Toeplitz algebras of semigroups
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Published:2022-07-29
Issue:10
Volume:375
Page:7443-7507
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ISSN:0002-9947
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Container-title:Transactions of the American Mathematical Society
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language:en
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Short-container-title:Trans. Amer. Math. Soc.
Author:
Laca Marcelo,Sehnem Camila
Abstract
To each submonoid
P
P
of a group we associate a universal Toeplitz
C
∗
\mathrm {C}^*
-algebra
T
u
(
P
)
\mathcal {T}_u(P)
defined via generators and relations;
T
u
(
P
)
\mathcal {T}_u(P)
is a quotient of Li’s semigroup
C
∗
\mathrm {C}^*
-algebra
C
s
∗
(
P
)
\mathrm {C}^*_s(P)
and they are isomorphic iff
P
P
satisfies independence. We give a partial crossed product realization of
T
u
(
P
)
\mathcal {T}_u(P)
and show that several results known for
C
s
∗
(
P
)
\mathrm {C}^*_s(P)
when
P
P
satisfies independence are also valid for
T
u
(
P
)
\mathcal {T}_u(P)
when independence fails. At the level of the reduced semigroup
C
∗
\mathrm {C}^*
-algebra
T
λ
(
P
)
\mathcal {T}_\lambda (P)
, we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of
P
P
, generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of
T
λ
(
P
)
\mathcal {T}_\lambda (P)
is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over
P
P
with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on
P
P
for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the
a
x
+
b
ax+b
-monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails.
Funder
Natural Sciences and Engineering Research Council of Canada
Publisher
American Mathematical Society (AMS)
Subject
Applied Mathematics,General Mathematics
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