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We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.
We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of states on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev’s main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs.
We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra
${\rm C}_{\rm r}^* (\Gamma)$
. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.
We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for
$n=m>0$
and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.
We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous
$\text{C}^{\ast }$
-algebras with exactly
$n$
Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.
We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalized gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.
We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding
$C^{\ast }$
-algebras on one-dimensional solenoids.
We introduce the concept of strong property
$(\mathbb{B})$
with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property
$(\mathbb{B})$
with a constant given by
$288\unicode[STIX]{x1D70B}(1+\sqrt{2})$
. We then use this result to find a concrete upper bound for the hyperreflexivity constant of
${\mathcal{Z}}^{n}(A,X)$
, the space of bounded
$n$
-cocycles from
$A$
into
$X$
, where
$A$
is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and
$X$
is a Banach
$A$
-bimodule for which
${\mathcal{H}}^{n+1}(A,X)$
is a Banach space. As another application, we show that for a locally compact amenable group
$G$
and
$1<p<\infty$
, the space
$CV_{P}(G)$
of convolution operators on
$L^{p}(G)$
is hyperreflexive with a constant given by
$384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$
. This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, Math. Scand.50(1) (1982), 111–122] for
$p=2$
.
We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.
In this article, we consider a twisted partial action
$\unicode[STIX]{x1D6FC}$
of a group
$G$
on an associative ring
$R$
and its associated partial crossed product
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
. We provide necessary and sufficient conditions for the commutativity of
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
when the twisted partial action
$\unicode[STIX]{x1D6FC}$
is unital. Moreover, we study necessary and sufficient conditions for the simplicity of
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
in the following cases: (i)
$G$
is abelian; (ii)
$R$
is maximal commutative in
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
; (iii)
$C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$
is simple; (iv)
$G$
is hypercentral. When
$R=C_{0}(X)$
is the algebra of continuous functions defined on a locally compact and Hausdorff space
$X$
, with complex values that vanish at infinity, and
$C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$
is the associated partial skew group ring of a partial action
$\unicode[STIX]{x1D6FC}$
of a topological group
$G$
on
$C_{0}(X)$
, we study the simplicity of
$C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$
by using topological properties of
$X$
and the results about the simplicity of
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
.
We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II
$_{1}$
factor is an enforceable II
$_{1}$
factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian
$\text{C}^{\ast }$
-algebras and use this to show that it is the prime model of its theory.
Let 𝔻n be the open unit polydisc in ℂn,
$n \ges 1$
, and let H2(𝔻n) be the Hardy space over 𝔻n. For
$n\ges 3$
, we show that if θ ∈ H∞(𝔻n) is an inner function, then the n-tuple of commuting operators
$(C_{z_1}, \ldots , C_{z_n})$
on the Beurling type quotient module
${\cal Q}_{\theta }$
is not essentially normal, where
Rudin's quotient modules of H2(𝔻2) are also shown to be not essentially normal. We prove several results concerning boundary representations of C*-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.
We characterize the class of RFD
$C^{\ast }$
-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the
$C^{\ast }$
-algebra is finite-dimensional, which is equivalent to the
$C^{\ast }$
-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of
$C^{\ast }$
-algebras whose norms in finite-dimensional representations fit certain prescribed properties.
with
$A_{n}=\bigoplus _{i=1}^{n_{i}}A_{[n,i]}$
, where all the
$A_{[n,i]}$
are Elliott–Thomsen algebras and
$\unicode[STIX]{x1D719}_{n,n+1}$
are homomorphisms. In this paper, we will prove that
$A$
can be written as another inductive limit
with
$B_{n}=\bigoplus _{i=1}^{n_{i}^{\prime }}B_{[n,i]^{\prime }}$
, where all the
$B_{[n,i]^{\prime }}$
are Elliott–Thomsen algebras and with the extra condition that all the
$\unicode[STIX]{x1D713}_{n,n+1}$
are injective.
In this paper, we study the boundary quotient
$\text{C}^{\ast }$
-algebras associated with products of odometers. One of our main
results shows that the boundary quotient
$\text{C}^{\ast }$
-algebra of the standard product of
$k$
odometers over
$n_{i}$
-letter alphabets
$(1\leqslant i\leqslant k)$
is always nuclear, and that it is a UCT Kirchberg algebra if and
only if
$\{\ln n_{i}:1\leqslant i\leqslant k\}$
is rationally independent, if and only if the associated
single-vertex
$k$
-graph
$\text{C}^{\ast }$
-algebra is simple. To achieve this, one of our main steps is to
construct a topological
$k$
-graph such that its associated Cuntz–Pimsner
$\text{C}^{\ast }$
-algebra is isomorphic to the boundary quotient
$\text{C}^{\ast }$
-algebra. Some relations between the boundary quotient
$\text{C}^{\ast }$
-algebra and the
$\text{C}^{\ast }$
-algebra
$\text{Q}_{\mathbb{N}}$
introduced by Cuntz are also investigated.
We give two characterisations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.
In this paper, we define the notion of monic representation for the
$C^{\ast }$
-algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative
$C^{\ast }$
-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the
$\unicode[STIX]{x1D6EC}$
-semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs. J. Math. Anal. Appl.434 (2015), 241–270) and also provide a universal representation model for non-negative monic representations.
We study the finite versus infinite nature of C
$^{\ast }$
-algebras arising from étale groupoids. For an ample groupoid
$G$
, we relate infiniteness of the reduced C
$^{\ast }$
-algebra
$\text{C}_{r}^{\ast }(G)$
to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid
$S(G)$
which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C
$^{\ast }$
-algebra of
$G$
in the sense that if
$G$
is ample, minimal, topologically principal, and
$S(G)$
is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for
$\text{C}_{r}^{\ast }(G)$
. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph
$\text{C}^{\ast }$
-algebras as well.
For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.