967 research outputs found
Amenability and paradoxicality in semigroups and C*-algebras
We analyze the dichotomy amenable/paradoxical in the context of (discrete,
countable, unital) semigroups and corresponding semigroup rings. We consider
also F{\o}lner's type characterizations of amenability and give an example of a
semigroup whose semigroup ring is algebraically amenable but has no F{\o}lner
sequence.
In the context of inverse semigroups we give a characterization of
invariant measures on (in the sense of Day) in terms of two notions:
and . Given a unital representation of
in terms of partial bijections on some set we define a natural
generalization of the uniform Roe algebra of a group, which we denote by
. We show that the following notions are then equivalent: (1)
is domain measurable; (2) is not paradoxical; (3) satisfies the
domain F{\o}lner condition; (4) there is an algebraically amenable dense
*-subalgebra of ; (5) has an amenable trace; (6)
is not properly infinite and (7) in the
-group of . We also show that any tracial state on
is amenable. Moreover, taking into account the localization
condition, we give several C*-algebraic characterizations of the amenability of
. Finally, we show that for a certain class of inverse semigroups, the
quasidiagonality of implies the amenability of . The
converse implication is false.Comment: 29 pages, minor corrections. Mistake in the statement of Proposition
4.19 from previous version corrected. Final version to appear in Journal of
Functional Analysi
Non-supramenable groups acting on locally compact spaces
Supramenability of groups is characterised in terms of invariant measures on
locally compact spaces. This opens the door to constructing interesting crossed
product C*-algebras for non-supramenable groups. In particular, stable
Kirchberg algebras in the UCT class are constructed using crossed products for
both amenable and non-amenable groups.Comment: Minor changes; to appear in Doc. Mat
Zeno meets modern science
``No one has ever touched Zeno without refuting him''. We will not refute
Zeno in this paper. Instead we review some unexpected encounters of Zeno with
modern science. The paper begins with a brief biography of Zeno of Elea
followed by his famous paradoxes of motion. Reflections on continuity of space
and time lead us to Banach and Tarski and to their celebrated paradox, which is
in fact not a paradox at all but a strict mathematical theorem, although very
counterintuitive. Quantum mechanics brings another flavour in Zeno paradoxes.
Quantum Zeno and anti-Zeno effects are really paradoxical but now experimental
facts. Then we discuss supertasks and bifurcated supertasks. The concept of
localization leads us to Newton and Wigner and to interesting phenomenon of
quantum revivals. At last we note that the paradoxical idea of timeless
universe, defended by Zeno and Parmenides at ancient times, is still alive in
quantum gravity. The list of references that follows is necessarily incomplete
but we hope it will assist interested reader to fill in details.Comment: 40 pages, LaTeX, 10 figure
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