A note on lattice ordered ‐algebras and Perron–Frobenius theory

Glück, Jochen

doi:10.1002/mana.201700404

Cited by 4 publications

(5 citation statements)

References 29 publications

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“…This interpretation of Theorem 1.3 was kindly brought to my attention by Wolfgang Arendt. A similar approach as in the proof above, using Perron–Frobenius theory and Gelfand's

T = id

theorem to show that a given semigroup generator equals 0, was used in [9, Section 2] to give a new proof of a classical result of Sherman about lattice ordered

C^{*}

‐algebras.The same comments as at the end of [9, Section 2] also apply to the proof above; in particular: (e)The Perron–Frobenius type theorem from [17, Corollary C‐III‐2.13] that we used in the proof relies on quite heavy machinery. However, we only use the result for semigroups with bounded generators, for which it is much simpler to prove — see, for instance, [9, Proposition 2.2]. (f)Our proof also uses Gelfand's

T = id

theorem for

C_{0}

‐semigroups which is not quite trivial. But again, we apply this theorem only for semigroups with bounded generator — and for these, it can be derived from the single operator version of Gelfand's

T = id

theorem, which is a bit simpler (see, for instance, [1, Theorem 1.1]).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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On the decoupled Markov group conjecture

Glück

2020

Bull. London Math. Soc.

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The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup T = (Tt) t∈[0,∞) on 1 has bounded generator if the operator T1 is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite-dimensional matrix semigroups of growing dimension-that is, it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite-dimensional systems. In this article, we show that such attempts must necessarily fail, that is, we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators Q of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite-dimensional Perron-Frobenius theory and Gelfand's T = id theorem.

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“…This interpretation of Theorem 1.3 was kindly brought to my attention by Wolfgang Arendt. A similar approach as in the proof above, using Perron–Frobenius theory and Gelfand's

T = id

theorem to show that a given semigroup generator equals 0, was used in [9, Section 2] to give a new proof of a classical result of Sherman about lattice ordered

C^{*}

T = id

theorem for

C_{0}

T = id

theorem, which is a bit simpler (see, for instance, [1, Theorem 1.1]).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…A similar approach as in the proof above, using Perron–Frobenius theory and Gelfand's

T = id

theorem to show that a given semigroup generator equals 0, was used in [9, Section 2] to give a new proof of a classical result of Sherman about lattice ordered

C^{*}

‐algebras.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

On the decoupled Markov group conjecture

Glück

2020

Bull. London Math. Soc.

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“…The self-adjoint part A sa := {a ∈ A : a * = a} is a real Banach space, and it becomes an ordered Banach space if we endow it with its usual cone A + sa := {a ∈ A sa : σ(a) ⊆ [0, ∞)}, where σ(a) denotes the spectrum of a. The ordered Banach space (A sa , A + sa ) is not lattice ordered unless A is commutative (this is a classical result of Sherman [47, Theorems 1 and 2], see [24] for a new approach to this result based on the spectral theory of positive operators). Now we describe the almost interior and interior points of A + sa :…”

Section: 3mentioning

confidence: 99%

Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups

Glück

Weber

2020

Studia Math.

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The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of "functions which are strictly positive almost everywhere" on L p -spaces and of "quasi-interior points" in Banach lattices.In the second part we study the long-term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs-de Leeuw-Glicksberg decomposition together with the theory presented in the first part of the paper we deduce sufficiency criteria for such semigroups to converge (strongly or in operator norm) as time tends to infinity. This generalises known results for semigroups on Banach lattices as well as on normally ordered Banach spaces with unit.

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“…In this section, we discuss C * -algebra's; for more details we refer the reader to [19][20][21]. Let A be a unital algebra and e A be its unit.…”

Section: * -Algebra-valued Fuzzy Metric Spacesmentioning

confidence: 99%