Idempotent measures on a compact topological semigroup

Heble, M. P.; Rosenblatt, Murray

doi:10.1090/s0002-9939-1963-0169971-2

Cited by 20 publications

(6 citation statements)

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“…Also we prove that if ¡x is idempotent, then F is completely simple. This was proved for compact 5 in [3] and [5] and under various compactness conditions in [4] and [6] and conjectured in [6]. As in the compact case, the result that F is completely simple enables us to formulate a complete characterization of idempotent measures on locally compact semigroups as products of a Haar measure and two regular Borel measures (see Theorem 3.1).…”

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confidence: 87%

Idempotent Measures on Locally Compact Semigroups

Mukherjea¹,

Tserpes²

1971

Proceedings of the American Mathematical Society

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Abstract.A conjecture that the support of an /■"'-invariant regular finite measure on a locally compact semigroup is a left group is proven. Moreover we also prove that the support of an idempotent measure on a locally compact semigroup is completely simple, thus extending a well-known result of Pym and HebleRosenblatt on compact semigroups to the locally compact case. These results are also shown to be true in a complete metric semigroup.

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confidence: 87%

Idempotent Measures on Locally Compact Semigroups

Mukherjea¹,

Tserpes²

1971

Proceedings of the American Mathematical Society

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“…Hence F -XxGxY. (vii) The fact that (i can be decomposed as a product measure ix x x n G x n Y onXxGxY can be proved by a method similar to one used in [3] (pp. 183 -184), using the fact that n(Bs~l) = constant for all se S a , for some a e Y.…”

Section: Theorem Let S Be a Countable Hausdorjf Semigroup And Let Fimentioning

confidence: 99%

“…For the case S is compact, the conjecture was already proved in [3] and [5]. For discrete S, it was proved in [4].…”

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confidence: 92%

A note on the idempotent measures on countable semigroups

Sun¹,

Tserpes²

1970

J. Aust. Math. Soc.

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In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

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“…His papers [12], [5], [13], [14] generalized the work of Levy [2] on random walks on the circle and the work of Kawada and ltd [4] on random walks on compact groups. In [14], he also completely characterized the structure of the limit measures for the special case of compact semigrougs of n x n stochastic matrices.…”

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confidence: 99%

“…In the next step, Heble and Rosenblatt in [5] determined the structure of this limit measure JJLOI±K = XXGXY.…”

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confidence: 99%

Rosenblatt’s Contributions to Random Walks on Compact Semigroups

Davis

Lii

Politis

2011

Selected Works of Murray Rosenblatt

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Murray Rosenblatt's interest in random walks on compact semigroups probably came from his work on representations of stationary processes as shifts of functions of independent random variables described in [11], where products of matrices were studied. His papers [12], [5], [13], [14] generalized the work of Levy [2] on random walks on the circle and the work of Kawada and ltd [4] on random walks on compact groups. In [14], he also completely characterized the structure of the limit measures for the special case of compact semigrougs of n x n stochastic matrices.To describe Rosenblatt's work in this area in more detail, we first introduce some definitions and basic facts about compact topological semigroups (see [15] and [3]).1. Every compact topological semigroup S has a minimal two sided ideal K which is called the kernel of S.2. The kernel if is a compact completely simple semigroup which has a Rees structure 1x6x7, where (a) X is a compact left-zero semigroup, Y is a compact right-zero semigroup and G is a compact group.(b) The topology on X x G x Y is the product topology.

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Idempotent measures on a compact topological semigroup

Cited by 20 publications

References 4 publications

Idempotent Measures on Locally Compact Semigroups

Idempotent Measures on Locally Compact Semigroups

A note on the idempotent measures on countable semigroups

Rosenblatt’s Contributions to Random Walks on Compact Semigroups

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