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Following is a research paper in which for that conclusion i sought above answers by BING/26.08.23
LEBESGUE DECOMPOSITION OF NON-COMMUTATIVE MEASURES
MICHAEL T. JURY AND ROBERT T.W. MARTIN
Abstract. The Riesz-Markov theorem identifies any positive, finite, and regular Borel
measure on the complex unit circle with a positive linear functional on the continuous
functions. By the Weierstrass approximation theorem, the continuous functions are obtained
as the norm closure of the Disk Algebra and its conjugates. Here, the Disk Algebra can be
viewed as the unital norm-closed operator algebra of the shift operator on the Hardy Space,
H2 of the disk.
Replacing square-summable Taylor series indexed by the non-negative integers, i.e. H2
of the disk, with square-summable power series indexed by the free (universal) monoid on d
generators, we show that the concepts of absolutely continuity and singularity of measures,
Lebesgue Decomposition and related results have faithful extensions to the setting of ‘noncommutative measures’ defined as positive linear functionals on a non-commutative multivariable ‘Disk Algebra’ and its conjugates.
1. Introduction
The results of this paper extend the Lebesgue decomposition of any finite, positive and regular
Borel measure, with respect to Lebesgue measure on the complex unit circle, from one to several non-commuting (NC) variables. In [1], we extended the concepts of absolute continuity and
singularity of positive measures with respect to Lebesgue measure, the Lebesgue Decomposition,
and the Radon-Nikodym formula of Fatou’s Theorem to the non-commutative, multi-variable setting of ‘NC measures’, i.e. positive linear functionals on a certain operator system, the Free Disk
System. Here, the free disk system, Ad + A∗
d
, is the operator system of the Free Disk Algebra,
Ad := Alg(I, L)
−k·k, the norm-closed operator algebra generated by the left free shifts on the NC
Hardy space. (Equivalently, the left creation operators on the full Fock space over C
d
.) We will
recall in some detail below why this is the appropriate (and even canonical) extension of the concept of a positive measure on the circle to several non-commuting variables. The primary goal of
this paper is to further develop the NC Lebesgue Decomposition Theory of an arbitrary (positive)
NC measure with respect to NC Lebesgue measure (the ‘vacuum state’ on the Fock space), by
proving that our concepts of absolutely continuous (AC) and singular NC measures define positive
hereditary cones, and hence that the Lebesgue Decomposition commutes with summation. That
is, the Lebesgue Decomposition of the sum of any two NC measures is the sum of the Lebesgue
Decompositions. (Here, we say a positive cone, P0 ⊂ P is hereditary in a larger positive cone P
if p0 ∈ P0, and p0 ≥ p for any p ∈ P implies that p ∈ P0. The sets of absolutely continuous and
First named author partially supported by NSF grant DMS-1900364.
1
singular positive, finite, regular Borel measures on the circle, ∂D, with respect to another fixed
positive measure, are clearly positive hereditary sub-cones.) In this paper we focus on positive NC
measures and their Lebesgue decomposition with respect to NC Lebesgue measure. The study of
more general complex NC measures, and the Lebesgue Decomposition of an arbitrary positive NC
measure with respect to another will be the subject of future research.
By the Riesz-Markov Theorem, any finite positive Borel measure, µ, on ∂D, can be identified
with a positive linear functional, ˆµ on C (∂D), the commutative C
∗−algebra of continuous functions
on the circle. By the Weierstrass Approximation Theorem, C (∂D) = (A(D) + A(D)
∗
)
−k·k∞ , where
A(D) is the Disk Algebra, the algebra of all analytic functions in the complex unit disk, D, with
continuous extensions to the boundary. In the above formula, elements of A(D) are identified with
their continuous boundary values and k · k∞ denotes the supremum norm for continuous functions
on the circle. The disk algebra can also be viewed as the norm-closed unital operator algebra
generated by the shift, S := Mz, A(D) = Alg(I, S)
−k·k (with equality of norms). The shift is the
isometry of multiplication by z on the Hardy space, H2
(D), and plays a central role in the theory
of Hardy spaces. Here recall that the Hardy Space, H2
(D), is the space of all analytic functions
in D with square-summable MacLaurin series coefficients (and with the ℓ
2
inner product of these
Taylor series coefficients at 0 ∈ D). The positive linear functional ˆµ is then completely determined
by the moments of the measure µ:
µˆ(S
k
) := Z
∂D
ζ
kµ(dζ).
The shift on H2
(D) is isomorphic to the unilateral shift on ℓ
2
(N0), where N0, the non-negative
integers, is the universal monoid on one generator. A canonical several-variable extension of ℓ
2
(N0)
is then ℓ
2
(F
d
), where F
d
is the free (and universal) monoid on d generators, the set of all words
in d letters. One can define a natural d−tuple of isometries on ℓ
2
(F
d
), the left free shifts, Lk,
1 ≤ k ≤ d defined by Lkeα = ekα where α ∈ F
d and {eα} is the standard orthonormal basis.
These left free shifts have pairwise orthogonal ranges so that the row operator L := (L1, ..., Ld) :
ℓ
2
(F
d
) ⊗ C
d → ℓ
2
(F
d
) is an isometry from d copies of ℓ
2
(F
d
) into one copy which we call the left
free shift. This Hilbert space of free square-summable sequences can also be identified with a ‘NC
Hardy Space’ of ‘non-commutative analytic functions’ in a non-commutative open unit disk or ball
of several matrix variables. Under this identification, the left free shifts become left multiplication
by independent matrix-variables, see Section 2. The immediate analogue of a positive measure
in this non-commutative (NC) multi-variable setting is then a positive linear functional, or NC
measure, on the Free Disk System:
(Ad + A
∗
d
)
−k·k
,
where Ad := Alg(I, L)
−k·k is the Free Disk Algebra, the operator norm-closed unital operator algebra
generated by the left free shifts. As in the classical theory, elements of the Free Disk Algebra can be
identified with bounded (matrix-valued) analytic functions (in several non-commuting matrix variables) which extend continuously from the interior to the boundary of a certain non-commutative
multi-variable open unit disk or ball.
2
There is a fundamental connection between this work and the theory of row isometries (isometries
from several copies of a Hilbert space into itself), or equivalently to the representation theory of
the Cuntz-Toeplitz C
∗−algebra, Ed = C
∗
(I, L) (the C
∗− algebra generated by the left free shifts),
the universal C
∗−algebra generated by a d−tuple of isometries with pairwise orthogonal ranges,
and of the Cuntz C
∗−algebra, Od, the universal C
∗−algebra of an onto row isometry [2]. Namely,
applying the Gelfand-Naimark-Segal (GNS) construction to (µ, Ad), where µ is any (positive) NC
measure yields a GNS Hilbert space, F
2
d
(µ), and a ∗−representation πµ of Ed so that Πµ := πµ(L)
is a row-isometry on F
2
d
(µ). A Lebesgue Decomposition for bounded linear functionals on the Free
Disk Algebra, Ad, has been developed by Davidson, Li and Pitts in the theory of Free Semigroup
Algebras, i.e. W OT−closed (weak operator topology closed) operator algebras generated by row
isometries [3, 4, 5]. Building on this, Kennedy has constructed a Lebesgue Decomposition for row
isometries [6], and we will explicitly work out the relationship between this theory and our Lebesgue
Decomposition.
1.1. Three approaches to classical Lebesgue Decomposition Theory. There are three approaches to classical Lebesgue Decomposition theory which will provide natural and equivalent
extensions to NC measures.
Let µ be an arbitrary finite, positive, and regular Borel measure on ∂D, and as before, m denotes
normalized Lebesgue measure on the circle. As we will prove, one can construct the Lebesgue
decomposition of µ with respect to m using reproducing kernel Hilbert space theory. Namely,
setting H2
(µ) to be the closure of the analytic polynomials in L
2
(µ, ∂D), let H +(Hµ) be the space
of all Cauchy Transforms of elements in H2
(µ): If h ∈ H2
(µ),
(Cµh)(z) := Z
∂D
1
1 − zζ
h(ζ)µ(dζ).
Equipped with the inner product of H2
(µ), this is a reproducing kernel Hilbert space of analytic
functions in D, the classical Herglotz Space with reproducing kernel:
Kµ
(z, w) = 1
2
Hµ(z) + Hµ(w)
∗
1 − zw∗
=
Z
∂D
1
1 − zζ
1
1 − ζw
µ(dζ),
and
Hµ(z) := Z
∂D
1 + zζ
1 − zζ
h(ζ)µ(dζ)
= 2(Cµ1)(z) − µ(∂D),
is the Riesz-Herglotz integral transform of µ, an analytic function with non-negative real part in D
(see [7, Chapter 1], or [8, Chapter 1, Section 5]). It is not hard to verify that domination of finite,
positive, and regular Borel measures is equivalent to domination of the Herglotz kernels for their
reproducing kernel Hilbert spaces of Cauchy Transforms:
0 ≤ µ ≤ t
2λ ⇔ Kµ ≤ t
2Kλ
; t > 0.
3
Moreover, by a classical result of Aronszajn, domination of the reproducing kernels Kµ ≤ t
2Kλ
is equivalent to bounded containment of the corresponding Herglotz spaces on D, H +(Hµ) ⊆
H +(Hλ), and the least such t > 0 is the norm of the embedding map eµ : H +(Hµ) ֒→ H +(Hλ)
[9, Theorem I, Section 7]. Absolute continuity of measures on ∂D can also be recast in terms of
containment of reproducing kernel Hilbert spaces. Namely, given two finite, positive, regular Borel
measures λ, µ, recall that µ is absolutely continuous with respect to λ if there is a non-decreasing
sequence of finite, positive, regular Borel measures µn, which are each dominated by λ, and increase
monotonically to µ:
0 ≤ µn ≤ µ, µn ↑ µ,
µn ≤ t
2
nλ, tn > 0.
Reproducing kernel Hilbert space theory then implies that each space of µn−Cauchy transforms
is contractively contained in the space of µ−Cauchy transforms, and their linear span is dense in
H +(Hµ) since the µn increase to µ,
_
H +(Hµn
) = H +(Hµ).
Since each µn ≤ t
2
nλ, is dominated by λ, it also follows that each space of µn−Cauchy transforms
is boundedly contained in the space of λ−Cauchy transforms, and the intersection space:
int(µ, λ) := H +(Hµ)
\
H +(Hλ),
is dense in the space of µ−Cauchy transforms. In the case where λ = m is normalized Lebesgue
measure, one can check that Hm ≡ 1 is constant, so that H +(Hm) = H2
(D) is the classical
Hardy space of the disk. It follows that one can take this as a starting point, and simply define a
measure, µ, to be absolutely continuous or singular (with respect to m) depending on whether the
intersection space
int(µ, m) := H +(Hµ)
\
H2
(D),
is dense or trivial, respectively, in the space of µ−Cauchy Transforms. In this way one can develop
Lebesgue Decomposition Theory using reproducing kernel techniques. It appears that this approach
is new, even in this classical setting, and as shown in Corollary 8.5, this recovers the Lebesgue
decomposition of any finite, positive and regular Borel measure on the unit circle with respect to
normalized Lebesgue measure.
As discussed in the introduction, any positive, finite, regular Borel measure, µ, on ∂D, can be
viewed as a positive linear functional, ˆµ, on A(D) + A(D)
∗
. Equivalently, µ (or ˆµ) can be identified
with the (generally unbounded) positive quadratic or sesquilinear form,
qµ(a1, a2) := Z
∂D
a1(ζ)a2(ζ)µ(dζ); a1, a2 ∈ A(D),
densely-defined in H2
(D). Applying the theory of Lebesgue Decomposition of quadratic forms due
to B. Simon yields:
qµ = qac + qs,
4
where qac is the maximal positive form absolutely continuous to qm, where m is normalized Lebesgue
measure, and qs is singular [10]. In this theory, a positive quadratic form with dense domain in a
Hilbert space, H, is said to be absolutely continuous if it is closable, i.e. it has a closed extension.
Here, a positive semi-definite quadratic form, q, is closed if its domain, Dom(q) is complete in the
norm
k · kq+1 := p
q(·, ·) + h·, ·iH.
Closed positive semi-definite forms obey an extension of the Riesz Representation Lemma: A
positive semi-definite densely-defined quadratic form, q, is closed if and only if q is the quadratic
form of a closed, densely-defined, positive semi-definite operator, T ≥ 0:
q(h, g) = h
√
T h, √
T giH; h, g ∈ Dom(√
T) = Dom(q),
see [11, Chapter VI, Theorem 2.21, Theorem 2.23]. If q = qµ, we will prove that qac = qµac
, and
qs = qµs where
µ = µac + µs,
is the classical Lebesgue Decomposition of µ with respect to m, see Corollary 8.5. Indeed, if one
instead defines qµ as a quadratic form densely-defined in L
2
(∂D), then it follows without difficulty
in this case that T is affiliated to L∞(∂D) so that
qµ(f, g) = Z
∂D
f(ζ)g(ζ)|h(ζ)|
2m(dζ),
where √
T1 = |h| ∈ L
2
(∂D). This shows that |h|
2 ∈ L
1
(∂D) is the Radon-Nikodym derivative of µ
with respect to normalized Lebesgue measure, m. The Lebesgue Decomposition of quadratic forms
in [10] is similar in this case to von Neumann’s proof of the Lebesgue Decomposition theory [12,
Lemma 3.2.3]. In [1], we applied this quadratic form decomposition to the quadratic form, qµ, of
any (positive) NC measure µ to construct an NC Lebesgue decomposition of µ, µ = µac + µs into
absolutely continuous and singular NC measures µac and µs, 0 ≤ µac, µs ≤ µ, where qµ = qµac +qµs
is the Lebesgue decomposition of the quadratic form qµ [1, Theorem 5.9].
A third approach to Lebesgue Decompositon theory is to define a positive, finite, regular,
Borel measure µ, on ∂D to be absolutely continuous if the corresponding linear functional ˆµ
on C (∂D) = (A(D) + A(D)
∗
)
−k·k has a weak−∗ continuous extension to a linear functional on
(H∞(D) + H∞(D)
∗
)
−wk−∗ = L∞(∂D).
This notion of absolute continuity for bounded linear functionals on Ad extends the classical
notion of absolute continuity of a measure with respect to normalized Lebesgue measure on ∂D, if
one identifies finite positive Borel measures on ∂D with positive linear functionals on the classical
Disk Algebra A1 = A(D) ⊂ H∞(D). Indeed in the case where d = 1, L∞
1 = H∞(D), and
(H
∞(D) + H
∞(D)
∗
)
−weak−∗ ≃ L
∞(∂D),
a commutative von Neumann algebra. In this case, if ˆµ ∈ (A(D)
†
)+ = C (∂D)
†
+ is any positive linear
functional, the Riesz-Markov-Kakutani Theorem implies it is given by integration against a positive
finite Borel measure, µ, on ∂D, and to say it has a weak−∗ continuous extension to (H∞(D) +
5
H∞(D)
∗
)
†
+ ≃ L∞(∂D)
†
+ is equivalent to ˆµ being the restriction of a positive ˆµ ∈ L∞(∂D)∗ ≃ L
1
(∂D).
Equivalently,
µ(dζ) = µ(dζ)
m(dζ)
m(dζ); m − a.e.,
µ(dζ)
m(dζ)
∈ L
1
(∂D),
i.e. µ is absolutely continuous with respect to Lebesgue measure.
This definition of absolute continuity has an obvious generalization to the non-commutative
setting of NC measures, i.e. positive linear functionals on the Free Disk System, and this gives
essentially the same definition of absolute continuity for linear functionals on the Free Disk Algebra
introduced by Davidson-Li-Pitts [3]. We will show that all three of these approaches extend naturally to the NC setting and yield the same Lebesgue Decomposition of any positive NC measure
with respect to NC Lebesgue measure.
