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Register a new userCondition (K) for inverse semigroups and the ideal structure of their $C^*$-algebras
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Inspired by results for graph $C^*$-algebras, we investigate connections between the ideal structure of an inverse semigroup $S$ and that of its tight $C^*$-algebra by relating ideals in $S$ to certain open invariant sets in the associated tight groupoid. We also develop analogues of Conditions (L) and (K) for inverse semigroups, which are related to certain congruences on $S$. We finish with applications to the inverse semigroups of self-similar graph actions and some relevant comments on the authors earlier uniqueness theorems for inverse semigroups.
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We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: one related to universal structures, and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of $P$ that reflect the space $mathcal{J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation $mathrm{C}^*_lambda(P)$ and we consider the boundary quotient $partial mathrm{C}^*_lambda(P)$ related to the minimal boundary space. We show that $partial mathrm{C}^*_lambda(P)$ is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of $P$; and (b) with respect to the equivariant C*-covers of the reduced nonselfadjoint semigroup algebra $mathcal{A}(P)$. If $P$ is an Ore semigroup, or if $G$ acts topologically freely on the minimal boundary space, then $partial mathrm{C}^*_lambda(P)$ coincides with the usual C*-envelope $mathrm{C}^*_{text{env}}(mathcal{A}(P))$ in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of nonzero algebraic integers, and the $ax+b$-semigroups over integral domains that are not a field. In particular, we show that $P$ is an Ore semigroup if and only if there exists a canonical $*$-isomorphism from $partial mathrm{C}^*_lambda(P)$, or from $mathrm{C}^*_{text{env}}(mathcal{A}(P))$, onto $mathrm{C}^*_lambda(G)$. If any of the above holds, then $mathcal{A}(P)$ is shown to be hyperrigid.
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also F{o}lners type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no F{o}lner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a natural generalization of the uniform Roe algebra of a group, which we denote by $mathcal{R}_X$. We show that the following notions are then equivalent: (1) $X$ is domain measurable; (2) $X$ is not paradoxical; (3) $X$ satisfies the domain F{o}lner condition; (4) there is an algebraically amenable dense *-subalgebra of $mathcal{R}_X$; (5) $mathcal{R}_X$ has an amenable trace; (6) $mathcal{R}_X$ is not properly infinite and (7) $[0] ot=[1]$ in the $K_0$-group of $mathcal{R}_X$. We also show that any tracial state on $mathcal{R}_X$ is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of $X$. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of $C_r^*left(Xright)$ implies the amenability of $X$. The converse implication is false.
We characterise, in several complementary ways, etale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property, assuming that the groupoid is either Hausdorff or $sigma$-compact. This leads directly to a characterisation of the simplicity of this C*-algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C*-algebra. Specifically, we prove that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that is equivalent to the uniqueness of equivariant pseudo-expectations and in the minimal case to an appropriate generalisation of Powers averaging property. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras that includes the essential groupoid C*-algebra itself. By considering a relative version of Powers averaging property, we obtain new examples of C*-irreducible inclusions in the sense of R{o}rdam. These arise from the inclusion of the C*-algebra generated by a suitable group representation into a simple groupoid C*-algebra. This is illustrated by the example of the C*-algebra generated by the quasi-regular representation of Thompsons group T with respect to Thompsons group F, which is contained C*-irreducibly in the Cuntz algebra $mathcal{O}_2$.
This paper introduces the notion of Rota-Baxter $C^{ast}$-algebras. Here a Rota-Baxter $C^{ast}$-algebra is a $C^{ast}$-algebra with a Rota-Baxter operator. Symmetric Rota-Baxter operators, as special cases of Rota-Baxter operators on $C^{ast}$-algebra, are defined and studied. A theorem of Rota-Baxter operators on concrete $C^{ast}$-algebras is given, deriving the relationship between two kinds of Rota-Baxter algebras. As a corollary, some connection between $ast$-representations and Rota-Baxter operators is given. The notion of representations of Rota-Baxter $C^{ast}$-algebras are constructed, and a theorem of representations of direct sums of Rota-Baxter representations is derived. Finally using Rota-Baxter operators, the notion of quasidiagonal operators on $C^{ast}$-algebra is reconstructed.
We initiate the study of real $C^*$-algebras associated to higher-rank graphs $Lambda$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $mathcal{CR}$ $K$-theory of $C^*_{mathbb R} (Lambda, gamma)$ for any involution $gamma$ on $Lambda$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $Lambda$ and the involution $gamma$. We provide a complete description of $K^{CR}(C^*_{mathbb R}(Lambda, gamma))$ for several examples of higher-rank graphs $Lambda$ with involution.
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