The boundary quotient for algebraic dynamical systems
The boundary quotient for algebraic dynamical systems
(based on joint work with Nathan Brownlowe and Nadia S. Larsen)
Workshop on C*-algebras: Structure and Classification
Nicolai Stammeier
WWU Münster
April 23, 2015
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The boundary quotient for algebraic dynamical systems
Algebraic dynamical systems
All semigroups are assumed countable, discrete, left cancellative and unital.
Definition
A semigroup is called right LCM if the intersection of two principal right
ideals is either empty or another principal right ideal.
Examples
There are numerous examples of right LCM semigroups, for instance,
(a) groups,
n
(b) free (abelian) monoids (e.g. F+
n and N ),
(c) braid monoids and Garside monoids,
(d) X ∗ ./ G for self-similar actions (G, X),
(e) N o N× , or R o R× for every principal ideal domain R, and
(f) semigroups with the property that incomparable elements have
disjoint principal right ideals.
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The boundary quotient for algebraic dynamical systems
Algebraic dynamical systems
Let P be a right LCM semigroup that acts on a group G by group
endomorphisms θ. Then pP ∩ qP = rP implies θp (G) ∩ θq (G) ⊃ θr (G).
Definition (Brownlowe-Larsen-S 2015)
An algebraic dynamical system (G, P, θ) consists of an action θ of a
right LCM semigroup P on a group G by injective endomorphisms such
that
θp (G) ∩ θq (G) = θr (G) holds whenever pP ∩ qP = rP.
Remark
a) If (G, P, θ) is an algebraic dynamical system, then G oθ P given by
(g, p)(h, q) = (gθp (h), pq) is a right LCM semigroup.
b) Every right LCM semigroup arises as G oθ P for some (G, P, θ).
Examples
Integer matrices acting on Zn (or Fn ), shift systems,...
build your own using your favourite group and a P from (a) - (f) .
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The boundary quotient for algebraic dynamical systems
Algebraic dynamical systems
Definition (Brownlowe-Larsen-S 2015)
For an algebraic dynamical system (G, P, θ), the C ∗ -algebra A[G, P, θ] is
the universal C ∗ -algebra generated by a unitary representation u of the
group G and a representation s of the semigroup P by isometries subject
to the relations
(A1)
sp ug = uθp (g) sp for all p ∈ P, g ∈ G, and

∗ ∗
−1

uk sp0 sq0 u` if g = θp (k)θq (` ) for some k, ` ∈ G
(A2) s∗p ug sq =
and pP ∩ qP = rP, pp0 = qq 0 = r,


0
otherwise.
Note that (A2) produces a Wick ordering on A[G, P, θ].
Proposition
Let Ug , Sp ∈ L(`2 (G oθ P )) be given by Ug ξ(h,q) := ξ(gh,q) and
Sp ξ(h,q) := ξ(θp (h),pq) . Then the map ug 7→ Ug , sp 7→ Sp defines a
representation A[G, P, θ] → L(`2 (G oθ P )).
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The boundary quotient for algebraic dynamical systems
Algebraic dynamical systems
The proposition suggests that A[G, P, θ] is related to semigroup
C ∗ -algebras of the right LCM semigroup G oθ P because we have
C ∗ ((Ug )g∈G , (Sp )p∈P ) = Cr∗ (G oθ P ) ⊂ L(`2 (G oθ P )). Indeed, using the
notion of a full semigroup C ∗ -algebra developed by Li, we get:
Theorem (Brownlowe-Larsen-S 2015)
For every algebraic dynamical system (G, P, θ), the map ug sp 7→ v(g,p)
defines an isomorphism between A[G, P, θ] and C ∗ (G oθ P ).
Viewing A[G, P, θ] as C ∗ (G oθ P ) indicates that this C ∗ -algebra tends to
be non-simple. This raises the question of the existence of a natural
quotient that is simple under mild assumptions.
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
Inspired by the work of Crisp and Laca, such a quotient was proposed for
right LCM semigroups in 2014:
Definition (Brownlowe-Ramagge-Robertson-Whittaker 2014)
Let S be right LCM. A finite subset F ⊂ S is called a foundation set if
for all s ∈ S there exists f ∈ F such that sS ∩ f S 6= ∅.
Definition (Brownlowe-Ramagge-Robertson-Whittaker 2014)
For a right LCM semigroup S, the boundary quotient Q(S) is the
quotient of C ∗ (S) by the relation
Q
(1 − ef S ) = 0 for every foundation set F.
(Q)
f ∈F
We say that F1 ⊂ S refines F2 ⊂ S if every f1 ∈ F1 belongs to f2 S for
some f2 ∈ F2 .
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
For G oθ P we can do better in many cases:
Definition
A finite subset F of G oθ P is called an elementary foundation set if
(1)
(n1 )
F = {(g1 , p1 ), . . . , (g1
satisfies:
(1)
(nm )
, p1 ), (g2 , p2 ), . . . , (gm
, pm )}
(i) G/θpi (G) is finite for all i.
(ii) {p1 , . . . , pm } is a foundation set for P with pi P ∩ pj P = ∅ for i 6= j.
F
(k)
(iii) G = 1≤k≤ni gi θpi (G) for each 1 ≤ i ≤ m.
Let Fe (G, P, θ) denote the collection of all elementary foundation sets.
Proposition
If P is directed or incomparable elements in P have disjoint principal right
ideals, then every foundation set for G oθ P can be refined by an
elementary foundation set.
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
Theorem
Suppose that every foundation set for G oθ P has an elementary
refinement. Then Q(G oθ P ) is the universal C ∗ -algebra generated by a
unitary representation ū of the group G and a representation s̄ of the
semigroup P by isometries subject to the relations:
s̄p ūg = ūθp (g) s̄p
for all p ∈ P, g ∈ G.

∗ ∗
0
0

ūk s̄p0 s̄q0 ū` if pP ∩ qP = rP, pp = qq = r and
(A2) s̄∗p ūg s̄q =
g = θp (k)θq (`−1 ) for some k, ` ∈ G,


0
otherwise.
P
(O)
1=
ēg,p for every F ∈ Fe (G, P, θ),
(A1)
(g,p)∈F
where ēg,p = ūg s̄p s̄∗p ū∗g .
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
An important ingredient in the proof of this theorem is
Lemma (B.H.Neumann 1954)
If G is a group,
Sn g1 , . . . , gn ∈ G and G1 , . . . , Gn are subgroups of G such
that G = i=1 gi Gi , then there is 1 ≤ i ≤ n such that G/Gi is finite and
the union of the cosets with finite G/Gi suffices to cover G.
Since we are dealing with an action on the group G, it is natural to
compare Q(G oθ P ) with the C ∗ -subalgebra of L(`2 (G)) generated by
Ũg ξh = ξgh and S̃p ξh = ξθp (h) `2 (G). However, we find that:
Proposition
For an algebraic dynamical system (G, P, θ), the assignment ūg s̄p 7→ Ũg S̃p
defines a representation of Q(G oθ P ) on `2 (G) if and only if P is directed.
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
Another guiding aspect of O[G, P, θ]: If G is amenable and
T
algebra. If G is
p∈P θp (G) = {1}, then O[G, P, θ] is a UCT Kirchberg
T
abelian and G/θp (G) is finite for all p ∈ P , then p∈P θp (G) = {1} is
known to be necessary.This leads to two questions:
(1) Can we discuss simplicity, pure infiniteness, and, possibly, being a
UCT Kirchberg algebra for the boundary quotient Q(G oθ P ) of
general algebraic dynamical systems (G, P, θ)?
(2) Can we improve the results obtained for O[G, P, θ] concerning the
necessity of the conditions on (G, P, θ)?
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
In principal, the answer to both questions is “Yes, we can!”, but there are
some limitations: Letting P ∗ denote the group of units in P , we demand
p ∈ P ∗ whenever θp ∈ Aut(G).
Theorem
Let (G, P, θ) be an algebraic dynamical system such that P is right
cancellative and P ∗ p ⊂ pP ∗ for all p ∈ P . Then Q(G oθ P ) is simple if
and only if
∼ C ∗ (Gtight (I(G oθ P ))),
(i) Q(G oθ P ) =
r
T
(ii)
kθr (G)k −1 = {1}, and
(k,r)∈Goθ P
(iii)
T
kθr (G)θp̃ (k)−1 is empty for all p̃ ∈ P ∗ arising from
(k,r)∈Goθ P
pq = q p̃ for some p ∈ P ∗ \ {1} and q ∈ P .
(i) relates to amenability since Q(S) is isomorphic to C ∗ (Gtight (I(S))) for
every right LCM semigroup S by the work of Starling, where Gtight (I(S))
denotes the tight groupoid for the inverse semigroup of S.
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The boundary quotient for algebraic dynamical systems
The boundary quotient - structure and classifiability
Starling: If Q(S) is simple (+ε) then Q(S) is either purely infinite or C.
With classical results due to Tu and Anantharaman-Delaroche, we get:
Theorem
Q(G oθ P ) is a UCT Kirchberg algebra provided that (G, P, θ) is an
algebraic dynamical system such that
(i) Gtight (I(G oθ P )) is amenable,
(ii) P is right cancellative and not a group,
(iii) P ∗ p ⊂ pP ∗ for all p ∈ P ,
T
(iv)
kθr (G)k −1 = {1}, and
(k,r)∈Goθ P
(v)
T
kθr (G)θp̃ (k)−1 is empty for all p̃ ∈ P ∗ arising from
(k,r)∈Goθ P
pq = q p̃ for some p ∈ P ∗ \ {1} and q ∈ P .
Q: What is K∗ (Q(G oθ P ))?
Troublesome task even for a set of relatively prime integers acting on Z
by multiplication (ongoing work with Selçuk Barlak and Tron Omland).
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