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• W1557732328 abstract "Suppose that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper A comma upper G comma alpha right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(A,G,alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript asterisk> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dynamical system such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of polynomial growth. 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 finite dimensional, we show that any element in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K left-parenthesis upper G semicolon upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>;</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K(G;A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has slow growth and that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript 1 Baseline left-parenthesis upper G comma upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^1(G, A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <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>-regular. Furthermore, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is discrete and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a “nice representation” 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>, we define a new Banach <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>-algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l Subscript pi Superscript 1 Baseline left-parenthesis upper G comma upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>π<!-- π --></mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>l^1_{pi }(G, A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which coincides with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l Superscript 1 Baseline left-parenthesis upper G semicolon upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>;</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>l^1(G;A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <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 finite dimensional. We also show that any element in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K left-parenthesis upper G semicolon upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>;</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K(G;A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has slow growth and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l Subscript pi Superscript 1 Baseline left-parenthesis upper G comma upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>π<!-- π --></mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>l^1_{pi }(G, A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <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>-regular." @default.
• W1557732328 created "2016-06-24" @default.
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• W1557732328 date "2005-07-19" @default.
• W1557732328 modified "2023-10-16" @default.
• W1557732328 title "Functional calculus and *-regularity of a class of Banach algebras" @default.
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