Difference between revisions of "Arveson spectrum"
(Importing text file) |
|||
Line 36: | Line 36: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Arveson, "On groups of automorphisms of operator algebras" ''J. Funct. Anal.'' , ''' | + | <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> |
Revision as of 17:33, 15 May 2014
Suppose, for initial discussion, that the unit circle is represented by a strongly continuous, isometric representation
on a Banach space
(cf. also Representation theory). The space
may be quite arbitrary, but for definiteness, consider
to be any Banach space of functions on
on which translation is continuous and then take translation for
. For
and an integer
, let
, where the integral is a vector-valued Riemann integral. Then
is an element of
that satisfies the equation
,
. Thus,
is a common eigenvector for all the operators
. If
is a Banach function space, then
, as a function, is the
th Fourier coefficient of
multiplied by the function
. The spectrum of
is defined to be
and is denoted by
. 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
is non-empty precisely when
; in fact, the series
is
-summable to
(cf. also Summation methods; Cesàro summation methods). Indeed, the
th arithmetic mean of the partial sums of this series is given by the vector-valued integral
, where
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
may be reconstructed from its spectral parts just as ordinary functions on
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 is represented by invertible operators
acting on a Banach space
such that
is finite. The assumption of continuity is also weakened. His primary applications concern the settings where:
a) is a Hilbert space and
is a strongly continuous unitary representation;
b) is a
-algebra and
is a strongly continuous representation of
as a group of automorphisms; and
c) is a von Neumann algebra and
is a representation of
as a group of automorphisms that is continuous with respect to the ultraweak topology on
. Since these groups are isometric, in this discussion it is assumed that
is isometric (cf. also Isometric mapping).
Arveson considers pairs of Banach spaces that are in duality via a pairing
. He assumes that
determines the norm on
in the sense that
![]() |
Further, calling the topology on determined by
the weak topology, he assumes that the weakly closed convex hull of every weakly compact set in
is weakly compact. These hypotheses guarantee that if
is an isometric representation of
that is continuous in the weak topology, then for each finite regular Borel measure
on
there is an operator
on
such that
,
.
Arveson also considers pairs of such pairs, and
, and places additional hypotheses on each to ensure that the space of weakly continuous mappings from
to
,
, with the operator norm, is in the same kind of duality with the closed linear span of the functionals of the form
, where
,
,
,
. (This space of the functionals will be denoted
.) The reason for this is that he wants to study representations of
,
and
on
and
, respectively, and wants to focus on the representation
of
on
that they induce via the formula
,
. The additional hypotheses that he assumes, then, are:
i) is a norm-closed subspace of the Banach space dual of
, and similarly for
and
; and
ii) relative to the -topology on
, the closed convex hull of every compact set in
is compact. He then restricts his attention to representations
of
on
such that for each
,
is continuous with respect to the norm on
. Under these assumptions,
satisfy the hypotheses of the previous paragraph and
is weakly continuous.
Returning to the case of the pair and
satisfying the hypotheses above, let
and consider the space
, where
is identified with the measure that is
times Haar measure. Then
is a closed ideal in
that is proper, if
, by an approximate identity argument. The hull of
, which, by definition, is the intersection of the zero sets of the Fourier transforms of the functions in
, is a closed subset of the dual group
that is non-empty if
, i.e., if
, by the Tauberian theorem (cf. also Tauberian theorems). This hull is called the (Arveson) spectrum of
and is denoted by
. A moment's reflection reveals that
coincides with the set discussed at the outset when
is a representation of
.
For each closed subset , let
![]() |
Then is a closed subspace of
that is invariant under
and is called the spectral subspace determined by
. It can be shown that if
is a Hilbert space, so that
is a unitary representation of
with spectral measure
on
, then
. Thus, the spectral subspaces
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
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 ,
, and
, in the setting described above. To state it, suppose
is a closed subset of
that contains
and let
. Then
is an additive semi-group, containing
and contained in
, that coincides with
if
is a sub-semi-group of
. Now assume the hypotheses i)–ii). Arveson proves [a1], Thm. 2.3, that if a closed subset
and an operator
are given, then
lies in
if and only if
maps
into
for every
.
The principal application of Arveson's theorem is to this very general set up: Suppose is a
-algebra (respectively, a von Neumann algebra) and that
is an action of
by automorphisms that is strongly continuous (respectively, ultraweakly continuous). Let
be a
-representation (that is ultraweakly continuous when
is a von Neumann algebra) and let
be a strongly continuous unitary representation of
on
. The problem is to determine when the pair
is a covariant representation in the sense that
for all
and
. Covariant representations play an important role throughout operator algebra and in particular in its applications to physics. In the particular case when
, one finds on the basis of Arveson's theorem that
is a covariant representation if and only if
for all
.
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) |
Arveson spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arveson_spectrum&oldid=32186