Quotient semigroups and extension semigroups

Rong, Xing; Wei, Changguo; Liu, Shudong

doi:10.1007/s12044-012-0086-3

Cited by 4 publications

(3 citation statements)

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“…Using exactly the same ideas one can prove that if a Banach space admits an invariant mean with respect to a group, then it also does so with respect to subgroups and quotients of the group (see [12,Theorem 3.12] and [7, Lemma 2.3]). We would like to get a similar result for quotients of semigroups (subsemigroups of an amenable group need not be amenable) but first we must say something about normal semigroups and quotients of semigroups (see also [17]). Let (S, +) be a semigroup, G be a subsemigroup of S. Then G is called a normal subsemigroup if x + G = G + x for every x ∈ S. Of course in a commutative semigroup each subsemigroup is normal.…”

Section: Preliminariesmentioning

confidence: 99%

Invariant vector means and complementability of Banach spaces in their second duals

Łukasik

2022

Aequat. Math.

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Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of $$X^{**}$$ X ∗ ∗ . We then show that X is complemented in $$X^{**}$$ X ∗ ∗ if and only if there exists an invariant mean $$M:\ell _\infty (S,X)\rightarrow X$$ M : ℓ ∞ ( S , X ) → X . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).

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Section: Preliminariesmentioning

confidence: 99%

Invariant vector means and complementability of Banach spaces in their second duals

Łukasik

2022

Aequat. Math.

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“…Firstly, we need to recall some definitions and notations of C * -algebra extension. One can see [1,10,[23][24][25][26][27][28][29][30][31][32] for more details.…”

Section: Preliminariesmentioning

confidence: 99%

Topologizing UCTs for unital extensions

Wei

Liu

2020

Banach J. Math. Anal.

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In this paper, we study the topological structure on Ext-groups of unital extensions. It is proved that the stable strong Ext-group and the stable weak Ext-group for unital extensions are pseudopolish groups if A is a unital C * -algebra in the bootstrap class N and B is a stable separable C * -algebra. Furthermore, we topologize certain UCTs and prove that these UCTs are exact sequences as topological groups.

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“…Firstly, we need to recall some definitions and notations of C * -algebra extension. One can see [2], [16], [17], [18], [19], [20], [21], [22] for more details.…”

Section: Topologies On Ext-semigroupsmentioning

confidence: 99%