![Page 1: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/1.jpg)
Leavitt path algebras: An Introduction
UW Algebra Seminar Pre-talk
May 20, 2014
Leavitt path algebras
![Page 2: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/2.jpg)
General path algebras
Let E be a directed graph. E = (E 0,E 1, r , s)
•s(e) e // •r(e)
The path algebra KE is the K -algebra with basis {pi} consisting ofthe directed paths in E . (View vertices as paths of length 0.)
In KE , p · q = pq if r(p) = s(q), 0 otherwise.
In particular, s(e) · e = e = e · r(e).
Note: E 0 is finite ⇔ KE is unital; in this case 1KE =∑
v∈E0 v .
Leavitt path algebras
![Page 3: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/3.jpg)
Building Leavitt path algebras
Start with E , build its double graph E .
Example:
E = •t •uh
~~||||||||
•v
e
>>||||||||
f//
g
==•wiQQ j
// •x
E = •te
��
•uh
h∗~~||||||||
•v
e∗>>||||||||
f//
g
==•w
LL
f ∗tt
g∗
WWiQQuu
i∗ j// •x
j∗
WW
Leavitt path algebras
![Page 4: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/4.jpg)
Building Leavitt path algebras
Start with E , build its double graph E . Example:
E = •t •uh
~~||||||||
•v
e
>>||||||||
f//
g
==•wiQQ j
// •x
E = •te
��
•uh
h∗~~||||||||
•v
e∗>>||||||||
f//
g
==•w
LL
f ∗tt
g∗
WWiQQuu
i∗ j// •x
j∗
WW
Leavitt path algebras
![Page 5: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/5.jpg)
Building Leavitt path algebras
Start with E , build its double graph E . Example:
E = •t •uh
~~||||||||
•v
e
>>||||||||
f//
g
==•wiQQ j
// •x
E = •te
��
•uh
h∗~~||||||||
•v
e∗>>||||||||
f//
g
==•w
LL
f ∗tt
g∗
WWiQQuu
i∗ j// •x
j∗
WW
Leavitt path algebras
![Page 6: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/6.jpg)
Building Leavitt path algebras
Construct the path algebra K E .
Consider these relations in K E :
(CK1) e∗e ′ = δe,e′r(e) for all e, e ′ ∈ E 1.
(CK2) v =∑{e∈E1|s(e)=v} ee∗ for all v ∈ E 0
(just at those vertices v which are not sinks, and which emit only
finitely many edges)
Definition
The Leavitt path algebra of E with coefficients in K
LK (E ) = K E / < (CK 1), (CK 2) >
Leavitt path algebras
![Page 7: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/7.jpg)
Building Leavitt path algebras
Construct the path algebra K E . Consider these relations in K E :
(CK1) e∗e ′ = δe,e′r(e) for all e, e ′ ∈ E 1.
(CK2) v =∑{e∈E1|s(e)=v} ee∗ for all v ∈ E 0
(just at those vertices v which are not sinks, and which emit only
finitely many edges)
Definition
The Leavitt path algebra of E with coefficients in K
LK (E ) = K E / < (CK 1), (CK 2) >
Leavitt path algebras
![Page 8: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/8.jpg)
Building Leavitt path algebras
Construct the path algebra K E . Consider these relations in K E :
(CK1) e∗e ′ = δe,e′r(e) for all e, e ′ ∈ E 1.
(CK2) v =∑{e∈E1|s(e)=v} ee∗ for all v ∈ E 0
(just at those vertices v which are not sinks, and which emit only
finitely many edges)
Definition
The Leavitt path algebra of E with coefficients in K
LK (E ) = K E / < (CK 1), (CK 2) >
Leavitt path algebras
![Page 9: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/9.jpg)
Building Leavitt path algebras
Construct the path algebra K E . Consider these relations in K E :
(CK1) e∗e ′ = δe,e′r(e) for all e, e ′ ∈ E 1.
(CK2) v =∑{e∈E1|s(e)=v} ee∗ for all v ∈ E 0
(just at those vertices v which are not sinks, and which emit only
finitely many edges)
Definition
The Leavitt path algebra of E with coefficients in K
LK (E ) = K E / < (CK 1), (CK 2) >
Leavitt path algebras
![Page 10: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/10.jpg)
Building Leavitt path algebras
Construct the path algebra K E . Consider these relations in K E :
(CK1) e∗e ′ = δe,e′r(e) for all e, e ′ ∈ E 1.
(CK2) v =∑{e∈E1|s(e)=v} ee∗ for all v ∈ E 0
(just at those vertices v which are not sinks, and which emit only
finitely many edges)
Definition
The Leavitt path algebra of E with coefficients in K
LK (E ) = K E / < (CK 1), (CK 2) >
Leavitt path algebras
![Page 11: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/11.jpg)
Leavitt path algebras: Examples
Some sample computations in LC(E ) from the Example:
E = •te
��
•uh
h∗~~||||||||
•v
e∗>>||||||||
f//
g
==•w
LL
f ∗tt
g∗
WWiQQuu
i∗ j// •x
j∗
WW
ee∗ + ff ∗ + gg∗ = v g∗g = w g∗f = 0
h∗h = w hh∗ = u ff ∗ = ... (no simplification)
But (ff ∗)2 = f (f ∗f )f ∗ = f · r(f ) · f ∗ = ff ∗.
Leavitt path algebras
![Page 12: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/12.jpg)
Leavitt path algebras: Examples
Some sample computations in LC(E ) from the Example:
E = •te
��
•uh
h∗~~||||||||
•v
e∗>>||||||||
f//
g
==•w
LL
f ∗tt
g∗
WWiQQuu
i∗ j// •x
j∗
WW
ee∗ + ff ∗ + gg∗ = v g∗g = w g∗f = 0
h∗h = w hh∗ = u ff ∗ = ... (no simplification)
But (ff ∗)2 = f (f ∗f )f ∗ = f · r(f ) · f ∗ = ff ∗.
Leavitt path algebras
![Page 13: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/13.jpg)
Leavitt path algebras: Examples
Standard algebras arising as Leavitt path algebras:
E = •v1 e1 // •v2 e2 // •v3 •vn−1en−1 // •vn
Then LK (E ) ∼= Mn(K ).
E = •v xff
Then LK (E ) ∼= K [x , x−1].
Leavitt path algebras
![Page 14: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/14.jpg)
Leavitt path algebras: Examples
Standard algebras arising as Leavitt path algebras:
E = •v1 e1 // •v2 e2 // •v3 •vn−1en−1 // •vn
Then LK (E ) ∼= Mn(K ).
E = •v xff
Then LK (E ) ∼= K [x , x−1].
Leavitt path algebras
![Page 15: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/15.jpg)
Leavitt path algebras: Examples
Standard algebras arising as Leavitt path algebras:
E = •v1 e1 // •v2 e2 // •v3 •vn−1en−1 // •vn
Then LK (E ) ∼= Mn(K ).
E = •v xff
Then LK (E ) ∼= K [x , x−1].
Leavitt path algebras
![Page 16: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/16.jpg)
Leavitt path algebras: Examples
E = Rn = •v y1ff
y2
ss
y3
��
yn
Then LK (E ) ∼= LK (1, n), the classical “Leavitt algebra of order n”.
LK (1, n) is generated by y1, ..., yn, x1, ..., xn, with relations
xiyj = δi ,j1K andn∑
i=1
yixi = 1K .
Note: A = LK (1, n) has AA ∼= AAn as left A-modules:
a 7→ (ay1, ay2, ..., ayn) and (a1, a2, ..., an) 7→n∑
i=1
aixi .
Leavitt path algebras
![Page 17: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/17.jpg)
Leavitt path algebras: Examples
E = Rn = •v y1ff
y2
ss
y3
��
yn
Then LK (E ) ∼= LK (1, n), the classical “Leavitt algebra of order n”.LK (1, n) is generated by y1, ..., yn, x1, ..., xn, with relations
xiyj = δi ,j1K andn∑
i=1
yixi = 1K .
Note: A = LK (1, n) has AA ∼= AAn as left A-modules:
a 7→ (ay1, ay2, ..., ayn) and (a1, a2, ..., an) 7→n∑
i=1
aixi .
Leavitt path algebras
![Page 18: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/18.jpg)
Leavitt path algebras: Examples
E = Rn = •v y1ff
y2
ss
y3
��
yn
Then LK (E ) ∼= LK (1, n), the classical “Leavitt algebra of order n”.LK (1, n) is generated by y1, ..., yn, x1, ..., xn, with relations
xiyj = δi ,j1K andn∑
i=1
yixi = 1K .
Note: A = LK (1, n) has AA ∼= AAn as left A-modules:
a 7→ (ay1, ay2, ..., ayn) and (a1, a2, ..., an) 7→n∑
i=1
aixi .
Leavitt path algebras
![Page 19: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/19.jpg)
Things we know about Leavitt path algebras
The main goal in the early years of the development: Establishresults of the form
LK (E ) has algebraic property P ⇔E has graph-theoretic property Q.
(Only recently has the structure of K played a role.)
Leavitt path algebras
![Page 20: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/20.jpg)
Things we know about Leavitt path algebras
We know precisely the graphs E for which LK (E ) has theseproperties: (No role played by the structure of K in any of these.)
1 simplicity
2 purely infinite simplicity
3 (one-sided) artinian; (one-sided) noetherian
4 (two-sided) artinian; (two-sided) noetherian
5 exchange (∗∗)6 prime
7 primitive
Leavitt path algebras
![Page 21: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/21.jpg)
The monoid V(R), and the Grothendieck group K0(R)
Isomorphism classes of finitely generated projective (left)R-modules, with operation ⊕, denoted V(R).
(Conical) monoid, with ‘distinguished’ element [R].
Theorem
(George Bergman, Trans. A.M.S. 1975) Given a field K andfinitely generated conical monoid S with a distinguished element I ,there exists a universal K -algebra R for which V(R) ∼= S, and forwhich, under this isomorhpism, [R] 7→ I .
The construction is explicit, uses amalgamated products.
Bergman included the algebras LK (1, n) as examples of theseuniversal algebras.
Leavitt path algebras
![Page 22: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/22.jpg)
The monoid V(R), and the Grothendieck group K0(R)
Isomorphism classes of finitely generated projective (left)R-modules, with operation ⊕, denoted V(R).
(Conical) monoid, with ‘distinguished’ element [R].
Theorem
(George Bergman, Trans. A.M.S. 1975) Given a field K andfinitely generated conical monoid S with a distinguished element I ,there exists a universal K -algebra R for which V(R) ∼= S, and forwhich, under this isomorhpism, [R] 7→ I .
The construction is explicit, uses amalgamated products.
Bergman included the algebras LK (1, n) as examples of theseuniversal algebras.
Leavitt path algebras
![Page 23: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/23.jpg)
The monoid V(LK (E ))
For any graph E construct the free abelian monoid ME .
generators {av | v ∈ E 0}; relations av =∑
r(e)=w
aw
Using Bergman’s construction,
Theorem
(Ara, Moreno, Pardo, Alg. Rep. Thy. 2007)
For any field K ,V(LK (E )) ∼= ME .
Under this isomorphism, [LK (E )] 7→∑
v∈E0 av .
Leavitt path algebras
![Page 24: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/24.jpg)
The monoid V(LK (E ))
For any graph E construct the free abelian monoid ME .
generators {av | v ∈ E 0}; relations av =∑
r(e)=w
aw
Using Bergman’s construction,
Theorem
(Ara, Moreno, Pardo, Alg. Rep. Thy. 2007)
For any field K ,V(LK (E )) ∼= ME .
Under this isomorphism, [LK (E )] 7→∑
v∈E0 av .
Leavitt path algebras
![Page 25: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/25.jpg)
The monoid V(LK (E ))
Example.1
�� $$3
EE
22 2rr
dd
Not hard to show: ME = {z ,A1,A2,A3,A1 + A2 + A3}
Note: ME \ {z} ∼= Z2 × Z2.
So V(LK (E )) \ {0} ∼= Z2 × Z2.
Here LK (E ) 7→ A1 + A2 + A3 = (0, 0) ∈ Z2 × Z2.
Leavitt path algebras
![Page 26: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/26.jpg)
The monoid V(LK (E ))
Example.1
�� $$3
EE
22 2rr
dd
Not hard to show: ME = {z ,A1,A2,A3,A1 + A2 + A3}
Note: ME \ {z} ∼= Z2 × Z2.
So V(LK (E )) \ {0} ∼= Z2 × Z2.
Here LK (E ) 7→ A1 + A2 + A3 = (0, 0) ∈ Z2 × Z2.
Leavitt path algebras
![Page 27: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/27.jpg)
The monoid V(LK (E ))
Example.1
�� $$3
EE
22 2rr
dd
Not hard to show: ME = {z ,A1,A2,A3,A1 + A2 + A3}
Note: ME \ {z} ∼= Z2 × Z2.
So V(LK (E )) \ {0} ∼= Z2 × Z2.
Here LK (E ) 7→ A1 + A2 + A3 = (0, 0) ∈ Z2 × Z2.
Leavitt path algebras
![Page 28: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/28.jpg)
The monoid V(LK (E ))
Example: For each n ∈ N let Cn denote the “directed cycle”graph with n vertices.
Then it’s easy to show that MCn = Z+, and so MCn \ {0} = N.
So in particular V(LK (Cn)) \ {0} is not a group.
Leavitt path algebras
![Page 29: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/29.jpg)
C∗-algebras
Theorem
(Cuntz, Comm. Math. Physics, 1977) There exist simpleC∗-algebras generated by partial isometries.
Denote by On.
Subsequently, a similar construction was produced of the “graphC∗-algebra” C ∗(E ), for any graph E . In this context,On∼= C ∗(Rn).
For any graph E ,
LC(E ) ⊆ C ∗(E )
as a dense ∗-subalgebra. In particular, LC(1, n) ⊆ On.
(But C ∗(E ) is usually “much bigger” than LC(E ).)
Leavitt path algebras
![Page 30: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/30.jpg)
C∗-algebras
Theorem
(Cuntz, Comm. Math. Physics, 1977) There exist simpleC∗-algebras generated by partial isometries.
Denote by On.
Subsequently, a similar construction was produced of the “graphC∗-algebra” C ∗(E ), for any graph E . In this context,On∼= C ∗(Rn). For any graph E ,
LC(E ) ⊆ C ∗(E )
as a dense ∗-subalgebra. In particular, LC(1, n) ⊆ On.
(But C ∗(E ) is usually “much bigger” than LC(E ).)
Leavitt path algebras
![Page 31: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/31.jpg)
C∗-algebras
Properties of C∗-algebras. These typically include topologicalconsiderations.
1 simple
2 purely infinite simple
3 stable rank, prime, primitive, exchange, etc....
Leavitt path algebras
![Page 32: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/32.jpg)
C∗-algebras
Properties of C∗-algebras. These typically include topologicalconsiderations.
1 simple
2 purely infinite simple
3 stable rank, prime, primitive, exchange, etc....
Leavitt path algebras
![Page 33: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/33.jpg)
C∗-algebras
For a vast number of (but not all) properties ..
LC(E ) has (algebraic) property P ⇐⇒C ∗(E ) has (topological) property P.
... if and only if LK (E ) has (algebraic) property P for every field K .
... if and only if E has some graph-theoretic property.
Still no good understanding as to Why.
Note: O2 ⊗O2∼= O2.
But (2011, three different proofs) LK (1, 2)⊗ LK (1, 2) 6∼= LK (1, 2).
Leavitt path algebras
![Page 34: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/34.jpg)
C∗-algebras
For a vast number of (but not all) properties ..
LC(E ) has (algebraic) property P ⇐⇒C ∗(E ) has (topological) property P.
... if and only if LK (E ) has (algebraic) property P for every field K .
... if and only if E has some graph-theoretic property.
Still no good understanding as to Why.
Note: O2 ⊗O2∼= O2.
But (2011, three different proofs) LK (1, 2)⊗ LK (1, 2) 6∼= LK (1, 2).
Leavitt path algebras
![Page 35: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/35.jpg)
C∗-algebras
For a vast number of (but not all) properties ..
LC(E ) has (algebraic) property P ⇐⇒C ∗(E ) has (topological) property P.
... if and only if LK (E ) has (algebraic) property P for every field K .
... if and only if E has some graph-theoretic property.
Still no good understanding as to Why.
Note: O2 ⊗O2∼= O2.
But (2011, three different proofs) LK (1, 2)⊗ LK (1, 2) 6∼= LK (1, 2).
Leavitt path algebras
![Page 36: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/36.jpg)
C∗-algebras
For a vast number of (but not all) properties ..
LC(E ) has (algebraic) property P ⇐⇒C ∗(E ) has (topological) property P.
... if and only if LK (E ) has (algebraic) property P for every field K .
... if and only if E has some graph-theoretic property.
Still no good understanding as to Why.
Note: O2 ⊗O2∼= O2.
But (2011, three different proofs) LK (1, 2)⊗ LK (1, 2) 6∼= LK (1, 2).
Leavitt path algebras
![Page 37: Leavitt path algebras: An Introduction...Building Leavitt path algebras Construct the path algebra KEb. Consider these relations in KEb: (CK1) ee0= e;e0r(e) for all e;e02E1. (CK2)](https://reader036.vdocuments.site/reader036/viewer/2022070904/5f6ffcc43a7780026d285bb2/html5/thumbnails/37.jpg)
C∗-algebras
For a vast number of (but not all) properties ..
LC(E ) has (algebraic) property P ⇐⇒C ∗(E ) has (topological) property P.
... if and only if LK (E ) has (algebraic) property P for every field K .
... if and only if E has some graph-theoretic property.
Still no good understanding as to Why.
Note: O2 ⊗O2∼= O2.
But (2011, three different proofs) LK (1, 2)⊗ LK (1, 2) 6∼= LK (1, 2).
Leavitt path algebras