Spectral theory of normal C*-algebras
(Learn how and when to remove this message) In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra
of bounded linear operators on some Hilbert space
This article describes the spectral theory of closed normal subalgebras of
. A subalgebra
of
is called normal if it is commutative and closed under the
operation: for all
, we have
and that
.[1]
Resolution of identity
Throughout,
is a fixed Hilbert space.
A projection-valued measure on a measurable space
where
is a σ-algebra of subsets of
is a mapping
such that for all
is a self-adjoint projection on
(that is,
is a bounded linear operator
that satisfies
and
) such that
(where
is the identity operator of
) and for every
the function
defined by
is a complex measure on
(that is, a complex-valued countably additive function).
A resolution of identity on a measurable space
is a function
such that for every
:
;
; - for every
is a self-adjoint projection on
; - for every
the map
defined by
is a complex measure on
;
; - if
then
;
If
is the
-algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:
- for every
the map
is a regular Borel measure (this is automatically satisfied on compact metric spaces).
Conditions 2, 3, and 4 imply that
is a projection-valued measure.
Properties
Throughout, let
be a resolution of identity. For all
is a positive measure on
with total variation
and that satisfies
for all
For every
:
(since both are equal to
). - If
then the ranges of the maps
and
are orthogonal to each other and ![{\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd55c983898cde2634ce37aaf69987ddcaf6b9b)
is finitely additive. - If
are pairwise disjoint elements of
whose union is
and if
for all
then ![{\displaystyle \pi (\omega )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1dfa71add4fad07951290949928ce618d848c2)
- However,
is countably additive only in trivial situations as is now described: suppose that
are pairwise disjoint elements of
whose union is
and that the partial sums
converge to
in
(with its norm topology) as
; then since the norm of any projection is either
or
the partial sums cannot form a Cauchy sequence unless all but finitely many of the
are ![{\displaystyle 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962)
- For any fixed
the map
defined by
is a countably additive
-valued measure on
- Here countably additive means that whenever
are pairwise disjoint elements of
whose union is
then the partial sums
converge to
in
Said more succinctly, ![{\displaystyle \sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi (\omega )x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44069fe8a5cd9e0405890c0cb30ea6308c42293c)
- In other words, for every pairwise disjoint family of elements
whose union is
, then
(by finite additivity of
) converges to
in the strong operator topology on
: for every
, the sequence of elements
converges to
in
(with respect to the norm topology).
L∞(π) - space of essentially bounded function
The
be a resolution of identity on
Essentially bounded functions
Suppose
is a complex-valued
-measurable function. There exists a unique largest open subset
of
(ordered under subset inclusion) such that
To see why, let
be a basis for
's topology consisting of open disks and suppose that
is the subsequence (possibly finite) consisting of those sets such that
; then
Note that, in particular, if
is an open subset of
such that
then
so that
(although there are other ways in which
may equal 0). Indeed,
The essential range of
is defined to be the complement of
It is the smallest closed subset of
that contains
for almost all
(that is, for all
except for those in some set
such that
). The essential range is a closed subset of
so that if it is also a bounded subset of
then it is compact.
The function
is essentially bounded if its essential range is bounded, in which case define its essential supremum, denoted by
to be the supremum of all
as
ranges over the essential range of
Space of essentially bounded functions
Let
be the vector space of all bounded complex-valued
-measurable functions
which becomes a Banach algebra when normed by
The function
is a seminorm on
but not necessarily a norm. The kernel of this seminorm,
is a vector subspace of
that is a closed two-sided ideal of the Banach algebra
Hence the quotient of
by
is also a Banach algebra, denoted by
where the norm of any element
is equal to
(since if
then
) and this norm makes
into a Banach algebra. The spectrum of
in
is the essential range of
This article will follow the usual practice of writing
rather than
to represent elements of
Theorem — Let
be a resolution of identity on
There exists a closed normal subalgebra
of
and an isometric *-isomorphism
satisfying the following properties:
for all
and
which justifies the notation
;
for all
and
; - an operator
commutes with every element of
if and only if it commutes with every element of ![{\displaystyle A=\operatorname {Im} \Psi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0909b09abdad2d8277a30bad33689703aa367c)
- if
is a simple function equal to
where
is a partition of
and the
are complex numbers, then
(here
is the characteristic function); - if
is the limit (in the norm of
) of a sequence of simple functions
in
then
converges to
in
and
;
for every ![{\displaystyle f\in L^{\infty }(\pi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9201ecac016f6d8ca8e0c2b61ed95d35efc88bdd)
Spectral theorem
The maximal ideal space of a Banach algebra
is the set of all complex homomorphisms
which we'll denote by
For every
in
the Gelfand transform of
is the map
defined by
is given the weakest topology making every
continuous. With this topology,
is a compact Hausdorff space and every
in
belongs to
which is the space of continuous complex-valued functions on
The range of
is the spectrum
and that the spectral radius is equal to
which is
Theorem — Suppose
is a closed normal subalgebra of
that contains the identity operator
and let
be the maximal ideal space of
Let
be the Borel subsets of
For every
in
let
denote the Gelfand transform of
so that
is an injective map
There exists a unique resolution of identity
that satisfies:
the notation
is used to summarize this situation. Let
be the inverse of the Gelfand transform
where
can be canonically identified as a subspace of
Let
be the closure (in the norm topology of
) of the linear span of
Then the following are true:
is a closed subalgebra of
containing ![{\displaystyle A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a71bf21ad35b8fe05555041d54d1e17eeb0f490)
- There exists a (linear multiplicative) isometric *-isomorphism
extending
such that
for all
- Recall that the notation
means that
for all
; - Note in particular that
for all ![{\displaystyle T\in A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1d8a1d5350fecd00574c51fe7ed8dfcfbeb4eb)
- Explicitly,
satisfies
and
for every
(so if
is real valued then
is self-adjoint).
- If
is open and nonempty (which implies that
) then ![{\displaystyle \pi (\omega )\neq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74e4dc57a4a6917d8f62a3b84a250c2936bdfa6)
- A bounded linear operator
commutes with every element of
if and only if it commutes with every element of ![{\displaystyle \operatorname {Im} \pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc690963b3956101d1d2e971de9b767391cf5c04)
The above result can be specialized to a single normal bounded operator.
See also
References
- ^ Rudin, Walter (1991). Functional Analysis (2nd ed.). New York: McGraw Hill. pp. 292–293. ISBN 0-07-100944-2.
- Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.
- Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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