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− | Suppose, for initial discussion, that the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202801.png" /> is represented by a strongly continuous, isometric representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202802.png" /> on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202803.png" /> (cf. also [[Representation theory|Representation theory]]). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202804.png" /> may be quite arbitrary, but for definiteness, consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202805.png" /> to be any Banach space of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202806.png" /> on which translation is continuous and then take translation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202807.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202808.png" /> and an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a1202809.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028010.png" />, where the integral is a vector-valued [[Riemann integral|Riemann integral]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028011.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028012.png" /> that satisfies the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028014.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028015.png" /> is a common eigenvector for all the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028017.png" /> is a [[Banach function space|Banach function space]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028018.png" />, as a function, is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028019.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028020.png" /> multiplied by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028021.png" />. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028022.png" /> is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028023.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028024.png" />. Thus, the spectrum generalizes the idea of the support of the [[Fourier transform|Fourier transform]] (i.e. Fourier series) of a function. It can be shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028025.png" /> is non-empty precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028026.png" />; in fact, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028027.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028028.png" />-summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028029.png" /> (cf. also [[Summation methods|Summation methods]]; [[Cesàro summation methods|Cesàro summation methods]]). Indeed, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028030.png" />th arithmetic mean of the partial sums of this series is given by the vector-valued integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028032.png" /> is the Fejér kernel (cf. also [[Fejér singular integral|Fejér singular integral]]), and the standard argument using this kernel that shows that the Cesàro means of the Fourier series of a continuous function converge uniformly to the function applies here, mutatis-mutandis, [[#References|[a7]]]. Thus, each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028033.png" /> may be reconstructed from its spectral parts just as ordinary functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028034.png" /> coming from spaces on which translation is continuous may be reconstructed from its Fourier series.
| + | <!--This article has been texified automatically. Since there was no Nroff source code for this article, |
| + | the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist |
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− | Building on a long tradition of [[Harmonic analysis|harmonic analysis]] that may be traced back to [[#References|[a6]]] and [[#References|[a3]]], W. Arveson [[#References|[a1]]] generalized and expanded the analysis just presented to cover cases when an arbitrary locally compact [[Abelian group|Abelian group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028035.png" /> is represented by invertible operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028036.png" /> acting on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028038.png" /> is finite. The assumption of continuity is also weakened. His primary applications concern the settings where:
| + | Out of 174 formulas, 170 were replaced by TEX code.--> |
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− | a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028039.png" /> is a [[Hilbert space|Hilbert space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028040.png" /> is a strongly continuous [[Unitary representation|unitary representation]]; | + | {{TEX|semi-auto}}{{TEX|partial}} |
| + | Suppose, for initial discussion, that the unit circle $\mathbf{T}$ is represented by a strongly continuous, isometric representation $\{ U _ { z } \} _ { z \in \mathbf T }$ on a [[Banach space|Banach space]] $\mathcal{X}$ (cf. also [[Representation theory|Representation theory]]). The space $\mathcal{X}$ may be quite arbitrary, but for definiteness, consider $\mathcal{X}$ to be any Banach space of functions on $\mathbf{T}$ on which translation is continuous and then take translation for $\{ U _ { z } \} _ { z \in \mathbf T }$. For $x \in \cal X$ and an integer $n$, let $\hat{x} ( n ) = \int _ { \mathbf{T} } \overline{z}^{n} U _ { z } ( x ) d z$, where the integral is a vector-valued [[Riemann integral|Riemann integral]]. Then $\hat{x} ( n )$ is an element of $\mathcal{X}$ that satisfies the equation $U _ { z } \hat { x } ( n ) = z ^ { n } \hat { x } ( n )$, $z \in \mathbf T$. Thus, $\hat{x} ( n )$ is a common eigenvector for all the operators $U _ { z }$. If $\mathcal{X}$ is a [[Banach function space|Banach function space]], then $\hat{x} ( n )$, as a function, is the $n$th Fourier coefficient of $x$ multiplied by the function $z \mapsto z ^ { n }$. The spectrum of $x$ is defined to be $\{ n : \tilde{x} ( n ) \neq 0 \}$ and is denoted by $\operatorname{sp} _ { U } ( x )$. Thus, the spectrum generalizes the idea of the support of the [[Fourier transform|Fourier transform]] (i.e. Fourier series) of a function. It can be shown that $\operatorname{sp} _ { U } ( x )$ is non-empty precisely when $x \neq 0$; in fact, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028027.png"/> is $C _ { 1 }$-summable to $x$ (cf. also [[Summation methods|Summation methods]]; [[Cesàro summation methods|Cesàro summation methods]]). Indeed, the $n$th arithmetic mean of the partial sums of this series is given by the vector-valued integral $\int k _ { n } ( z ) U _ { z } ( x ) d z$, where $k _ { n } ( z )$ is the Fejér kernel (cf. also [[Fejér singular integral|Fejér singular integral]]), and the standard argument using this kernel that shows that the Cesàro means of the Fourier series of a continuous function converge uniformly to the function applies here, mutatis-mutandis, [[#References|[a7]]]. Thus, each element of $\mathcal{X}$ may be reconstructed from its spectral parts just as ordinary functions on $\mathbf{T}$ coming from spaces on which translation is continuous may be reconstructed from its Fourier series. |
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− | b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028041.png" /> is a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028042.png" />-algebra]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028043.png" /> is a strongly continuous representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028044.png" /> as a group of automorphisms; and
| + | Building on a long tradition of [[Harmonic analysis|harmonic analysis]] that may be traced back to [[#References|[a6]]] and [[#References|[a3]]], W. Arveson [[#References|[a1]]] generalized and expanded the analysis just presented to cover cases when an arbitrary locally compact [[Abelian group|Abelian group]] $G$ is represented by invertible operators $\{ U _ { t } \} _ { t \in G }$ acting on a Banach space $\mathcal{X}$ such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028038.png"/> is finite. The assumption of continuity is also weakened. His primary applications concern the settings where: |
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− | c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028045.png" /> is a [[Von Neumann algebra|von Neumann algebra]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028046.png" /> is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028047.png" /> as a group of automorphisms that is continuous with respect to the ultraweak topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028048.png" />. Since these groups are isometric, in this discussion it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028049.png" /> is isometric (cf. also [[Isometric mapping|Isometric mapping]]).
| + | a) $\mathcal{X}$ is a [[Hilbert space|Hilbert space]] and $\{ U _ { t } \} _ { t \in G }$ is a strongly continuous [[Unitary representation|unitary representation]]; |
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− | Arveson considers pairs of Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028050.png" /> that are in duality via a pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028051.png" />. He assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028052.png" /> determines the norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028053.png" /> in the sense that
| + | b) $\mathcal{X}$ is a [[C*-algebra|$C ^ { * }$-algebra]] and $\{ U _ { t } \} _ { t \in G }$ is a strongly continuous representation of $G$ as a group of automorphisms; and |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028054.png" /></td> </tr></table>
| + | c) $\mathcal{X}$ is a [[Von Neumann algebra|von Neumann algebra]] and $\{ U _ { t } \} _ { t \in G }$ is a representation of $G$ as a group of automorphisms that is continuous with respect to the ultraweak topology on $\mathcal{X}$. Since these groups are isometric, in this discussion it is assumed that $\{ U _ { t } \} _ { t \in G }$ is isometric (cf. also [[Isometric mapping|Isometric mapping]]). |
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− | Further, calling the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028055.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028056.png" /> the weak topology, he assumes that the weakly closed [[Convex hull|convex hull]] of every weakly compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028057.png" /> is weakly compact. These hypotheses guarantee that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028058.png" /> is an isometric representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028059.png" /> that is continuous in the weak topology, then for each finite regular [[Borel measure|Borel measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028060.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028061.png" /> there is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028063.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028065.png" />.
| + | Arveson considers pairs of Banach spaces $( \mathcal{X} , \mathcal{X}_{*} )$ that are in duality via a pairing $\langle \, .\, ,\, . \, \rangle$. He assumes that $\mathcal{X}_{*}$ determines the norm on $\mathcal{X}$ in the sense that |
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− | Arveson also considers pairs of such pairs, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028067.png" />, and places additional hypotheses on each to ensure that the space of weakly continuous mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028068.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028070.png" />, with the operator norm, is in the same kind of duality with the closed linear span of the functionals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028075.png" />. (This space of the functionals will be denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028076.png" />.) The reason for this is that he wants to study representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028079.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028081.png" />, respectively, and wants to focus on the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028083.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028084.png" /> that they induce via the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028086.png" />. The additional hypotheses that he assumes, then, are:
| + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028054.png"/></td> </tr></table> |
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− | i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028087.png" /> is a norm-closed subspace of the Banach space dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028088.png" />, and similarly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028090.png" />; and
| + | Further, calling the topology on $\mathcal{X}$ determined by $\mathcal{X}_{*}$ the weak topology, he assumes that the weakly closed [[Convex hull|convex hull]] of every weakly compact set in $\mathcal{X}$ is weakly compact. These hypotheses guarantee that if $\{ U _ { t } \} _ { t \in G }$ is an isometric representation of $G$ that is continuous in the weak topology, then for each finite regular [[Borel measure|Borel measure]] $\mu$ on $G$ there is an operator $U _ { \mu }$ on $\mathcal{X}$ such that $\langle U _ { \mu } ( x ) , \rho \rangle = \int \langle U _ { t } ( x ) , \rho \rangle d \mu ( t )$, $\rho \in {\cal X}_{*}$. |
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− | ii) relative to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028091.png" />-topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028092.png" />, the closed convex hull of every compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028093.png" /> is compact. He then restricts his attention to representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028094.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028095.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028096.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028098.png" /> is continuous with respect to the norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a12028099.png" />. Under these assumptions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280100.png" /> satisfy the hypotheses of the previous paragraph and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280101.png" /> is weakly continuous.
| + | Arveson also considers pairs of such pairs, $( \mathcal{X} , \mathcal{X}_{*} )$ and $( \mathcal{Y} , \mathcal{Y}_{ *} )$, and places additional hypotheses on each to ensure that the space of weakly continuous mappings from $\mathcal{X}$ to $\mathcal{Y}$, $\mathcal{L} _ { W } ( \mathcal{X} , \mathcal{Y} )$, with the operator norm, is in the same kind of duality with the closed linear span of the functionals of the form $\rho \otimes x$, where $\rho \otimes x ( A ) = \langle A x , \rho \rangle$, $A \in \mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )$, $x \in \cal X$, $\rho \in \cal Y_{*}$. (This space of the functionals will be denoted $\mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )_{*}$.) The reason for this is that he wants to study representations of $G$, $U$ and $V$ on $\mathcal{X}$ and $\mathcal{Y}$, respectively, and wants to focus on the representation $\phi$ of $G$ on $\mathcal{L} _ { W } ( \mathcal{X} , \mathcal{Y} )$ that they induce via the formula $\phi _ { t } ( A ) = U _ { t } A V _ { - t }$, $A \in \mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )$. The additional hypotheses that he assumes, then, are: |
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− | Returning to the case of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280103.png" /> satisfying the hypotheses above, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280104.png" /> and consider the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280106.png" /> is identified with the [[Measure|measure]] that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280107.png" /> times [[Haar measure|Haar measure]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280108.png" /> is a closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280109.png" /> that is proper, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280110.png" />, by an approximate identity argument. The hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280111.png" />, which, by definition, is the intersection of the zero sets of the Fourier transforms of the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280112.png" />, is a closed subset of the dual group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280113.png" /> that is non-empty if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280114.png" />, i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280115.png" />, by the Tauberian theorem (cf. also [[Tauberian theorems|Tauberian theorems]]). This hull is called the (Arveson) spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280116.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280117.png" />. A moment's reflection reveals that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280118.png" /> coincides with the set discussed at the outset when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280119.png" /> is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280120.png" />.
| + | i) $\mathcal{X}_{*}$ is a norm-closed subspace of the Banach space dual of $\mathcal{X}$, and similarly for $\mathcal{Y} _ { * }$ and $\mathcal{Y}$; and |
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− | For each closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280121.png" />, let
| + | ii) relative to the $\mathcal{X}$-topology on $\mathcal{X}_{*}$, the closed convex hull of every compact set in $\mathcal{X}_{*}$ is compact. He then restricts his attention to representations $V$ of $G$ on $\mathcal{Y}$ such that for each $\rho \in \cal Y_{*}$, $t \mapsto V _ { t } ^ { * } \rho$ is continuous with respect to the norm on $\mathcal{Y} _ { * }$. Under these assumptions, $\cal ( L _ { w } ( X , Y ) , L _ { w } ( X , Y ) * )$ satisfy the hypotheses of the previous paragraph and $\{ \phi _ { t } \} _ { t \in G }$ is weakly continuous. |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280122.png" /></td> </tr></table>
| + | Returning to the case of the pair $( \mathcal{X} , \mathcal{X}_{*} )$ and $\{ U _ { t } \} _ { t \in G }$ satisfying the hypotheses above, let $x \in \cal X$ and consider the space ${\cal I} = \{ f \in L ^ { 1 } ( G ) : U _ { f } ( x ) = 0 \}$, where $f$ is identified with the [[Measure|measure]] that is $f$ times [[Haar measure|Haar measure]]. Then $\cal I$ is a closed ideal in $L ^ { 1 } ( G )$ that is proper, if $x \neq 0$, by an approximate identity argument. The hull of $\cal I$, which, by definition, is the intersection of the zero sets of the Fourier transforms of the functions in $\cal I$, is a closed subset of the dual group $\hat { C }$ that is non-empty if $\mathcal{I} \neq L ^ { 1 } ( G )$, i.e., if $x \neq 0$, by the Tauberian theorem (cf. also [[Tauberian theorems|Tauberian theorems]]). This hull is called the (Arveson) spectrum of $x$ and is denoted by $\operatorname{sp} _ { U } ( x )$. A moment's reflection reveals that $\operatorname{sp} _ { U } ( x )$ coincides with the set discussed at the outset when $U$ is a representation of $\mathbf{T}$. |
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− | Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280123.png" /> is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280124.png" /> that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280125.png" /> and is called the spectral subspace determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280126.png" />. It can be shown that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280127.png" /> is a Hilbert space, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280128.png" /> is a unitary representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280129.png" /> with spectral measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280130.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280131.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280132.png" />. Thus, the spectral subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280133.png" /> generalize to arbitrary Banach spaces and isometry groups, satisfying the basic assumptions above, giving the familiar spectral subspaces of unitary representations. They are defined, however, only for closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280134.png" /> and do not, in general, have the nice lattice-theoretic properties of the spectral subspaces for unitary representations. Nevertheless, they have proved to be immensely useful in analyzing group representations of Abelian groups.
| + | For each closed subset $E \subseteq \hat { G }$, let |
| | | |
− | The principal contribution of Arveson in this connection is a result that generalizes a theorem of F. Forelli [[#References|[a5]]] that relates the spectral subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280136.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280137.png" />, in the setting described above. To state it, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280138.png" /> is a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280139.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280140.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280141.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280142.png" /> is an additive [[Semi-group|semi-group]], containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280143.png" /> and contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280144.png" />, that coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280145.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280146.png" /> is a sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280147.png" />. Now assume the hypotheses i)–ii). Arveson proves [[#References|[a1]]], Thm. 2.3, that if a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280148.png" /> and an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280149.png" /> are given, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280150.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280151.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280152.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280153.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280154.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280155.png" />.
| + | \begin{equation*} M ^ { U } ( E ) = \{ x \in \mathcal{X} : \operatorname { sp } _ { U } ( x ) \subseteq E \}. \end{equation*} |
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− | The principal application of Arveson's theorem is to this very general set up: Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280156.png" /> is a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280157.png" />-algebra]] (respectively, a [[Von Neumann algebra|von Neumann algebra]]) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280158.png" /> is an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280159.png" /> by automorphisms that is strongly continuous (respectively, ultraweakly continuous). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280160.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280161.png" />-representation (that is ultraweakly continuous when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280162.png" /> is a von Neumann algebra) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280163.png" /> be a strongly continuous [[Unitary representation|unitary representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280164.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280165.png" />. The problem is to determine when the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280166.png" /> is a covariant representation in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280167.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280169.png" />. Covariant representations play an important role throughout operator algebra and in particular in its applications to physics. In the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280170.png" />, one finds on the basis of Arveson's theorem that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280171.png" /> is a covariant representation if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280172.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120280/a120280173.png" />. | + | Then $M ^ { U } ( E )$ is a closed subspace of $\mathcal{X}$ that is invariant under $\{ U _ { t } \} _ { t \in G }$ and is called the spectral subspace determined by $E$. It can be shown that if $\mathcal{X}$ is a Hilbert space, so that $\{ U _ { t } \} _ { t \in G }$ is a unitary representation of $G$ with spectral measure $P$ on $\hat { C }$, then $M ^ { U } ( E ) = P ( E )\cal X$. Thus, the spectral subspaces $M ^ { U } ( E )$ generalize to arbitrary Banach spaces and isometry groups, satisfying the basic assumptions above, giving the familiar spectral subspaces of unitary representations. They are defined, however, only for closed subsets of $\hat { C }$ and do not, in general, have the nice lattice-theoretic properties of the spectral subspaces for unitary representations. Nevertheless, they have proved to be immensely useful in analyzing group representations of Abelian groups. |
| + | |
| + | The principal contribution of Arveson in this connection is a result that generalizes a theorem of F. Forelli [[#References|[a5]]] that relates the spectral subspaces of $U$, $V$, and $\phi$, in the setting described above. To state it, suppose $E$ is a closed subset of $\hat { C }$ that contains $0$ and let $S _ { E } = \{ \omega \in \hat { G } : E + \omega \subseteq E \}$. Then $S _ { E }$ is an additive [[Semi-group|semi-group]], containing $0$ and contained in $E$, that coincides with $E$ if $E$ is a sub-semi-group of $\hat { C }$. Now assume the hypotheses i)–ii). Arveson proves [[#References|[a1]]], Thm. 2.3, that if a closed subset $E \subseteq \hat { G }$ and an operator $A \in \mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )$ are given, then $A$ lies in $M ^ { \phi } ( S _ { E } )$ if and only if $A$ maps $M ^ { U } ( E + \omega )$ into $M ^ { V } ( E + \omega )$ for every $\omega \in \hat { G }$. |
| + | |
| + | The principal application of Arveson's theorem is to this very general set up: Suppose $A$ is a [[C*-algebra|$C ^ { * }$-algebra]] (respectively, a [[Von Neumann algebra|von Neumann algebra]]) and that $\{ \alpha _ { t } \} _ { t \in G }$ is an action of $G$ by automorphisms that is strongly continuous (respectively, ultraweakly continuous). Let $\pi : A \rightarrow B ( H )$ be a $C ^ { * }$-representation (that is ultraweakly continuous when $A$ is a von Neumann algebra) and let $\{ U _ { t } \} _ { t \in G }$ be a strongly continuous [[Unitary representation|unitary representation]] of $G$ on $H$. The problem is to determine when the pair $( \pi , \{ U _ { t } \} _ { t \in G } )$ is a covariant representation in the sense that $\pi ( \alpha _ { t } ( a ) ) = U _ { t } \pi ( a ) U _ { t } ^ { * }$ for all $a \in A$ and $t \in G$. Covariant representations play an important role throughout operator algebra and in particular in its applications to physics. In the particular case when $G = \mathbf{R}$, one finds on the basis of Arveson's theorem that $( \pi , \{ U _ { t } \} _ { t \in \mathbf{R} } )$ is a covariant representation if and only if $\pi ( a ) M ^ { U } ( [ t , \infty ) ) \subseteq M ^ { U } ( [ t + s , \infty ) )$ for all $a \in M ^ { \alpha } ( [ s , \infty ) )$. |
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| Arveson applied this theorem to re-prove and improve a number of theorems in the literature. It has come to be a standard tool and nowadays (1998) spectral subspaces are ubiquitous in operator algebra. (See [[#References|[a2]]] for an expanded survey.) Of particular note is the notion of the Connes spectrum of an automorphism group [[#References|[a4]]], which is based on the Arveson spectrum. The Connes spectrum is a very powerful conjugacy invariant of the group that has played a fundamental role in the classification of von Neumann algebras. | | Arveson applied this theorem to re-prove and improve a number of theorems in the literature. It has come to be a standard tool and nowadays (1998) spectral subspaces are ubiquitous in operator algebra. (See [[#References|[a2]]] for an expanded survey.) Of particular note is the notion of the Connes spectrum of an automorphism group [[#References|[a4]]], which is based on the Arveson spectrum. The Connes spectrum is a very powerful conjugacy invariant of the group that has played a fundamental role in the classification of von Neumann algebras. |
| | | |
| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Arveson, "On groups of automorphisms of operator algebras" ''J. Funct. Anal.'' , '''15''' (1974) pp. 217–243</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Arveson, "The harmonic analysis of automorphism groups" , ''Operator Algebras and Automorphisms'' , ''Proc. Symp. Pure Math.'' , '''38: 1''' , Amer. Math. Soc. (1982) pp. 199–269</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Beurling, "On the spectral synthesis of bounded functions" ''Acta Math.'' , '''81''' (1949) pp. 225–238</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Connes, "Une classification des facteurs de type III" ''Ann. Sci. Ecole Norm. Sup. 4'' , '''6''' (1973) pp. 133–252</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Forelli, "Analytic and quasi-invariant measures" ''Acta Math.'' , '''118''' (1967) pp. 33–59</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R. Godement, "Théorèmes taubériens et théorie spectrale" ''Ann. Sci. Ecole Norm. Sup. 3'' , '''63''' (1947) pp. 119–138</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Y. Katznelson, "An introduction to harmonic analysis" , Wiley (1968)</TD></TR></table> | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> W. Arveson, "On groups of automorphisms of operator algebras" ''J. Funct. Anal.'' , '''15''' (1974) pp. 217–243</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> W. Arveson, "The harmonic analysis of automorphism groups" , ''Operator Algebras and Automorphisms'' , ''Proc. Symp. Pure Math.'' , '''38: 1''' , Amer. Math. Soc. (1982) pp. 199–269</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> A. Beurling, "On the spectral synthesis of bounded functions" ''Acta Math.'' , '''81''' (1949) pp. 225–238</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A. Connes, "Une classification des facteurs de type III" ''Ann. Sci. Ecole Norm. Sup. 4'' , '''6''' (1973) pp. 133–252</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> F. Forelli, "Analytic and quasi-invariant measures" ''Acta Math.'' , '''118''' (1967) pp. 33–59</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> R. Godement, "Théorèmes taubériens et théorie spectrale" ''Ann. Sci. Ecole Norm. Sup. 3'' , '''63''' (1947) pp. 119–138</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> Y. Katznelson, "An introduction to harmonic analysis" , Wiley (1968)</td></tr></table> |
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| Other references : | | Other references : |
Suppose, for initial discussion, that the unit circle $\mathbf{T}$ is represented by a strongly continuous, isometric representation $\{ U _ { z } \} _ { z \in \mathbf T }$ on a Banach space $\mathcal{X}$ (cf. also Representation theory). The space $\mathcal{X}$ may be quite arbitrary, but for definiteness, consider $\mathcal{X}$ to be any Banach space of functions on $\mathbf{T}$ on which translation is continuous and then take translation for $\{ U _ { z } \} _ { z \in \mathbf T }$. For $x \in \cal X$ and an integer $n$, let $\hat{x} ( n ) = \int _ { \mathbf{T} } \overline{z}^{n} U _ { z } ( x ) d z$, where the integral is a vector-valued Riemann integral. Then $\hat{x} ( n )$ is an element of $\mathcal{X}$ that satisfies the equation $U _ { z } \hat { x } ( n ) = z ^ { n } \hat { x } ( n )$, $z \in \mathbf T$. Thus, $\hat{x} ( n )$ is a common eigenvector for all the operators $U _ { z }$. If $\mathcal{X}$ is a Banach function space, then $\hat{x} ( n )$, as a function, is the $n$th Fourier coefficient of $x$ multiplied by the function $z \mapsto z ^ { n }$. The spectrum of $x$ is defined to be $\{ n : \tilde{x} ( n ) \neq 0 \}$ and is denoted by $\operatorname{sp} _ { U } ( x )$. Thus, the spectrum generalizes the idea of the support of the Fourier transform (i.e. Fourier series) of a function. It can be shown that $\operatorname{sp} _ { U } ( x )$ is non-empty precisely when $x \neq 0$; in fact, the series
is $C _ { 1 }$-summable to $x$ (cf. also Summation methods; Cesàro summation methods). Indeed, the $n$th arithmetic mean of the partial sums of this series is given by the vector-valued integral $\int k _ { n } ( z ) U _ { z } ( x ) d z$, where $k _ { n } ( z )$ is the Fejér kernel (cf. also Fejér singular integral), and the standard argument using this kernel that shows that the Cesàro means of the Fourier series of a continuous function converge uniformly to the function applies here, mutatis-mutandis, [a7]. Thus, each element of $\mathcal{X}$ may be reconstructed from its spectral parts just as ordinary functions on $\mathbf{T}$ coming from spaces on which translation is continuous may be reconstructed from its Fourier series.
Building on a long tradition of harmonic analysis that may be traced back to [a6] and [a3], W. Arveson [a1] generalized and expanded the analysis just presented to cover cases when an arbitrary locally compact Abelian group $G$ is represented by invertible operators $\{ U _ { t } \} _ { t \in G }$ acting on a Banach space $\mathcal{X}$ such that
is finite. The assumption of continuity is also weakened. His primary applications concern the settings where:
a) $\mathcal{X}$ is a Hilbert space and $\{ U _ { t } \} _ { t \in G }$ is a strongly continuous unitary representation;
b) $\mathcal{X}$ is a $C ^ { * }$-algebra and $\{ U _ { t } \} _ { t \in G }$ is a strongly continuous representation of $G$ as a group of automorphisms; and
c) $\mathcal{X}$ is a von Neumann algebra and $\{ U _ { t } \} _ { t \in G }$ is a representation of $G$ as a group of automorphisms that is continuous with respect to the ultraweak topology on $\mathcal{X}$. Since these groups are isometric, in this discussion it is assumed that $\{ U _ { t } \} _ { t \in G }$ is isometric (cf. also Isometric mapping).
Arveson considers pairs of Banach spaces $( \mathcal{X} , \mathcal{X}_{*} )$ that are in duality via a pairing $\langle \, .\, ,\, . \, \rangle$. He assumes that $\mathcal{X}_{*}$ determines the norm on $\mathcal{X}$ in the sense that
Further, calling the topology on $\mathcal{X}$ determined by $\mathcal{X}_{*}$ the weak topology, he assumes that the weakly closed convex hull of every weakly compact set in $\mathcal{X}$ is weakly compact. These hypotheses guarantee that if $\{ U _ { t } \} _ { t \in G }$ is an isometric representation of $G$ that is continuous in the weak topology, then for each finite regular Borel measure $\mu$ on $G$ there is an operator $U _ { \mu }$ on $\mathcal{X}$ such that $\langle U _ { \mu } ( x ) , \rho \rangle = \int \langle U _ { t } ( x ) , \rho \rangle d \mu ( t )$, $\rho \in {\cal X}_{*}$.
Arveson also considers pairs of such pairs, $( \mathcal{X} , \mathcal{X}_{*} )$ and $( \mathcal{Y} , \mathcal{Y}_{ *} )$, and places additional hypotheses on each to ensure that the space of weakly continuous mappings from $\mathcal{X}$ to $\mathcal{Y}$, $\mathcal{L} _ { W } ( \mathcal{X} , \mathcal{Y} )$, with the operator norm, is in the same kind of duality with the closed linear span of the functionals of the form $\rho \otimes x$, where $\rho \otimes x ( A ) = \langle A x , \rho \rangle$, $A \in \mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )$, $x \in \cal X$, $\rho \in \cal Y_{*}$. (This space of the functionals will be denoted $\mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )_{*}$.) The reason for this is that he wants to study representations of $G$, $U$ and $V$ on $\mathcal{X}$ and $\mathcal{Y}$, respectively, and wants to focus on the representation $\phi$ of $G$ on $\mathcal{L} _ { W } ( \mathcal{X} , \mathcal{Y} )$ that they induce via the formula $\phi _ { t } ( A ) = U _ { t } A V _ { - t }$, $A \in \mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )$. The additional hypotheses that he assumes, then, are:
i) $\mathcal{X}_{*}$ is a norm-closed subspace of the Banach space dual of $\mathcal{X}$, and similarly for $\mathcal{Y} _ { * }$ and $\mathcal{Y}$; and
ii) relative to the $\mathcal{X}$-topology on $\mathcal{X}_{*}$, the closed convex hull of every compact set in $\mathcal{X}_{*}$ is compact. He then restricts his attention to representations $V$ of $G$ on $\mathcal{Y}$ such that for each $\rho \in \cal Y_{*}$, $t \mapsto V _ { t } ^ { * } \rho$ is continuous with respect to the norm on $\mathcal{Y} _ { * }$. Under these assumptions, $\cal ( L _ { w } ( X , Y ) , L _ { w } ( X , Y ) * )$ satisfy the hypotheses of the previous paragraph and $\{ \phi _ { t } \} _ { t \in G }$ is weakly continuous.
Returning to the case of the pair $( \mathcal{X} , \mathcal{X}_{*} )$ and $\{ U _ { t } \} _ { t \in G }$ satisfying the hypotheses above, let $x \in \cal X$ and consider the space ${\cal I} = \{ f \in L ^ { 1 } ( G ) : U _ { f } ( x ) = 0 \}$, where $f$ is identified with the measure that is $f$ times Haar measure. Then $\cal I$ is a closed ideal in $L ^ { 1 } ( G )$ that is proper, if $x \neq 0$, by an approximate identity argument. The hull of $\cal I$, which, by definition, is the intersection of the zero sets of the Fourier transforms of the functions in $\cal I$, is a closed subset of the dual group $\hat { C }$ that is non-empty if $\mathcal{I} \neq L ^ { 1 } ( G )$, i.e., if $x \neq 0$, by the Tauberian theorem (cf. also Tauberian theorems). This hull is called the (Arveson) spectrum of $x$ and is denoted by $\operatorname{sp} _ { U } ( x )$. A moment's reflection reveals that $\operatorname{sp} _ { U } ( x )$ coincides with the set discussed at the outset when $U$ is a representation of $\mathbf{T}$.
For each closed subset $E \subseteq \hat { G }$, let
\begin{equation*} M ^ { U } ( E ) = \{ x \in \mathcal{X} : \operatorname { sp } _ { U } ( x ) \subseteq E \}. \end{equation*}
Then $M ^ { U } ( E )$ is a closed subspace of $\mathcal{X}$ that is invariant under $\{ U _ { t } \} _ { t \in G }$ and is called the spectral subspace determined by $E$. It can be shown that if $\mathcal{X}$ is a Hilbert space, so that $\{ U _ { t } \} _ { t \in G }$ is a unitary representation of $G$ with spectral measure $P$ on $\hat { C }$, then $M ^ { U } ( E ) = P ( E )\cal X$. Thus, the spectral subspaces $M ^ { U } ( E )$ generalize to arbitrary Banach spaces and isometry groups, satisfying the basic assumptions above, giving the familiar spectral subspaces of unitary representations. They are defined, however, only for closed subsets of $\hat { C }$ and do not, in general, have the nice lattice-theoretic properties of the spectral subspaces for unitary representations. Nevertheless, they have proved to be immensely useful in analyzing group representations of Abelian groups.
The principal contribution of Arveson in this connection is a result that generalizes a theorem of F. Forelli [a5] that relates the spectral subspaces of $U$, $V$, and $\phi$, in the setting described above. To state it, suppose $E$ is a closed subset of $\hat { C }$ that contains $0$ and let $S _ { E } = \{ \omega \in \hat { G } : E + \omega \subseteq E \}$. Then $S _ { E }$ is an additive semi-group, containing $0$ and contained in $E$, that coincides with $E$ if $E$ is a sub-semi-group of $\hat { C }$. Now assume the hypotheses i)–ii). Arveson proves [a1], Thm. 2.3, that if a closed subset $E \subseteq \hat { G }$ and an operator $A \in \mathcal{L} _ { w } ( \mathcal{X} , \mathcal{Y} )$ are given, then $A$ lies in $M ^ { \phi } ( S _ { E } )$ if and only if $A$ maps $M ^ { U } ( E + \omega )$ into $M ^ { V } ( E + \omega )$ for every $\omega \in \hat { G }$.
The principal application of Arveson's theorem is to this very general set up: Suppose $A$ is a $C ^ { * }$-algebra (respectively, a von Neumann algebra) and that $\{ \alpha _ { t } \} _ { t \in G }$ is an action of $G$ by automorphisms that is strongly continuous (respectively, ultraweakly continuous). Let $\pi : A \rightarrow B ( H )$ be a $C ^ { * }$-representation (that is ultraweakly continuous when $A$ is a von Neumann algebra) and let $\{ U _ { t } \} _ { t \in G }$ be a strongly continuous unitary representation of $G$ on $H$. The problem is to determine when the pair $( \pi , \{ U _ { t } \} _ { t \in G } )$ is a covariant representation in the sense that $\pi ( \alpha _ { t } ( a ) ) = U _ { t } \pi ( a ) U _ { t } ^ { * }$ for all $a \in A$ and $t \in G$. Covariant representations play an important role throughout operator algebra and in particular in its applications to physics. In the particular case when $G = \mathbf{R}$, one finds on the basis of Arveson's theorem that $( \pi , \{ U _ { t } \} _ { t \in \mathbf{R} } )$ is a covariant representation if and only if $\pi ( a ) M ^ { U } ( [ t , \infty ) ) \subseteq M ^ { U } ( [ t + s , \infty ) )$ for all $a \in M ^ { \alpha } ( [ s , \infty ) )$.
Arveson applied this theorem to re-prove and improve a number of theorems in the literature. It has come to be a standard tool and nowadays (1998) spectral subspaces are ubiquitous in operator algebra. (See [a2] for an expanded survey.) Of particular note is the notion of the Connes spectrum of an automorphism group [a4], which is based on the Arveson spectrum. The Connes spectrum is a very powerful conjugacy invariant of the group that has played a fundamental role in the classification of von Neumann algebras.
References
[a1] | W. Arveson, "On groups of automorphisms of operator algebras" J. Funct. Anal. , 15 (1974) pp. 217–243 |
[a2] | W. Arveson, "The harmonic analysis of automorphism groups" , Operator Algebras and Automorphisms , Proc. Symp. Pure Math. , 38: 1 , Amer. Math. Soc. (1982) pp. 199–269 |
[a3] | A. Beurling, "On the spectral synthesis of bounded functions" Acta Math. , 81 (1949) pp. 225–238 |
[a4] | A. Connes, "Une classification des facteurs de type III" Ann. Sci. Ecole Norm. Sup. 4 , 6 (1973) pp. 133–252 |
[a5] | F. Forelli, "Analytic and quasi-invariant measures" Acta Math. , 118 (1967) pp. 33–59 |
[a6] | R. Godement, "Théorèmes taubériens et théorie spectrale" Ann. Sci. Ecole Norm. Sup. 3 , 63 (1947) pp. 119–138 |
[a7] | Y. Katznelson, "An introduction to harmonic analysis" , Wiley (1968) |
Other references :
- - "Operator algebras theory of C* algebras and von Neumann algebras", B. Blackadar, p. 282 (just the definition)
- - "Theory of operator algebras" vol. 2, Takesaki, Chap. XI
- - "C* algebras and their automorphism groups", G.K. Pedersen, Chap. 8