scholarly journals Topological Wiener–Wintner theorems for amenable operator semigroups

2013 ◽  
Vol 34 (5) ◽  
pp. 1674-1698 ◽  
Author(s):  
MARCO SCHREIBER
Keyword(s):  
Compact Group ◽  
Skew Product ◽  
Operator Semigroups ◽  
Group Extensions ◽  
Markov Operators ◽  
Mean Ergodicity ◽  
The Mean ◽  
Amenable Semigroups

AbstractInspired by topological Wiener–Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.

scholarly journals Lower bounds and the asymptotic behaviour of positive operator semigroups

2017 ◽  
Vol 38 (8) ◽  
pp. 3012-3041 ◽  
Author(s):  
MORITZ GERLACH ◽  
JOCHEN GLÜCK
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Lower Bounds ◽  
Simple Proof ◽  
Continuous Operator ◽  
Banach Lattices ◽  
Operator Semigroups ◽  
Markov Operators ◽  
Discrete Semigroups ◽  
Analytical Tools

If $(T_{t})$ is a semigroup of Markov operators on an $L^{1}$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t\rightarrow \infty$. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius–Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.

The structure of ergodic measures for compact group extensions

1974 ◽  
Vol 18 (4) ◽  
pp. 363-389 ◽  
Author(s):  
H. B. Keynes’ ◽  
D. Newton
Keyword(s):  
Compact Group ◽  
Group Extensions ◽  
Ergodic Measures

scholarly journals Group actions on topological graphs

2011 ◽  
Vol 32 (5) ◽  
pp. 1527-1566 ◽  
Author(s):  
VALENTIN DEACONU ◽  
ALEX KUMJIAN ◽  
JOHN QUIGG
Keyword(s):  
Compact Group ◽  
Universal Covering ◽  
Locally Compact Group ◽  
Skew Product ◽  
Topological Graph ◽  
Topological Graphs ◽  
Locally Compact ◽  
Natural Action ◽  
Morita Equivalent ◽  
Dual Action

AbstractWe define the action of a locally compact groupGon a topological graphE. This action induces a natural action ofGon theC*-correspondence ℋ(E) and on the graphC*-algebraC*(E). If the action is free and proper, we prove thatC*(E)⋊rGis strongly Morita equivalent toC*(E/G) . We define the skew product of a locally compact groupGby a topological graphEvia a cocyclec:E1→G. The group acts freely and properly on this new topological graphE×cG. IfGis abelian, there is a dual action onC*(E) such that$C^*(E)\rtimes \hat {G}\cong C^*(E\times _cG)$. We also define the fundamental group and the universal covering of a topological graph.

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