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3 The Left Inverse Hull, and Full Semigroup C*-Algebras
Semigroup C -Algebras
The reduced C*-algebra C .S/ of S is the sub-C*-algebra of L .`2 S / generated by
f s W s 2 Sg. The reader may find more information about inverse semigroups and
their C*-algebras in [19–21, 40, 42].
Now assume that P is a left cancellative semigroup. We construct the left inverse
hull of P as follows: Let S be the inverse semigroup of partial bijections of P
generated by p W P ! P; x 7! px (p 2 P). The semilattice of idempotents E
of S is given by the set of constructible ideals
J D fp1 1 q1 : : : pn 1 qn P W pi ; qi 2 Pg [ f;g:
As explained in [40, Corollary 3.2.13], the isometry `2 P ! `2 S ; ıp 7! ıp
induces a surjective homomorphism C .S/
To define the full semigroup C*-algebra of P, we set C .P/ WD C .S/. By
definition, C .S/ is the C*-algebra which is universal for representations of our
inverse semigroup S by partial isometries, with the extra requirement that 0 2 S
is representated by 0 in the target C*-algebra. Our definition of full semigroup
C*-algebras differs from the definitions in [25, 40], where two versions of full semigroup C*-algebras (denoted by C .P/ and Cs .P/) were introduced. In particular
cases, however, it is possible to identify our full semigroup C*-algebra with the one
from [25, 40] (compare [40, Proposition 3.3.1 and Proposition 3.3.2]). Furthermore,
we let W C .P/
C .P/ be the composite C .P/ D C .S/
3 The Toeplitz Condition
From now on, let us assume that P is a subsemigroup of a group G such that P
contains the identity e. Let D .P/ WD C .P/ \ `1 .P/, where we view `1 .P/ as
multiplication operators on `2 P. As P is a subsemigroup of G, it turns out that
we always have a partial crossed product description for C .P/, i.e., there exists a
canonically given partial action G Õ D .P/ such that C .P/ Š D .P/Ìr
is already helpful. For instance, if we let ˝P be the spectrum of D .P/, and if GË˝P
is the partial transformation groupoid attached to (the dual action corresponding to)
G Õ D .P/, then we get C .P/ Š Cr .G Ë ˝P /, a description of our semigroup
C*-algebra as a groupoid C*-algebra.
The Toeplitz condition will allow us to write C .P/ as a crossed product attached
to a global action, at least up to Morita equivalence.
Recall that S is the left inverse hull of P.
Definition 3.1 We say that P Â G is Toeplitz if for every g 2 G, the partial bijection
P \ .g 1 P/ ! .gP/ \ P; x 7! gx is in S.
Let us now give an operator algebraic reformulation of the Toeplitz condition,
which will immediately lead us to the desired crossed product description. Starting
with our embedding P Â G, we view `2 P as a subspace of `2 G. Let EP 2 L .`2 G/
be the orthogonal projection onto `2 P. Moreover, as `2 P is a subspace of `2 G, we
may view C .P/ as a sub-C*-algebra of L .`2 G/. Furthermore, let be the left
regular representation of G. Then we have Vp D EP p EP for every p 2 P. Clearly,
if we think of `1 .G/ as multiplication operators on `2 G, then EP corresponds to
1P 2 `1 .G/. In addition, we have a canonical action G Õ `1 .G/. Now let DPÂG
be the smallest G-invariant sub-C*-algebra of `1 .G/ containing EP . Because we
have Vp D EP p EP for every p 2 P, we obtain
C .P/ Â EP .DPÂG Ìr G/EP :
Moreover, EP is a full projection in DPÂG Ìr G. Therefore,
EP .DPÂG Ìr G/EP
DPÂG Ìr G:
If C .P/ Ã EP .DPÂG Ìr G/EP , then we get desired crossed product description.
It turns out that P Â G is Toeplitz if and only if for every g 2 G with EP g EP Ô
0, there exist p1 ; q1 ; : : : ; pn ; qn in P such that EP g EP D Vp1 Vq1 Vpn Vqn .
In particular, if P Â G is Toeplitz, then C .P/ D EP .DPÂG Ìr G/EP , and the
latter C*-algebra is Morita equivalent to DPÂG Ìr G.
For example, if P is left Ore, then P embeds into its group of left quotients P 1 P,
and P Â P 1 P is Toeplitz. A left Ore semigroup is defined as follows: A semigroup
is called right reversible if every pair of non-empty left ideals has a non-empty
intersection. A semigroup is said to satisfy the left Ore condition if it is cancellative
and right reversible. The following result can be found in [2, Theorem 1.23] or [23,
§1.1]: A semigroup P can be embedded into a group G such that G D P 1 P D
fq 1 p W p; q 2 Pg if and only if P satisfies the left Ore condition. It is easy to check
that P Â P 1 P is Toeplitz. For instance, this covers the case of cancellative, abelian
semigroups. It also covers the case of ax C b-semigroups over integral domains.
We mention two examples of subsemigroups of groups for which the Toeplitz
condition does not hold. Let N N be the free semigroup on two generators. We have
a canonical embedding of N N into F2 , the free group on two generators. It turns
out that if we compose that map with the canonical quotient map F2
F2 =F002 , then
we still get an embedding N N ,! F2 =F2 . This pair, however, does not satisfy the
Toeplitz condition. Roughly speaking, the argument is as follows: Let P D N N,
which we view as a subsemigroup in G D F2 =F002 . If P Â G were Toeplitz, then it
would follow that for every g 2 G, we must have that either gP\P is empty or gP\P
is of the form pP for some p 2 P. However, in our particular example, it is easy to
come up with a; b; c; d 2 P with ab 1 cd 1 2 F002 such that e … ba 1 P (in G) and
b; c 2 pP ) p D e. Then ba 1 P \ P Ô P and b; c 2 ba 1 P \ P, so that ba 1 P \ P
cannot be of the form pP for some p 2 P. Another example, which can be treated
using similar ideas as our first one, is given as follows: Consider the Thompson
group F, which is defined by the same generators and relations as the Thompson
monoid. This means that F is the universal group generated by ˙ D fx0 ; x1 ; : : :g
subject to the relations xn xk D xk xnC1 for k < n. Let a and b be the canonical free
Semigroup C -Algebras
generators of N N. It turns out that we obtain an embedding N N ,! F by a 7! x0 ,
b 7! x1 , and that N N ,! F does not satisfy the Toeplitz condition.
4 The Independence Condition
As above, we assume that P is a subsemigroup of a group G, and that P contains the
identity element of G.
The independence condition is a condition on the ideal structure of our semigroup
P. Recall that we have introduced the set of constructible ideals
J D fp1 1 q1 : : : pn 1 qn P W pi ; qi 2 Pg [ f;g:
Definition 4.1 J is called independent if for all X; X1 ; : : : ; Xn 2 J , X D
implies X D Xi for some 1 Ä i Ä n.
We say that P satisfies the independence condition if J is independent.
We would like to give equivalent characterizations of independence. To do so,
recall that S is the left inverse hull of P. Let E be the semilattice of idempotents of
C .P/ restricts to a canonical map C .E/
D .P/. It was shown
S. W C .P/
in [40, Theorem 3.2.14] that the following are equivalent:
• the canonical map C .S/
• the canonical map C .E/
• J is independent.
C .P/ is injective,
D .P/ is injective,
For example, if every constructible ideal is principal, i.e., J D fpP W p 2
Pg[f;g, then P satisfies the independence condition. This covers the case of positive
cones in quasi-lattice ordered groups (see [26, 39]). Moreover, ax C b-semigroups
over Dedekind domains, or more generally, Krull rings, satisfy the independence
condition (see [25, 29]).
Let us also present two (classes of) counterexamples, i.e., semigroups which
do not satisfy the independence condition. For instance, consider the numerical
semigroup Nnf1g. It turns out that it does not satisfy independence. More generally,
every numerical semigroup of the form N n F, where F is a non-empty, finite
subset of N, does not satisfy independence. Another class of examples is given as
follows: For certain integral domains which are not Krull rings, their multiplicative
or ax C b-semigroups do not satisfy independence. For example, R D
ZŒi 3 is such an example. For more details, we refer the reader to [29, §7].
5 Amenability and Nuclearity
Let P be a subsemigroup of a group G. The following result explains the connection
between amenability (of underlying dynamical systems) and nuclearity for semigroup C*-algebras.
Theorem 5.1 Consider the following statements:
(i) C .P/ is nuclear,
(ii) C .P/ is nuclear,
(iii) G Ë ˝P is amenable,
(iv) W C .P/
C .P/ is an isomorphism.
We always have (i) ) (ii) , (iii). If P satisfies the independence condition, then
(iii) ) (iv) and (iii) ) (i), so that (i), (ii) and (iii) are equivalent.
Here G Ë ˝P is the partial translation groupoid from Sect. 3. A proof of this theorem
will appear in , but the reader may also consult  for a special case of that
Moreover, it was observed in , and also follows immediately from the
partial crossed product description discussed in Sect. 3 together with  or [21,
Theorem 20.7 and Theorem 25.10] that if P is a subsemigroup of an amenable
group, then C .P/ is nuclear.
To give a particular class of examples, it was shown in , and also in , that
semigroup C*-algebras attached to right-angled Artin monoids are nuclear.
Let P be a subsemigroup of a group G. The following result gives a formula for the
K-theory of semigroup C*-algebras.
Theorem 6.1 Assume that P satisfies the independence condition, and that P Â G
is Toeplitz. If G satisfies the Baum-Connes conjecture with coefficients, then
K .C .P// Š
K .Cr .GX //:
Here JPÂG WD f niD1 gi P W g1 ; : : : ; gn 2 Gg [ f;g, and GX D fg 2 G W gX D Xg.
We refer the reader to [10, 11] for the proof of this theorem as well as further
details concerning K-theory for semigroup C*-algebras.
Let us apply our general K-theoretic formula to ax C b-semigroups over integral
domains. Let R be a countable Krull ring with group of multiplicative units R and
divisor class group C.R/. For every k 2 C.R/, let ak be a divisorial ideal which
Semigroup C -Algebras
represents k. Then
K .C .R Ì R // Š
K .C .ak Ì R //:
The reader may consult  for more details. In particular, if R is the ring of
algebraic integers in a number field K, we get
K .C .R Ì R // Š
K .C .ak Ì R //:
Here ClK is the class group of K, and R is the group of invertible elements in R
(i.e., the units in R or K).
If P is the positive cone in a totally ordered group G, then it is easy to see that P
satisfies the independence condition, and also that P Â G is Toeplitz. Moreover, in
that case, we have JPÂG D fgP W g 2 Gg [ f;g, and GP D P D feg. Therefore, if
G satisfies the Baum-Connes conjecture with coefficients, then we get in K-theory
K .C .P// Š K .C/:
The ideas in [10, 11] have been taken up in [34, 41] in order to obtain more
general K-theoretic computations which go beyond semigroup C*-algebras.
7 Classification Results
We start with a classification result for semigroup C*-algebras attached to rightangled Artin monoids. We need to introduce the following terminology: A graph
D .V; E/ is called co-reducible if there exist non-empty subsets V1 and V2 of
V with V D V1 t V2 such that V1 V2 Â E.
is called co-irreducible if
is not co-reducible. In general, we can always decompose
components, i.e., co-irreducible subgraphs i D .Vi ; Ei /. Let t. / be the number of
those co-irreducible components which are singletons. Moreover, let us define the
Euler characteristic of a graph 0 D .V 0 ; E0 /. To this end, we view 0 as a simplicial
complex by defining for every n D 0; 1; 2; : : : the set of n-simplices by
Kn WD ffv0 ; : : : ; vn g Â V 0 W .vi ; vj / 2 E0 for all i; j 2 f0; : : : ; ng; i Ô jg:
and k 2 Z, let Nk . /
Then we set . 0 / WD 1
nD0 . 1/ jKn j. Given a graph
be the number of co-irreducible components of
with Euler characteristic equal
Theorem 7.1 Let
be finite graphs. The following are equivalent:
1. C .A / Š C .A /
2. a. t. / D t. /
b. Nk . / C N k . / D Nk . / C N k . / for all k 2 Z
c. N0 . / > 0 or
Nk . / Á
Nk . / mod 2:
We remark that it is not necessary to restrict to finite graphs, and refer to  for
the general result and further details.
Here is another classification result, this time for semigroup C*-algebras attached
to ax C b-semigroups over rings of algebraic integers in number fields.
Theorem 7.2 Let K and L be number fields with rings of algebraic integers R and
S. Assume that K and L have the same number of roots of unity. If C .R Ì R / Š
C .S Ì S / then K and L are arithmetically equivalent, i.e., K D L .
Assume, in addition, that K and L are Galois extensions. In that case, we have
C .R Ì R / Š C .S Ì S / if and only if K Š L.
The reader may find the proof, and further information, in . [43, 44] contain
more details about arithmetic equivalence.
If we ask for isomorphism of semigroup C*-algebras preserving the canonical
sub-C*-algebras, then we get a stronger conclusion:
Theorem 7.3 Let K and L be number fields with rings of algebraic integers R and
S. If there exists an isomorphism C .R Ì R / Š C .S Ì S / which restricts to an
isomorphism D .R Ì R / Š D .S Ì S /, then K and L are arithmetically equivalent
and ClK Š ClL (as groups).
Here D denotes the sub-C*-algebra defined in Sect. 3. This result is proven in .
Our conclusion is really stronger than the one in the previous theorem, see .
Such sub-C*-algebras like D are Cartan subalgebras in the C*-algebraic sense
(see ). They are studied in a general context in .
Furthermore, we mention that there are more classification results for ax C bsemigroups over more general integral domains in .
For Baumslag-Solitar monoids, Spielberg obtained classification results in .
Acknowledgements Research supported by EPSRC grant EP/M009718/1.
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Topological Full Groups of Étale Groupoids
Abstract This is a survey of the recent development of the study of topological full
groups of étale groupoids on the Cantor set. Étale groupoids arise from dynamical
systems, e.g. actions of countable discrete groups, equivalence relations. Minimal Zactions, minimal ZN -actions and one-sided shifts of finite type are basic examples.
We are interested in algebraic, geometric and analytic properties of topological
full groups. More concretely, we discuss simplicity of commutator subgroups,
abelianization, finite generation, cohomological finiteness properties, amenability,
the Haagerup property, and so on. Homology groups of étale groupoids, groupoid
C -algebras and their K-groups are also investigated.
We discuss various properties of topological full groups of topological dynamical
systems on Cantor sets. The study of full groups in the setting of topological
dynamics was initiated by Giordano, Putnam and Skau . For a minimal action
' W Z Õ X on a Cantor set X, they defined several types of full groups and showed
that these groups completely determine the orbit equivalence class, the strong orbit
equivalence class and the flip conjugacy class of ', respectively.
The notion of topological full groups was later generalized to the setting of
essentially principal étale groupoids G on Cantor sets in . Étale groupoids
(called r-discrete groupoids in ) provide us a natural framework for unified
treatment of various topological dynamical systems. The topological full group
ŒŒG of G is a subgroup of Homeo.G .0/ / consisting of all homeomorphisms of
G .0/ whose graph is ‘contained’ in the groupoid G as a compact open subset (see
Definition 4.1). From an action ' of a discrete group on a Cantor set X, we can
construct the étale groupoid G' , which is called the transformation groupoid (see
Example 2.3). The topological full group ŒŒG' of G' is the group of ˛ 2 Homeo.X/
for which there exists a continuous map c W X ! such that ˛.x/ D ' c.x/ .x/ for all
H. Matui ( )
Graduate School of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan
© Springer International Publishing Switzerland 2016
T.M. Carlsen et al. (eds.), Operator Algebras and Applications, Abel Symposia 12,
x 2 X. Many other examples of étale groupoids and topological full groups will be
provided in later sections.
One of the most fundamental result for topological full groups is the isomorphism
theorem (Theorem 5.1), which says that G1 is isomorphic to G2 if and only if
ŒŒG1 is isomorphic to ŒŒG2 . In general, it is often difficult to distinguish two
discrete groups. But, the étale groupoids have rich information about the topological
dynamical systems, and so the isomorphism theorem helps us to determine the
isomorphism class of the topological full groups.
The homology groups Hn .G / for n
0 are defined for étale groupoids G (see
Definition 3.1). When G is a transformation groupoid G' , the homology Hn .G' /
agrees with the group homology (Example 3.3 (2)). In many examples, we can
check that the homology
L groups ‘coincide’ with the K-groups of Cr .G /. Thus, we
have isomorphisms n H2nCi .G / Š Ki .Cr .G // for i D 0; 1. This phenomenon is
formulated as the HK conjecture (Conjecture 3.5).
In many cases, it is known that the commutator subgroup D.ŒŒG / of ŒŒG
becomes simple (Theorem 6.5 (1), Theorem 7.3 (1)). So, it is natural to consider the
abelianization ŒŒG ab D ŒŒG =D.ŒŒG /. It turns out that the abelian group ŒŒG ab is
closely related to the homology groups of G . This relation is formulated as the AH
conjecture (Conjecture 4.7).
In addition to these two conjectures, we are interested in several properties of
ŒŒG . In , it was shown that D.ŒŒG' / is finitely generated if ' W Z Õ X
is a minimal subshift (see Theorem 8.3 (1)). In , it was shown that, for any
SFT groupoid GA (see Example 2.5), ŒŒGA is of type F1 and D.ŒŒGA / is finitely
generated (Theorem 8.8). Such finiteness conditions of topological full groups are
important problems. In , it was shown that, for any minimal action ' W Z Õ
X, ŒŒG' is amenable (see Theorem 8.4). In , it was shown that ŒŒGA has
the Haagerup property for any SFT groupoid GA (Theorem 8.9). Such analytic
properties of topological full groups are also our main concern.
2.1 Étale Groupoids
The cardinality of a set A is written #A and the characteristic function of A is written
1A . The finite cyclic group of order n is denoted by Zn D fNr j r D 1; 2; : : : ; ng.
We say that a subset of a topological space is clopen if it is both closed and
open. A topological space is said to be totally disconnected if its topology is
generated by clopen subsets. By a Cantor set, we mean a compact, metrizable,
totally disconnected space with no isolated points. It is known that any two such
spaces are homeomorphic. The homeomorphism group of a topological space X is
written Homeo.X/. The commutator subgroup of a group is denoted by D. /.
We let ab denote the abelianization =D. /.
Topological Full Groups of Étale Groupoids
In this article, by an étale groupoid we mean a second countable locally compact
Hausdorff groupoid such that the range map is a local homeomorphism. We refer the
reader to [35, 36] for background material on étale groupoids. Roughly speaking,
a groupoid G is a ‘group-like’ object, in which the product may not be defined
for all pairs in G . An étale groupoid G is equipped with locally compact Hausdorff
topology, which is compatible with the groupoid structure, and the map g 7! gg 1 is
a local homeomorphism. For an étale groupoid G , we let G .0/ denote the unit space
and let s and r denote the source and range maps, i.e. s.g/ D g 1 g and r.g/ D gg 1 .
An element g 2 G can be thought of as an arrow from s.g/ to r.g/. For x 2 G .0/ ,
G .x/ D r.G x/ is called the G -orbit of x. When every G -orbit is dense in G .0/ , G is
said to be minimal. For a subset Y G .0/ , the reduction of G to Y is r 1 .Y/\s 1 .Y/
and denoted by G jY. If Y is clopen, then the reduction G jY is an étale subgroupoid
of G in an obvious way. For x 2 G .0/ , we write Gx D r 1 .x/ \ s 1 .x/ and call it the
isotropy group of x. The isotropy bundle of G is G 0 D fg 2 G j r.g/ D s.g/g D
. When the interior of G 0 is G .0/ ,
x2G .0/ Gx . We say that G is principal if G D G
we say that G is essentially principal.
A subset U
G is called a G -set if rjU; sjU are injective. Any open G -set U
induces the homeomorphism .rjU/ ı .sjU/ 1 from s.U/ to r.U/. We write Â.U/ D
.rjU/ ı .sjU/ 1 . When U; V are G -sets,
D fg 2 G j g
UV D fgg0 2 G j g 2 U; g0 2 V; s.g/ D r.g0 /g
are also G -sets. A probability measure
on G .0/ is said to be G -invariant if
.r.U// D .s.U// holds for every open G -set U. The set of all G -invariant
probability measures is denoted by M.G /.
For an étale groupoid G , we denote the reduced groupoid C -algebra of G by
Cr .G / and identify C0 .G .0/ / with a subalgebra of Cr .G /. J. Renault obtained the
following theorem (see also [27, Theorem 5.1]).
Theorem 2.1 ([36, Theorem 5.9]) Two essentially principal étale groupoids G1
and G2 are isomorphic if and only if there exists an isomorphism ' W Cr .G1 / !
Cr .G2 / such that '.C0 .G1 // D C0 .G2 /.
In this subsection, we present several examples of étale groupoids. Throughout this
subsection, by an étale groupoid, we mean a second countable étale groupoid whose
unit space is the Cantor set.