Cocycles on groupoids arising from -actions

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

We show spectral invariance for faithful $$*$$ ∗ -representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group $$G_c$$ G c is $$C^*$$ C ∗ -unique and symmetric, then the twisted convolution algebra $$L^1 (G,c)$$ L 1 ( G , c ) is spectrally invariant in $${\mathbb {B}}({\mathcal {H}})$$ B ( H ) for any faithful $$*$$ ∗ -representation of $$L^1 (G,c)$$ L 1 ( G , c ) as bounded operators on a Hilbert space $${\mathcal {H}}$$ H . As an application of this result we give a proof of the statement that if $$\Delta $$ Δ is a closed cocompact subgroup of the phase space of a locally compact abelian group $$G'$$ G ′ , and if g is some function in the Feichtinger algebra $$S_0 (G')$$ S 0 ( G ′ ) that generates a Gabor frame for $$L^2 (G')$$ L 2 ( G ′ ) over $$\Delta $$ Δ , then both the canonical dual atom and the canonical tight atom associated to g are also in $$S_0 (G')$$ S 0 ( G ′ ) . We do this without the use of periodization techniques from Gabor analysis.

Modulation spaces as a smooth structure in noncommutative geometry

We demonstrate how to construct spectral triples for twisted group $$C^*$$ C ∗ -algebras of lattices in phase space of a second-countable locally compact abelian group using a class of weights appearing in time–frequency analysis. This yields a way of constructing quantum $$C^k$$ C k -structures on Heisenberg modules, and we show how to obtain such structures using Gabor analysis and certain weighted analogues of Feichtinger’s algebra. We treat the standard spectral triple for noncommutative 2-tori as a special case, and as another example we define a spectral triple on noncommutative solenoids and a quantum $$C^k$$ C k -structure on the associated Heisenberg modules.

Banach space valued weak second order stochastic processes

Banach space valued stochastic processes of weak second order on a locally compact abelian group G G is considered. These processes are recognized as operator valued processes on G G . More fully, letting U \mathfrak {U} be a Banach space and H \mathfrak {H} a Hilbert space, we study B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes. Since B ( U , H ) B(\mathfrak {U},\mathfrak {H}) has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued gramian, every B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued process has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator V V -boundedness for B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.

On the Existence of an Extremal Function in the Delsarte Extremal Problem

This paper is concerned with a Delsarte-type extremal problem. Denote by $${\mathcal {P}}(G)$$ P ( G ) the set of positive definite continuous functions on a locally compact abelian group G. We consider the function class, which was originally introduced by Gorbachev, $$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$ G ( W , Q ) G = f ∈ P ( G ) ∩ L 1 ( G ) : f ( 0 ) = 1 , supp f + ⊆ W , supp f ^ ⊆ Q where $$W\subseteq G$$ W ⊆ G is closed and of finite Haar measure and $$Q\subseteq {\widehat{G}}$$ Q ⊆ G ^ is compact. We also consider the related Delsarte-type problem of finding the extremal quantity $$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$ D ( W , Q ) G = sup ∫ G f ( g ) d λ G ( g ) : f ∈ G ( W , Q ) G . The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem $${\mathcal {D}}(W,Q)_G$$ D ( W , Q ) G . The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $$G={\mathbb {R}}^d$$ G = R d . So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

On the convergence of iterates of convolution operators in Banach spaces

Let $G$ be a locally compact abelian group and let $M(G)$ be the measure algebra of $G$. A measure $\mu \in M(G)$ is said to be power bounded if $\sup _{n\geq 0}\lVert \mu ^{n} \rVert _{1}<\infty $. Let $\mathbf {T} = \{ T_{g}:g\in G\}$ be a bounded and continuous representation of $G$ on a Banach space $X$. For any $\mu \in M(G)$, there is a bounded linear operator on $X$ associated with µ, denoted by $\mathbf {T}_{\mu }$, which integrates $T_{g}$ with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences $\{ \mathbf {T}_{\mu }^{n}x\}$ $(x\in X)$ in the case when µ is power bounded. Some related problems are also discussed.

On a group analogue of the Heyde theorem

Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analogue of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

A bridge theorem for the entropy of semigroup actions

The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.