Arveson spectrum

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