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• W2078183835 abstract "We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Subscript upper V> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {M}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the multiplier algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Drury-Arveson space to a holomorphic subvariety <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit ball <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper B Subscript d> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>B</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathbb {B}_d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We find that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Subscript upper V> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {M}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is completely isometrically isomorphic to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Subscript upper W> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mi>W</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {M}_W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding=application/x-tex>W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the image of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding=application/x-tex>W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are each a finite union of irreducible varieties and a discrete variety, when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, an isomorphism between <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Subscript upper V> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {M}_V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M Subscript upper W> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mi>W</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {M}_W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=asterisk> <mml:semantics> <mml:mo>∗<!-- ∗ --></mml:mo> <mml:annotation encoding=application/x-tex>*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold—particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences." @default.
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• W2078183835 date "2014-07-17" @default.
• W2078183835 modified "2023-10-17" @default.
• W2078183835 title "Operator algebras for analytic varieties" @default.
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• W2078183835 doi "https://doi.org/10.1090/s0002-9947-2014-05888-1" @default.
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