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- W2963482021 abstract "A scalar sequence <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to be admissible for a positive operator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A equals sigma-summation xi Subscript j Baseline upper P Subscript j> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>A= sum xi _jP_j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some rank-one projections <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P Subscript j> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>P_j</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, or, equivalently, if diag <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the diagonal of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V upper A upper V Superscript asterisk> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>VAV^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some partial isometry <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> having as domain the closure of the range of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main result of this paper is that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the sum of infinitely many projections (converging in the strong operator topology) and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a nonsummable sequence in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 0 comma 1 right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that satisfies the Kadison condition that requires that either <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation left-brace xi Subscript i Baseline bar xi Subscript i Baseline less-than-or-equal-to one half right-brace plus sigma-summation left-brace left-parenthesis 1 minus xi Subscript i Baseline right-parenthesis bar xi Subscript i Baseline greater-than one half right-brace equals normal infinity> <mml:semantics> <mml:mrow> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∣<!-- ∣ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>+</mml:mo> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∣<!-- ∣ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sum {xi _i mid xi _ile frac {1}{2}}+ sum {(1-xi _i) mid xi _i> frac {1}{2}} = infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or the difference <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation left-brace xi Subscript i Baseline bar xi Subscript i Baseline less-than-or-equal-to one half right-brace minus sigma-summation left-brace left-parenthesis 1 minus xi Subscript i Baseline right-parenthesis bar xi Subscript i Baseline greater-than one half right-brace> <mml:semantics> <mml:mrow> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∣<!-- ∣ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∣<!-- ∣ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sum {xi _i mid xi _ile frac {1}{2}}- sum {(1-xi _i) mid xi _i> frac {1}{2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an integer, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is admissible for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result extends Kadison’s carpenter’s theorem and provides an independent proof of it." @default.
- W2963482021 created "2019-07-30" @default.
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- W2963482021 date "2018-11-26" @default.
- W2963482021 modified "2023-09-24" @default.
- W2963482021 title "Admissible sequences of positive operators" @default.
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