (Cantor space) topology. Topologise R by taking the sets

Ekij as a base for the topology.

Likewise, if C ⊆ A ⊆ Fn then A, and the given subalgebra chain,

determines a subset R(A) of R, namely

R(A) = ∪{Ekij : ek

ij ∈ Ak}.

With the relative topology, R(A) is known as the topological binary

relation of A. In fact the topological binary relation R(A) is deter-

mined by the pair (A, C) and serves as the analogue of the graph of

a digraph algebra.

15

Proposition As a set, R(A) is the set of points (x, y) in X × X

for which there exists a ∈ A and δ > 0 such that

‖paq‖ ≥ δ

for all projections p, q in C with x(p) = y(q) = 1.

Theorem. Let A1 and A2 be norm-closed subalgebras of Fn with

Ai ∩ A∗i = C for i = 1, 2. (Such algebras are said to be triangular.)

Then the following statements are equivalent

(i) A1 and A2 are isometrically isomorphic operator algebras.

(ii) The topological binary relations R(A1), R(A2) are isomorphic,

that is, there is a homeomorphism α : M(C) → M(C) such that

α× α : R(A1) → R(A2) is a homeomorphism.

16

Sketch proof of (i) =⇒ (iii) Let Φ : A1 → A2 be an isometric

isomorphism. By triangularity, the set of projections p in Ai gen-

erate Ci. Since projections are the norm one idempotents it follows

that Φ(C1) = C2. By the zero-one characterisation Φ maps the C1-

normalising partial isometries to C2-normalising partial isometries.

Now considering the support of a normalising partial isometry as a

closed-open set in M(Ci) ×M(Ci) it follows that Φ maps a base of

closed open sets to a base of closed-open sets. Moreover Φ induces

a point bijection α : R(A1) → R(A2). (Take intersections of neigh-

bourhoods.) The point bijection is a topological homeomorphism

since its map on sets extends the original bijection of closed-open

sets.

17

Part II

Toeplitz algebras for countable directed graphs

Prelude: Context A: The classical Toeplitz context :

The Hardy Hilbert space H2 for the unit circle.

The unilateral shift operator S, with dim(I − SS∗) = 1.

The disc algebra A(D) and the function algebra H∞(D).

Realisations of A(D) and H∞(D) as ”analytic” Toeplitz algebras.

The classical Toeplitz C*-algebra TZ+.

`2(Z+) is unitarily equivalent to the Hardy space H2 of the Hilbert

space L2(T) of square integrable functions on the circle. The basis

matching unitary U : `2(Z+) → H2, with Uen := zn for n ≥ 0)

effects a unitary equivalence between the unilateral shift S and the

multiplication operator Tz : f → zf, f ∈ H2. That is

USU ∗ = Tz

The Toeplitz operator Tφ with symbol function φ in C(T) or L∞(T)

is given by Tφ : f → PH2φf .

18

A good exercise : show that the Toeplitz algebra TZ+ = C∗(I, Tz)

( = UC∗(I, S)U ∗) contains every compact operator on H2. (Start

with the rank one operator I − TzT∗z and ”move it around” with

the shifts Tz, T∗z to obtain every rank one operator of the form f →

〈f, zk〉zl. Then take linear combinations to approximate any finite

rank operator.)

Theorem.(i) For each Toeplitz operator Tφ and compact operator

K we have

‖Tφ + K‖ ≥ ‖Tφ‖.

(ii) The Toeplitz algebra TZ+ is equal to the set of operators Tφ+K

with φ in C(T) and K compact and the quotient TZ+/K is naturally

isomorphic to C(T)

The norm closed (resp. weakly closed) operator algebras generated

by Tz are abelian:

{Tφ : φ ∈ A(D)},

{Tφ : φ ∈ H∞(D)}.

The weak operator topology is the weakest topology for which the

spatial linear functionals T → 〈Tf, g〉 are continuous.

19

Toeplitz algebras from graphs.

G : a finite or countable directed graph.

F+(G), semigroupoid of G : set of finite directed paths.

Partially defined product = concatenation of paths

HG = `2(F+(G)) : Hilbert space with

orthonormal basis of vectors ξw, w ∈ F+(G).

For each edge e ∈ E(G) and vertex x ∈ V (G) define partial isome-

tries and projections on HG by left-sided actions on basis vectors:

Leξw =

ξew if ew ∈ F+(G)

0 otherwise

and

Lxξw =

ξxw = ξw if w = xw ∈ F+(G)

0 otherwise

Eg. G with single vertex and two loop edges. F+(G) = F+2 .

20

.

Definition. (i) The free semigroupoid algebra LG is the weak

operator topology closed algebra generated by the projections Lx and

the (partial) shift operators Le;

LG = wot–Alg {Le, Lx : e ∈ E(G), x ∈ V (G)}

= wot–span {Lw : words/paths in the edges }.

(ii) The algebraAG is the norm closed algebra generated by {Le, Lx :

e ∈ E(G), x ∈ V (G)}. This algebra is also referred to as the tensor

algebra for G.

21

Example 1.

G = ({x1, x2, x3}, {e, f} with e = x2ex1, f = x3fx1.

Fock space = span {ξx1, ξx2, ξx3, ξe, ξf}). Typical operator is X =

αLx1 + βLx2 + γLx3 + λLe + µLf . Represented by the matrix

X '

α

β

γ

λ β

µ γ

.

Here, LG is isometrically isomorphic to (but not unitarily equivalent

to) the usual digraph algebra of G consisting of the matricesα 0 0

λ β 0

µ 0 γ

.

22

Example 2.

G = ({x, y}, {e, f}), e a loop edge e = xex and edge f = yex

directed from x to y. Tree graph for Fock space :

Figure 3. Fock space graph.

LG is generated by {Le, Lf , Px, Py}. Identify

HG = PxHG ⊕ PyHG ' H2 ⊕H2. Then

Le '

Tz 0

0 0

, Lf '

0 0

Tz 0

, Px '

I 0

0 0

, Py '

0 0

0 I

.

Thus, LG is unitarily equivalent to a matrix function algebraH∞ 0

H∞0 CI

where H∞

0 is the subalgebra of H∞ formed by functions h with

h(0) = 0.

23

Example 3.

The cycle graph Cn = ({x1, . . . , xn}, {e1, . . . , en})

en = x1enxn and ek = xk+1ekxk.

LxiHG = H2 for each i and

HG = Lx1HG ⊕ . . .⊕ LxnHG ' Cn ⊗H2

Operator α1Le1 + . . . + αnLen is

0 αnTz

α1Tz 0

α2Tz 0

. . . . . .

αn−1Tz 0

.

LCn is isomorphic to the matrix function algebraH∞

n zn−1H∞n . . . zH∞

n

zH∞n H∞

n...

... . . .

zn−1H∞n . . . H∞

n

.

where H∞n = {h(zn) : h ∈ H∞}.

24

Generalised Fourier Series (once more)

Proposition. Let A ∈ Ln and aw ∈ C the coefficients for

which Aξ0 =∑

w∈F2awξw. Then the Cesaro sums associated with

the formal sum ∑awLw

given by

Σk(A) =∑

w

(1− |w|

k

)awLw

converge in the strong operator topology to A.

The right regular representation yields partial isometries for w ∈

F+(G) acting on HG defined by the equations Rw′ξv = ξvw, where

w′ is the word w in reverse order. The corresponding algebra is

RG = wot–Alg {Re, Rx : e ∈ E(G), x ∈ V (G)}.

Note that LeRfξw = ξewf = RfLeξw, so that LeRf = RfLe.

Fourier series arguments lead to

Proposition. The commutant L′G of LG is equal to RG.

LG is equal to its second commutant.

25

LG remembers the graph G

An ”eigenvector” for L∗n :

L∗eiν = αiν, , 1 ≤ i ≤ n,

for some complex numbers αi ∈ C. .

Theorem. The eigenvectors for L∗n are complex multiples of the

unit vectors

νλ = (1− ‖λ‖2)1/2∑w∈F+

n

w(λ)ξw,

for λ = (λ1, . . . , λn) in the open unit ball Bn ⊆ Cn. Furthermore

L∗eiνi = λiνλ, for each i.

w(λ) means substitute λi for ei in the word w.

Note that

‖n∑1

λiLei‖2 =

∑|λi|2 = ‖λ‖2 < 1

so that I −∑n

1 λiLeiis invertible, with inverse(

I −∑

e

λiLei

)−1

=∑k≥0

(∑e

λiLei

)k

=∑

w

w(λ)Lw.

26

Eigenvectors relate to characters, ie., multiplicative linear func-

tionals φ : Ln → C, φ : An → C:

The map φν : An → C defined by

φν(A) = 〈Aνλ, νλ〉

satisfies

φν(p(Le1, . . . , Len)) = 〈νλ, p(L∗e1, . . . , L∗en

)νλ〉

= 〈νλ, p(λ1, . . . , λn)νλ〉 = p(λ1, . . . , λn).

and the vector functional defines a character. The character space of

An is homeomorphic to closed unit ball.

The dimension of the character space serves as a classifying in-

variant for the algebras An and Ln. More generally one has the

following theorem, and an analogous result for the weakly closed free

semigroup algebras.

Theorem. Let G, G′ be directed graphs. Then the following

assertions are equivalent.

(i) G and G′ are isomorphic graphs.

(ii) AG and AG′ are unitarily equivalent.

(ii) AG and AG′ are isometrically isomorphic.

27

Invariant Subspaces

Beurling’s Thm. LatL1 = {uH2 : |u(z)| = 1a.e. (u is inner)}

Wandering vectors ξ for the semigroup {Lw : w ∈ F+n }:

{Lwξ : w ∈ F+n } is an orthonormal set.

They can be found in M (span{LwM : Lw 6= I}) when M is

invariant for Ln.

If M has such a ξ as a cyclic vector then one can first define an

isometry Rξ via

Rξξw := Lwξ

One can then check that Rξ is in the commutant of Ln and that

M = RξHn.

Theorem. LatLn = {UHn : U ∈ Rn, U∗U = I}

{........Automorphisms, Sketch proof of $L_G$

remembers $G$ }

28

Part III

Subalgebras of On

Definition. (a) A unital matricial star algebra is a complex alge-

bra B with spanning set

{ekij : 1 ≤ i, j ≤ nk, k = 1, 2, . . . }

such that

(i) for each k the set {ekij : 1 ≤ i, j ≤ nk} is a m.u.s. for Mnk

,

(ii) for each k, Mnk⊆ Mnk+1 and moreover the inclusion map is a

C*-algebra injection which maps each ekij to a sum of matrix units

from {ek+1ij : 1 ≤ i, j ≤ nk}.

(b) A regular matricial algebra is a complex algebra A which is a

unital subalgebra of a matricial star algebra containing the diagonal

subalgebra C = span{ekij}.

The algebra A in B is the union of the algebras

Ak = A ∩Mnk= span{ek

ij : ekij ∈ A}

and each Ak is a digraph algebra (= A(Gk)) relative to the given

m.u.s.

Important open (algebraic) problem. ”Is C unique in A

up to automorphisms of A.” ?

29

Examples. (i) Fn∞ is a unital matricial star algebra.

(ii) Towers of inclusions maps,

A(G1) → A(G2) → A(G3) . . .

Algebraic direct limits provide B and A and the diagonal algebra C,

with C ⊆ A ⊆ B.

Given

C ⊆ A ⊆ B

take operator norm closures:

C ⊆ A ⊆ B

Here B is a UHF C*-algebra, C is a ”particularly nice” maximal

abelian subalgebra (masa) in B and A is an instance of a limit

algebra A = limk(A(Gk), φk) where inclusion maps φk : A(Gk) →

A(Gk+1) are very nice (ie. star-extendible and regular).

There is infinite variety, and there are many papers that analyse and

classify special classes of these algebras. (eg Lexicographic products.)

Normalising p.i.s and R(A) are key tools.

30

Key theorems for the intermediate algebras

C ⊆ A ⊆ B

Theorem. (”Inductivity principle”, or ”local recovery”)

A is the closed union of the finite dimensional digraph algebras Ak =

A ∩Mnk, k = 1, 2, . . . .

Theorem. (Spectral theorem for subalgebras.) Let A1,A2 be

norm closed subalgebras of Fn both of which contain the canonical

diagonal algebra C. If R(A1) = R(A2) then A1 = A2.

Theorem. (Classification.) Let A1 and A2 be norm-closed sub-

algebras of Fn with Ai ∩ A∗i = C for i = 1, 2. (Such algebras are

said to be triangular.) Then the following statements are equivalent

(i) A1 ∩ Fn∞ and A2 ∩ Fn

∞ are isometrically isomorphic normed al-

gebras.

(ii) A1 and A2 are isometrically isomorphic operator algebras.

(iii) The topological binary relations R(A1), R(A2) are isomorphic,

that is, there is a homeomorphism α : M(C) → M(C) such that

α× α : R(A1) → R(A2) is a homeomorphism.

31

Topological semigroupoids

(binary relation invariants revisited)

Recall : Cantor interval picture and the partial homeomorphisms

αµν. Dilation factor of αµν is k = |ν| − |µ|.

The Cuntz groupoid is

R(On) = {(x, k, y) : x = αµν(y) for some αµν}

(ie., support of the algebra in the Cantorised interval picture)

together with

(i) the totally disconnected topology: (as before) this has the graphs

Eµν for the αµν as a base.

(ii) partially defined multiplication from composition of partial

homeomorphisms.

Now it is natural to obtain for subalgebras of On analogous results

to those in Section 2, using R(On).

32

Complication : ”synthesis” may fail. However, it is the gauge

invariant closed subalgebras containing C that are determined by

their ”support” :

Theorem. (Synthesis.) Let A be a closed subalgebra of On

containing the canonical diagonal masa C. Then A is generated

by the partial isometries SµS∗ν belonging to A if and only if A is

invariant under the gauge automorphisms γz for |z| = 1.

For such an algebra A define an associated topological semi-

groupoid R(A):

R(A) = {(x, k, y) : x = αµν(y) for µ, ν such that SµS∗ν ∈ A}

with relative topology and the partially defined multiplication. With

the characterisation of normalising partial isometries it is possible

show that gauge invariant triangular subalgebras of On remember

their semigroupoids and are classified by them:

Theorem. Let A1 and A2 be norm-closed subalgebras of On

with Ai ∩ A∗i = C for i = 1, 2. Then the following statements are

equivalent

(i) A1 and A2 are isometrically isomorphic operator algebras.

(ii) The semigroupoids R(A1), R(A2) are isomorphic, that is, there is

a homeomorphism α : R(A1) → R(A2) which respects the partially

defined multiplication.

33

Normalising partial isometries (once more

Recall

Definition. A partial isometry in Fn (or, more generally, in On)

is C-normalising if vCv∗ ⊆ C and v∗Cv ⊆ C.

Theorem. Let v be a contraction in On. Then the following

assertions are equivalent:

(i) v is a C-normalizing partial isometry.

(ii) v is an orthogonal sum of a finite number of partial isometries of

the form dSµS∗ν , where d ∈ C.

(iii) For all projections p, q ∈ C, the norm ‖qvp‖ is equal to 0 or 1.

Intuitive Proof of (iii) =⇒ (ii): Contemplate Fourier series of v:

E0(v) satisfies the 0,1 condition, because v is almost a generalised

Cesaro polynomial v′and ”support of E0(v) is disjoint almost every-

where from support of v′ − E0(v).”

Thus E0(v) is a normalising p.i.

WLOG, E0(v) = 0. Induction argument, until v = v′ is a finite sum

of n.p.is.

34

Subalgebras of On containing the masa C

(i) Generator constraints: Let S be a semigroup of operators SµS∗ν

which contains all the projections SµS∗µ. Take the norm closed linear

span. Note that this algebra is left invariant by the gauge automor-

phisms of On.

(ii) Fourier series constraints : Let A ⊆ Fn be a triangular subal-

gebra with A ∩ A∗ = C. Then

A = {a ∈ On : E0(a) ∈ A, Ek(a) = 0, k < 0}

is a triangular subalgebra of On.

(iii) Extrinsic constraints: LetN ⊆ C be a maximal totally ordered

family of projections. For example,N could consist of the projections

corresponding to the intervals [0, k/2n] in the Cantorised interval

picture. Assign the nest subalgebra

A = On ∩ AlgN = {a ∈ On : (1− p)ap = 0, for all p ∈ N}.

35

perhaps skip:

Some (algebraic) Toeplitz contexts.

B: The (spatial) free semigroup context :

The Fock space `2(F+n ) for the free semigroup on n generators.

The freely noncommuting shifts L1, . . . , Ln with

dim(I − (L1L∗1 + . . . LnL

∗n)) = 1.

The noncommutative disc algebra An

and the free semigroup algebra Ln.

The Cuntz-Toeplitz C*-algebra on `2(F+n ).

C: The (spatial) graph context :

The Fock space of a directed graph G = (V, E).

The freely noncommuting partial isometries Le, e ∈ E.

The tensor algebra AG and the free semigroupoid algebras LG.

The Cuntz-Krieger-Toeplitz C*-algebras TG = C∗(AG).

D: The (universal) free semigroup context :

The freely noncommuting isometries S1, . . . , Sn

with S1S∗1 + · · · + SnS

∗n = I.

The Cuntz algebras On = C∗(S1, . . . , Sn).

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E: The (universal) graph context :

The (universal) graph C*-algebra C*(G) of a countable directed

graph G = (V, E) with partial isometry generators Se, for e ∈ E,

and relations

Σr(e)=xSeS∗e = Px, S∗eSe = Ps(e),

where {Px : x ∈ V } is a family of orthogonal projections and e =

(r(e), s(e)).

F: Higher rank algebras

TO DO