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• W1485656779 abstract "In this paper we study the nest representations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho colon script upper A long right-arrow upper A l g script upper N> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo>:</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo stretchy=false>⟶<!-- ⟶ --></mml:mo> <mml:mi>Alg</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>rho : mathcal {A} longrightarrow operatorname {Alg} mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a strongly maximal TAF algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose ranges contain non-zero compact operators. We introduce a particular class of such representations, the essential nest representations, and we show that their kernels coincide with the completely meet irreducible ideals. From this we deduce that there exist enough contractive nest representations, with non-zero compact operators in their range, to separate the points in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using nest representation theory, we also give a coordinate-free description of the fundamental groupoid for strongly maximal TAF algebras. For an arbitrary nest representation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho colon script upper A long right-arrow upper A l g script upper N> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo>:</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo stretchy=false>⟶<!-- ⟶ --></mml:mo> <mml:mi>Alg</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>rho : mathcal {A} longrightarrow operatorname {Alg} mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that the presence of non-zero compact operators in the range of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho> <mml:semantics> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:annotation encoding=application/x-tex>rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is similar to a completely atomic nest. If, in addition, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho left-parenthesis script upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>rho (mathcal {A} )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is closed, then every compact operator in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho left-parenthesis script upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>rho (mathcal {A} )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be approximated by sums of rank one operators <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho left-parenthesis script upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>rho (mathcal {A} )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the case of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ordered nest representations, we show that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho left-parenthesis script upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>rho ( mathcal {A})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains finite rank operators iff <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kernel rho> <mml:semantics> <mml:mrow> <mml:mi>ker</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>ker rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fails to be a prime ideal." @default.
• W1485656779 created "2016-06-24" @default.
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• W1485656779 date "2007-01-04" @default.
• W1485656779 modified "2023-10-18" @default.
• W1485656779 title "Compact operators and nest representations of limit algebras" @default.
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• W1485656779 doi "https://doi.org/10.1090/s0002-9947-07-04071-8" @default.
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