A non-commutative F&M Riesz Theorem

The Riesz–Markov theorem identifies any finite and regular Borel measure on the complex unit circle with a bounded linear functional on the Banach space of continuous functions. By the Weierstrass approximation theorem, the Banach space of continuous functions on the circle is the supremum norm–closure of the linear span of the disk algebra and its conjugate algebra. The disk algebra is often defined as the unital Banach algebra of analytic functions in the complex unit disk with continuous boundary values. This is completely isometrically isomorphic to the unital norm–closed operator algebra generated by the shift operator, $S=M_{z}$, of multiplication by the independent variable $z$ on $H^{2}$. Here, $H^{2}$ denotes the Hardy space, the Hilbert space of analytic functions in the complex unit disk that have square–summable Taylor series coefficients at $0$, equipped with the $\ell^{2}-$inner product of these coefficients.

An immediate non-commutative (NC) multivariate generalization of $H^{2}$ is then $\mathbb{H}^{2}_{d}$, the NC Hardy space or full Fock space, which consists of square–summable power series in several formal NC variables $\mathfrak{z}=(\mathfrak{z}_{1},\cdots,\mathfrak{z}_{d})$. Elements of $\mathbb{H}^{2}_{d}$ are power series,

$$h(\mathfrak{z})=\sum_{\omega\in\mathbb{F}^{d}}\hat{h}_{\omega}\mathfrak{z}^{\omega},$$

with square–summable coefficients $\hat{h}_{\omega}\in\mathbb{C}$. The free monoid, $\mathbb{F}^{d}$, is the set of all words, $\omega=i_{1}\cdots i_{n}$, $1\leq i_{k}\leq d$, in the $d$ letters $\{1,\cdots,d\}$. This is a monoid with product given by concatenation of words and the unit $\varnothing$ is the empty word containing no letters. Given any $\omega=i_{1}\cdots i_{n}\in\mathbb{F}^{d}$, the free monomial $\mathfrak{z}^{\omega}$ is $\mathfrak{z}_{i_{1}}\cdots\mathfrak{z}_{i_{n}}$ and $\mathfrak{z}^{\varnothing}=1$, viewed as a constant NC function. As in the classical setting, left multiplication $L_{k}:=M^{L}_{\mathfrak{z}_{k}}$ by any of the $d$ independent NC variables defines an isometry on $\mathbb{H}^{2}_{d}$, and we call these isometries the left free shifts. These play the role of the shift in this NC Hardy space theory.

The NC analogues of the disk algebra, the continuous functions (equivalently, the disk operator system, the supremum norm–closed linear span of the disk algebra and its conjugates) and positive measures are then the free disk algebra, $\mathbb{A}_{d}:=\mathrm{Alg}\{I,L_{1},\cdots,L_{d}\}^{-\|\cdot\|}$, the free disk system,

$$\mathscr{A}_{d}:=\left(\mathbb{A}_{d}+\mathbb{A}_{d}^{*}\right)^{-\|\cdot\|},$$

and NC measures, i.e. bounded linear functionals on the free disk system. We denote the Banach space of all NC measures by $\mathscr{A}_{d}^{\dagger}$ and the cone of positive NC measures by $\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$. In [15, 14], the first two named authors constructed the Lebesgue decomposition of any positive NC measure with respect to a canonical NC Lebesgue measure and showed that the sets of absolutely continuous and singular NC measures are positive and hereditary cones, in syzygy with classical measure theory.

The isometric left free shifts $L_{k}=M^{L}_{\mathfrak{z}_{k}}$ on $\mathbb{H}^{2}_{d}$ have pairwise orthogonal ranges and it follows that the linear map $L:=\left(L_{1},\cdots,L_{d}\right):\mathbb{H}^{2}_{d}\otimes\mathbb{C}^{d}\rightarrow\mathbb{H}^{2}_{d}$ is an isometry from several copies of $\mathbb{H}^{2}_{d}$ into itself. Such an isometry is called a row isometry. By the classical Wold decomposition theorem, any isometry on Hilbert space decomposes as the direct sum of a pure isometry, i.e. an isometry unitarily equivalent to copies of the shift on $H^{2}$, and a unitary operator. There is an exact analogue of the Wold decomposition theorem for row isometries, established by G. Popescu; any row isometry decomposes as the direct sum of a pure row isometry, unitarily equivalent to several copies of the left free shift and a surjective or Cuntz row isometry, the multivariate analogue of a unitary operator [21]. The $C^{*}$-algebra $\mathscr{E}_{d}=C^{*}\{I,L_{1},\cdots,L_{d}\}$ and its quotient, $\mathscr{O}_{d}$, by the compact operators are the celebrated Cuntz–Toeplitz and Cuntz algebras [4], respectively. These are universal $C^{*}$-algebras for row isometries and are important objects in $C^{*}$-algebra theory. Any $*-$representation, $\pi$, of $\mathscr{E}_{d}$ determines and is uniquely determined by the row isometry $\Pi:=\pi(L)=(\pi(L_{1}),\cdots,\pi(L_{d}))$.

The Lebesgue–von Neumann–Wold decomposition of a single isometry on Hilbert space further splits the unitary direct summand into the direct sum of a unitary with absolutely continuous spectral measure (with respect to Lebesgue measure) and a singular unitary. In [18], M. Kennedy extended the Lebesgue–von Neumann–Wold decomposition for single isometries to row isometries. A new feature in this decomposition is that the Cuntz direct summand of any row isometry generally splits as the direct sum of three different types which we will call absolutely continuous (AC) Cuntz, von Neumann type (called singular in [18]) and dilation–type.

A Gelfand–Naimark–Segal (GNS) construction applied to the free disk algebra and any positive NC measure $\mu$ produces a Hilbert space $\mathbb{H}^{2}_{d}(\mu)$ and a row isometry $\Pi_{\mu}$ acting on $\mathbb{H}^{2}_{d}(\mu)$. Any cyclic row isometry (or $*-$representation of $\mathscr{E}_{d}$) can be obtained as the GNS row isometry of a positive NC measure, as recorded in Lemma 2.2. The role of normalized Lebesgue measure on the circle is played by the so-called vacuum state of the Fock space, $m(L^{\omega})=\langle{1},{L^{\omega}1}\rangle_{\mathbb{H}^{2}}$ where $1$ is identified with the vacuum vector $\mathfrak{z}^{\varnothing}$. We call $m$ NC Lebesgue measure. Observe that if we identify normalized Lebesgue measure, $m$, with a positive linear functional on the norm-closure of the linear span of the disk algebra and its conjugate algebra, via the Riesz–Markov Theorem, then $m(S^{k})=\langle{1},{S^{k}1}\rangle_{H^{2}}$. Thus, our definition of NC Lebesgue measure recovers normalized Lebesgue measure when $d=1$. In [15, 14], the first two named authors constructed the Lebesgue decomposition of any positive NC measure, $\mu\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$, $\mu=\mu_{ac}+\mu_{s}$, where $\mu_{ac},\mu_{s}\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ are absolutely continuous and singular, respectively, with respect to NC Lebesgue measure in the sense of [14, Corollary 8.12, Corollary 8.13]. In particular, it is shown that the sets of absolutely continuous and singular NC measures are positive cones that are hereditary in the sense that if $\mu,\lambda\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ and $\lambda$ is absolutely continuous or singular, then $\mu\leq\lambda$ implies that $\mu$ is also absolutely continuous or singular, respectively, in parallel with classical measure theory. In our NC Lebesgue decomposition, $\mu$ is absolutely continuous if and only if its GNS row isometry $\Pi_{\mu}$ is the direct sum of pure and AC Cuntz row isometries, while $\mu$ is singular if and only if $\Pi_{\mu}$ is the direct sum of von Neumann–type and dilation–type row isometries [14, Corollary 8.12, Corollary 8.13].

In this paper we further refine the NC Lebesgue decomposition of [14] by proving that the sets of dilation–type and von Neumann type NC measures (NC measures whose GNS row isometries are dilation or von Neumann type) are both positive hereditary cones in Theorem 3.18. This yields, in Corollary 3.19, a refined Kennedy–Lebesgue–von Neumann decomposition of any positive NC measure. We then apply this decomposition to obtain analogues of classical results due to Frigyes and Marcel Riesz characterizing analytic (complex) NC measures, which are those bounded linear functionals on the free disk system which annihilate $\{L^{\beta}1|\ \beta\in\mathbb{F}^{d}\backslash\{\varnothing\}\}$. In particular, in Theorem 5.2, we show that a complex NC measure is analytic if and only if its absolutely continuous, singular, dilation and von Neumann parts are each also analytic. We apply this to establish an analogue of the classical F&M Riesz theorem in Theorem 5.5. The classical theorem states that analytic measures are absolutely continuous [25], [11, Chapter 4]. We remark that the NC F&M Riesz theorem obtained here is related, but not equivalent to, one previously developed in [3, Theorem A] using different techniques. Both Theorem 5.5 of the present paper and Theorem A of [3] conclude that an analytic NC measure need not be absolutely continuous, however the results of this paper and those of [3] describe the obstruction in different ways. We discuss the relationship between these two results in Remark 5.10.

Throughout, given $h,h^{\prime}$ in a Hilbert space $\mathcal{H}$, we denote the inner product of $h$ and $h^{\prime}$ by $\langle h,h^{\prime}\rangle$, with $\langle\cdot,\cdot\rangle$ being conjugate linear in the first argument and linear in the second. We borrow notation from [15, 14]. The Fock space or NC Hardy space is

$$\mathbb{H}^{2}_{d}:=\left\{\left.h(\mathfrak{z})=\sum_{\omega\in\mathbb{F}^{d}}\hat{h}_{\omega}\mathfrak{z}^{\omega}\right|\ \hat{h}_{\omega}\in\mathbb{C},\ \sum_{\omega\in\mathbb{F}^{d}}|\hat{h}_{\omega}|^{2}<+\infty\right\},$$

equipped with the $\ell^{2}-$inner product of the power series coefficients $\hat{h}_{\omega}$. Elements of the NC Hardy space can be viewed as free non-commutative functions in the NC unit row–ball, $\mathbb{B}^{d}_{\mathbb{N}}$, of all finite–dimensional strict row contractions:

$$\mathbb{B}^{d}_{\mathbb{N}}=\bigsqcup_{n=1}^{\infty}\mathbb{B}^{d}_{n};\quad\mathbb{B}^{d}_{n}:=\left\{\left.Z=(Z_{1},\cdots,Z_{d})\in\mathbb{C}^{n\times n}\otimes\mathbb{C}^{1\times d}\right|\ ZZ^{*}=Z_{1}Z_{1}^{*}+\cdots Z_{d}Z_{d}^{*}<I_{n}\right\};$$

see, for example, [16, 15, 14, 23, 26, 1] for details.

The left free shift $L=(L_{1},\cdots,L_{d})$ is the row isometry on the full Fock space $\mathbb{H}^{2}_{d}$ whose component operators act by left multiplication by the independent variables, $L_{k}=M^{L}_{\mathfrak{z}_{k}}$. The free or NC disk algebra is $\mathbb{A}_{d}:=\mathrm{Alg}\{I,L_{1},\cdots,L_{d}\}^{-\|\cdot\|}$, the free disk system is $\mathscr{A}_{d}:=\left(\mathbb{A}_{d}+\mathbb{A}_{d}^{*}\right)^{-\|\cdot\|}$. This is a self-adjoint unital norm-closed subspace of operators, i.e. an operator system. A (complex) NC measure is a bounded linear functional on the free disk system. The set of all complex NC measures is denoted by $\mathscr{A}_{d}^{\dagger}$, and the positive NC measures by $\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$. We remark that any $\mu\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ is uniquely determined by the moments $(\mu(L^{\alpha}))_{\alpha\in\mathbb{F}^{d}}$.

Given $a\in\mathbb{A}_{d}$, we write $a=a(L)=M^{L}_{a}$ for the operator of left multiplication by $a(\mathfrak{z})$ on $\mathbb{H}^{2}_{d}$, where

$$a(\mathfrak{z})=\sum_{\omega\in\mathbb{F}^{d}}\hat{a}_{\omega}\mathfrak{z}^{\omega},$$

is the NC function determined by $a$. The partial Cesàro sums of the series for $a(L)$ converge in the strong operator topology to $a(L)=M^{L}_{a(\mathfrak{z})}$, as shown in [7, Lemma 1.1]. More generally, given a row isometry $\Pi$, we write $a\mapsto a(\Pi)$ for the unique representation of $\mathbb{A}_{d}$ satisfying $\mathfrak{z}^{\alpha}(\Pi)=\Pi^{\alpha}$ for all $\alpha\in\mathbb{F}^{d}$.

The free disk system has the semi-Dirichlet property [9]:

$$\mathbb{A}_{d}^{*}\mathbb{A}_{d}\subseteq(\mathbb{A}_{d}+\mathbb{A}_{d}^{*})^{-\|\cdot\|}=\mathscr{A}_{d}.$$

The semi-Dirichlet property enables one to apply a Gelfand–Naimark–Segal (GNS)-type construction to $(\mu,\mathbb{A}_{d})$, where $\mu\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ is any positive NC measure. One obtains a GNS–Hilbert space $\mathbb{H}^{2}_{d}(\mu)$ as the norm closure of the free disk algebra (modulo vectors of zero length) with respect to the pre-inner product

$$(a_{1},a_{2})\mapsto\mu\left(a_{1}^{*}a_{2}\right).$$

Elements of this Hilbert space are equivalence classes $a+N_{\mu}$, $a\in\mathbb{A}_{d}$, where $N_{\mu}$ is the left ideal of all those $a\in\mathbb{A}_{d}$ such that $\mu(a^{*}a)=0$. This construction provides a representation $\pi_{\mu}:\mathbb{A}_{d}\rightarrow\mathcal{L}(\mathbb{H}^{2}_{d})$, where

$$\pi_{\mu}(a)\,a^{\prime}+N_{\mu}:=aa^{\prime}+N_{\mu}.$$

This is a unital completely isometric isomorphism, so that the image of the left free shifts,

$$\Pi_{\mu}:=\left(\Pi_{\mu;1},\cdots,\Pi_{\mu;d}\right),\quad\quad\Pi_{\mu;k}:=\pi_{\mu}(L_{k}),\quad k=1,\ldots,d,$$

defines a row isometry, which we call the GNS row isometry of $\mu$, acting on the GNS space $\mathbb{H}^{2}_{d}(\mu)$. The original positive NC measure $\mu\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ then has the spacial representation

$$\mu(L^{\omega})=\langle{I+N_{\mu}},{\Pi_{\mu}^{\omega}\,I+N_{\mu}}\rangle.$$

If $\mu,\lambda\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ and $\mu\leq\lambda$, then the map

$$p+N_{\lambda}\,\mapsto\,p+N_{\mu},$$

extends by continuity to a contraction $E_{\mu,\lambda}:\mathbb{H}^{2}_{d}(\lambda)\rightarrow\mathbb{H}^{2}_{d}(\mu)$ with dense range. In this case, setting $D_{\mu,\lambda}:=E^{*}_{\mu,\lambda}E_{\mu,\lambda}$, we have

$$\mu(L^{\omega})=\langle{I+N_{\lambda}},{D_{\mu,\lambda}\,\Pi_{\lambda}^{\omega}(I+N_{\lambda})}\rangle,$$

and $D\geq 0$ can be viewed as the ‘NC Radon–Nikodym derivative’ of $\mu$ with respect to $\lambda$.

We refer to $E_{\mu,\lambda}$ as the co-embedding determined by the inequality $\mu\leq\lambda$, since its adjoint is injective. Given $\mu,\nu,\lambda\in\left(\mathscr{A}_{d}\right)^{\dagger}_{+}$ satisfying $\mu\leq\nu\leq\lambda$, it follows that

$$E_{\mu,\nu}E_{\nu,\lambda}=E_{\mu,\lambda}.$$

We remark that that $E_{\mu,\lambda}^{*}$ is unitarily equivalent to an embedding of NC reproducing kernel Hilbert spaces; see [15, Lemma 3][14].

We now record the fact that any cyclic row isometry is unitarily equivalent to the GNS row isometry of a positive NC measure. The proof is straightforward and thus omitted.

Let $\mathfrak{L}^{\infty}_{d}:=\mathrm{Alg}\{I,L_{1},\cdots,L_{d}\}^{-\text{weak-}*}$ denote the left free analytic Toeplitz algebra or the Free Hardy Algebra. From a result of Davidson–Pitts [7, Corollary 2.12], it follows that $\mathfrak{L}^{\infty}_{d}=\mathrm{Alg}\{I,L_{1},\cdots,L_{d}\}^{-\text{WOT}}$, the closure of $\mathrm{Alg}\{I,L_{1},\cdots,L_{d}\}$ in the weak operator topology (WOT). That is, $\mathfrak{L}^{\infty}_{d}$ is a free semigroup algebra, the unital WOT–closed operator algebra generated by a row isometry [5]. The algebra $\mathfrak{L}^{\infty}_{d}$ can also be identified with the left multiplier algebra of $\mathbb{H}^{2}_{d}$, viewed as a non-commutative reproducing kernel Hilbert space (RKHS) [26, 16, 1]. We remark that this left multiplier algebra is equal to the unital Banach algebra $\mathbb{H}^{\infty}_{d}$ of all free NC functions in the NC unit row-ball $\mathbb{B}^{d}_{\mathbb{N}}$ that are uniformly bounded in supremum norm [26, 23]. The NC or free Toeplitz system is

$$\mathscr{T}_{d}:=\left(\mathfrak{L}^{\infty}_{d}+(\mathfrak{L}^{\infty}_{d})^{*}\right)^{-\text{weak-}*}=\mathscr{A}_{d}^{-\text{weak-}*}.$$

We also use the right free shift $R=(R_{1},\cdots,R_{d})$, the row isometry of right multiplications $R_{k}:=M^{R}_{\mathfrak{z}_{k}}$ by the independent NC variables on the Fock space. The right free analtyic Toeplitz algebra is $\mathfrak{R}^{\infty}_{d}:=\mathrm{Alg}\{I,R_{1},\cdots,R_{d}\}^{-\mathrm{WOT}}$.

If $0\leq\mu\leq\lambda$ are positive NC measures, it is natural to ask whether the contractive co-embedding $E_{\mu,\lambda}:\mathbb{H}^{2}_{d}(\lambda)\to\mathbb{H}^{2}_{d}(\mu)$ intertwines the various structure projections of $\mu$ and $\lambda$. That is, do we generally have that $E_{\mu,\lambda}P_{\lambda;\mathfrak{t}}=P_{\mu;\mathfrak{t}}E_{\mu,\lambda}$, where $\mathfrak{t}\in\mathrm{Types}$? By [14, Corollary 8.11] the sets of absolutely continuous (AC) and singular positive NC measures are positive hereditary cones. It is therefore also natural to ask whether the sets of von Neumann type and dilation type NC measures are also positive hereditary cones, as we prove in Theorem 3.18 at the end of this section.