1 Introduction
A family
$m=(m_{ij})_{i,j \in \mathbb {N}}$
of complex numbers is called a Schur multiplier if for any matrix
$[a_{ij}] \in \mathcal {B}(\ell _2)$
, the Schur product
$T_m(a) = [m_{ij}a_{ij}]$
is the matrix of an element of
$\mathcal {B}(\ell _2)$
. Schur multipliers are an important tool in analysis, and play for instance a fundamental role in Perturbation Theory. See below for more information and references.
There is a well-known characterization of Schur multipliers due to Grothendieck in terms of factorization of the symbol m, see [Reference Pisier18, Theorem 5.1]. It turns out, using the theory of operator spaces, that bounded Schur multipliers are completely bounded and in that case, the norm of
$T_m$
is equal to its complete norm. It is not yet known whether this is true for Schur multipliers defined on the Schatten classes. We refer the reader to [Reference Caspers and Wildschut12] for recent developments regarding this question.
In this paper, we are interested in Schur multipliers in the multilinear setting. Effros and Ruan [Reference Effros and Ruan11] introduced a Schur product as a multilinear map
$T \colon M_n(\mathbb {C}) \times \cdots \times M_n(\mathbb {C}) \rightarrow M_n(\mathbb {C})$
defined on the product of n copies of
$M_n(\mathbb {C})$
and characterized the mappings T that extend to a complete contraction on the Haagerup tensor product
$M_n(\mathbb {C}) \overset {h}{\otimes } \cdots \overset {h}{\otimes } M_n(\mathbb {C})$
. This result was generalized by Juschenko, Todorov and Turowska in [Reference Juschenko, Todorov and Turowska13] where they considered measurable multilinear Schur multipliers. They are defined as follows: let
$n\in \mathbb {N}$
and let
$(\Omega _1, \mu _1), \ldots , (\Omega _n, \mu _n)$
be
$\sigma $
-finite measure spaces. Let
$\phi \in L^{\infty }(\Omega _1 \times \cdots \times \Omega _n)$
. We will identify
$L^2(\Omega _i\times \Omega _j)$
with the space
$\mathcal {S}^2(L^2(\Omega _i), L^2(\Omega _j))$
of Hilbert-Schmidt operators from
$L^2(\Omega _i)$
into
$L^2(\Omega _j)$
. If
$K_i \in L^2(\Omega _i\times \Omega _{i+1})$
,
$1\leq i \leq n-1$
, we let
$\Lambda (\phi )(K_1,\ldots , K_{n-1})$
to be the Hilbert-Schmidt operator with kernel
$$ \begin{align*} \int \phi(t_1,\ldots,t_n)K_1(t_1,t_2)\ldots K_{n-1}(t_{n-1},t_n) \,\text{d}\mu_2(t_2) \ldots \text{d}\mu_{n-1}(t_{n-1}) \in L^2(\Omega_1 \times \Omega_n), \end{align*} $$
which defines a multilinear mapping
$$ \begin{align*} \Lambda(\phi) \colon \mathcal{S}^2(L^2(\Omega_{n-1}), L^2(\Omega_n)) \times \cdots \times \mathcal{S}^2(L^2(\Omega_1), L^2(\Omega_2)) \rightarrow S^2(L^2(\Omega_1), L^2(\Omega_n)). \end{align*} $$
Using the notion of multilinear module mappings, the authors proved that if
$\Lambda (\phi )$
extends to a bounded map on the Haagerup tensor product
$\mathcal {S}^{\infty }(L^2(\Omega _{n-1}), L^2(\Omega _n)) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(L^2(\Omega _1), L^2(\Omega _2))$
into
$\mathcal {S}^{\infty }(L^2(\Omega _1), L^2(\Omega _n))$
, the extension is completely bounded [Reference Juschenko, Todorov and Turowska13, Lemma 3.3]. Using this fact, they characterized the functions
$\phi $
that give rise to a (completely) bounded
$\Lambda (\phi )$
in terms of the extended Haagerup tensor product
$L^{\infty }(\Omega _1) \otimes _{eh} \cdots \otimes _{eh} L^{\infty }(\Omega _n)$
, see [Reference Juschenko, Todorov and Turowska13, Theorem 3.4] and the remark following the theorem. We also refer the reader to [Reference Spronk21] for more results on the case
$n=2$
.
Let
$A_1, \ldots , A_n$
be normal operators and let
$\lambda _{A_1}, \ldots , \lambda _{A_n}$
be scalar-valued spectral measures associated with these operators, that is,
$\lambda _{A_i}$
is a finite measure on the Borel subsets of
$\sigma (A_i)$
such that
$\lambda _{A_i}$
and
$E^{A_i}$
, the spectral measure of
$A_i$
, have the same sets of measure
$0$
. For
$\phi \in L^{\infty }(\lambda _{A_1} \times \cdots \times \lambda _{A_n})$
and
$X_1, \ldots , X_{n-1} \in \mathcal {S}^2(\mathcal {H})$
, we formally define a multiple operator integral by
$$ \begin{align*} & \left[\Gamma^{A_1, \ldots, A_n}(\phi)\right](X_1, \ldots, X_{n-1})\\ & \ \ \ \ \ \ \ \ = \int_{\sigma(A_1)\times \cdots \times \sigma(A_n)} \phi(s_1, \ldots, s_n)\, \text{d}E^{A_1}(s_1)\, X_1\, \text{d}E^{A_2}(s_2) \ldots X_{n-1} \, \text{d}E^{A_n}(s_n). \end{align*} $$
The theory of double operator integral (case
$n=2$
) was developed by Birman and Solomyak in a series of three papers [Reference Birman and Solomyak1, Reference Birman and Solomyak2, Reference Birman and Solomyak3] and was then generalized to the case of multiple operator integrals [Reference Pavlov15, Reference Stenkin22]. They play a prominent role in operator theory, especially in perturbation theory where they are a fundamental tool in the study of differentiability of operator functions. See [Reference Coine, Le Merdy, Sukochev and Skripka6, Reference Coine7, Reference Le Merdy and Skripka14, Reference Peller16] where Fréchet and Gâteaux-differentiability of the mapping
$f \mapsto f(A)$
are studied in the Schatten norms.
The definition of multiple operator integrals we will use in this paper is the one given in [Reference Coine, Le Merdy and Sukochev5] and which is based on the construction of Pavlov [Reference Pavlov15]. See [Reference Peller16, Reference Potapov, Skripka and Sukochev19] for other constructions of multiple operator integrals. The advantage of this definition is the property of
$w^*$
-continuity of the mapping
$\phi \mapsto \Gamma ^{A_1, \ldots , A_n}(\phi )$
which allows to prove certain identities by simply checking them for functions with separated variables, see [Reference Coine, Le Merdy and Sukochev5, Reference Coine, Le Merdy, Sukochev and Skripka6] and the proof of Theorem 4.1.
In this paper, we prove that a characterization similar to the one established for measurable Schur multipliers in [Reference Juschenko, Todorov and Turowska13] holds in the setting of multiple operator integrals. Namely, we prove that if a multiple operator integral
$\Gamma ^{A_1, \ldots , A_n}$
extends to a bounded mapping on the Haagerup tensor product
$\mathcal {S}^{\infty }(\mathcal {H}) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(\mathcal {H})$
then the extension is completely bounded and that we have such extension if and only if
$\phi $
has the following factorization: there exist separable Hilbert spaces
$H_1, \ldots , H_{n-1}$
,
$a_1\in L^{\infty }(\lambda _{A_1} ; H_1), a_n \in L^{\infty }(\lambda _{A_n} ; H_{n-1})$
and
$a_i\in L_{\sigma }^{\infty }(\lambda _{A_i} ; \mathcal {B}(H_i, H_{i-1})), 2\leq i \leq n-1$
such that
$$ \begin{align*} \phi(t_1,\ldots,t_n)= \left\langle a_1(t_1), [a_2(t_2)\ldots a_{n-1}(t_{n-1})](a_n(t_n)) \right\rangle. \end{align*} $$
Our proof rests on several properties of the Haagerup tensor product (Section 2.1) and the connection between multiple operator integrals and measurable multilinear Schur multipliers that we will present in Section 3.
2 Preliminaries
2.1 Operator spaces and the Haagerup tensor product
We refer to [Reference Pisier17] and [Reference Effros and Ruan20] for the theory of operator spaces. If
$E \subset \mathcal {B}(H)$
and
$F \subset \mathcal {B}(K)$
are two operator spaces, we denote by
$CB(E,F)$
the Banach space of completely bounded maps from E into F equipped with the c.b. norm. If
$\mathcal {H}$
is a Hilbert space, we will denote by
$\mathcal {H}_c = \mathcal {B}(\mathbb {C}, \mathcal {H})$
its column structure.
In [Reference Pisier17, Chapter 5], Pisier defines the Haagerup tensor product
$E_1 \overset {h}{\otimes } \cdots \overset {h}{\otimes } E_N$
of N operator spaces
$E_1, \ldots , E_N$
. We will recall a few properties of the Haagerup tensor product that we will use in Section 4. The first one is the factorization of multilinear maps. If E is an operator space, then by [Reference Effros and Ruan20, Proposition 9.2.2], a multilinear mapping
$v : E_1 \times \cdots \times E_n \to E$
is completely bounded (in the sense of [Reference Effros and Ruan20, Section 9.1], see also [Reference Blecher and Le Merdy4]) if and only if v extends to a completely bounded map
$v : E_1 \overset {h}{\otimes } \cdots \overset {h}{\otimes } E_N \to E$
. The following important theorem describes those maps.
Theorem 2.1 Let
$E_1, \ldots , E_n$
be operator spaces and let
$H_0$
and
$H_n$
be Hilbert spaces. A linear mapping
$u \colon E_1 \overset {h}{\otimes } \cdots \overset {h}{\otimes } E_n \rightarrow \mathcal {B}(H_n, H_0)$
is completely bounded if and only if there exist Hilbert spaces
$H_1, \ldots , H_{n-1}$
and completely bounded mappings
$\phi _i \colon E_i \rightarrow \mathcal {B}(H_i, H_{i-1}), 1\leq i \leq n,$
such that
$$ \begin{align*}u(x_1 \otimes \cdots \otimes x_n) = \phi_1(x_1) \ldots \phi_n(x_n).\end{align*} $$
In this case we can choose
$\phi _i, 1\leq i \leq n$
, such that
$$ \begin{align*}\|u\|_{\text{cb}} = \|\phi_1\|_{\text{cb}} \cdots \|\phi_n\|_{\text{cb}}.\end{align*} $$
Remark 2.2 When
$H_0=H_n=\mathbb {C}$
we can reformulate as follows: a linear functional
$u \colon E_1 \overset {h}{\otimes } \cdots \overset {h}{\otimes } E_n \rightarrow \mathbb {C}$
is bounded (and therefore completely bounded) if and only if there exist Hilbert spaces
$H_1, \ldots , H_{n-1}$
,
$\alpha _1 \colon E_1 \rightarrow (H_c)^*$
linear,
$\alpha _i \colon E_i \rightarrow \mathcal {B}(H_i, H_{i-1}), 2\leq i\leq n-1$
and
$\alpha _n \colon E_n \rightarrow (H_{n-1})_c$
antilinear such that the
$\alpha _j$
are completely bounded and
$$ \begin{align*}u(x_1, \ldots, x_n) = \left\langle \alpha_1(x_1), [\alpha_2(x_2) \ldots \alpha_{n-1}(x_{n-1})]\alpha_n(x_n) \right\rangle.\end{align*} $$
Recall that a map
$s \colon X \rightarrow Y$
between two Banach spaces is called a quotient map if the injective map
$\hat {s} \colon X/\ker (s) \rightarrow Y$
induced by s is a surjective isometry. If
$E_1 \subset E_2$
are operator spaces, we equip
$E_2/E_1$
with the quotient operator space structure (see e.g. [Reference Pisier17, Section 2.4]). When E and F are operator spaces, a quotient map
$u \colon E \rightarrow F$
is said to be a complete metric surjection if the associated mapping
$\hat {u} \colon E/\ker (u) \rightarrow F$
is a completely isometric isomorphism.
Proposition 2.3 Let
$E_1, E_2, F_1, F_2$
be operator spaces.
-
(i) If
$q_i \colon E_i \rightarrow F_i$
is completely bounded, then
$q_1 \otimes q_2 \colon E_1 \otimes E_2 \rightarrow F_1 \overset {h}{\otimes } F_2$
defined by
$(q_1 \otimes q_2)(e_1 \otimes e_2) = q_1(e_1) \otimes q_2(e_2)$
extends to a completely bounded map
$$ \begin{align*}q_1 \otimes q_2 \colon E_1 \overset{h}{\otimes} E_2 \rightarrow F_1 \overset{h}{\otimes} F_2.\end{align*} $$
-
(ii) If
$E_i \subset F_i$
completely isometrically, then
$E_1 \overset {h}{\otimes } E_2 \subset F_1 \overset {h}{\otimes } F_2$
completely isometrically. -
(iii) If
$q_i \colon E_i \rightarrow F_i$
is a complete metric surjection, then
$q_1 \otimes q_2 \colon E_1 \overset {h}{\otimes } E_2 \rightarrow F_1 \overset {h}{\otimes } F_2.$
-
(iv) If
$E_i \subset F_i$
are subspaces, let
$p_i \colon F_i \rightarrow F_i/E_i$
be the canonical mappings. Then the induced map
$p_1 \otimes p_2 \colon F_1 \overset {h}{\otimes } F_2 \rightarrow F_1/E_1 \overset {h}{\otimes } F_2/E_2$
satisfies
$$ \begin{align*}\ker (p_1\otimes p_2) = \overline{E_1 \otimes F_2 + F_1 \otimes E_2}.\end{align*} $$
The second property is called the injectivity and the third one the projectivity of the Haagerup tensor product.
Proof We refer to [Reference Effros and Ruan20, Proposition 9.2.5] for the proof of
$(i)$
and to [Reference Pisier17, Corollary 5.7] for the proof of
$(ii)$
and
$(iii)$
.
Let us prove
$(iv)$
. Write
$N = \overline {E_1 \otimes F_2 + F_1 \otimes E_2}$
. Note that the inclusion
$N \,{\subset}\, \ker (p_1 \otimes p_2).$
is clear. Therefore, to show the result, it is enough to show that
$$ \begin{align*}N^{\perp} \subset \ker (p_1 \otimes p_2)^{\perp}.\end{align*} $$
Let
$\sigma \colon F_1 \overset {h}{\otimes } F_2 \rightarrow \mathbb {C}$
be such that
$\sigma _{|N}=0$
. By Remark 2.2, there exist a Hilbert space H,
$\alpha \colon F_1 \rightarrow (H_c)^*$
linear and
$\beta \colon F_2 \rightarrow H_c$
antilinear,
$\alpha $
and
$\beta $
completely bounded such that
$$ \begin{align*}\sigma(x, y) = \left\langle \alpha(x), \beta(y) \right\rangle, x\in F_1, y\in F_2.\end{align*} $$
Let
$K = \overline {\alpha (F_1)}$
and denote by
$P_K$
the orthogonal projection onto K. Then we have, for any x and y,
$$ \begin{align*}\sigma(x,y) = \left\langle P_K \alpha(x), \beta(y) \right\rangle = \left\langle P_K \alpha(x), P_K \beta(y) \right\rangle.\end{align*} $$
Thus, by changing
$\alpha $
into
$P_K \alpha $
and
$\beta $
into
$P_K \beta $
, we can assume that
$\alpha $
has a dense range. Similarly, setting
$L = \overline {\beta (F_2)}$
and considering
$P_L$
, we may assume that
$\beta $
has a dense range.
By assumption, for any
$e\in E_2$
and any
$x\in E_1$
, we have
$$ \begin{align*}0= \sigma(x,e) = \left\langle \alpha(x), \beta(e) \right\rangle.\end{align*} $$
This implies that
$\beta _{|E_2}=0$
. Similarly, we show that
$\alpha _{|E_1}=0$
. Thus, we can consider
$$ \begin{align*}\widehat{\alpha} \colon F_1/E_1 \rightarrow H \quad \text{and} \quad\widehat{\beta} \colon F_2/E_2 \rightarrow H\end{align*} $$
such that
$\alpha = \widehat {\alpha } \circ p_1$
and
$\beta = \widehat {\beta } \circ p_2$
and where
$F_1/E_1$
and
$F_2/E_2$
are equipped with their quotient structure. Now, define
$\widehat {\sigma } \colon F_1/E_1 \overset {h}{\otimes } F_2/E_2 \rightarrow \mathbb {C}$
by
$$ \begin{align*}\widehat{\sigma}(s,t) = \left\langle \widehat{\alpha}(s), \widehat{\beta}(t) \right\rangle.\end{align*} $$
Then
$\sigma = \widehat {\sigma } \circ (p_1 \otimes p_2)$
, so that
$\sigma \in \ker (p_1\otimes p_2)^{\perp }$
. ▪
Finally, we recall the following [Reference Effros and Ruan20, Proposition 9.3.3] which will be important in the last section.
Proposition 2.4 Let E be an operator space and let
$\mathcal {H}$
and
$\mathcal {K}$
be Hilbert spaces. For any
$T\in CB(E, \mathcal {B}(\mathcal {H}, \mathcal {K}))$
we define a mapping
$\sigma _T \colon \mathcal {K}^* \otimes E \otimes \mathcal {H} \rightarrow \mathbb {C}$
by setting
$$ \begin{align*}\sigma_T(k^* \otimes e \otimes h) = \left\langle T(e)h, k \right\rangle.\end{align*} $$
Then, the mapping
$T \mapsto \sigma _T$
induces a complete isometry
$$ \begin{align*}CB(E, \mathcal{B}(\mathcal{H}, \mathcal{K})) = \Big( (\mathcal{K}_c)^* \overset{h}{\otimes} E \overset{h}{\otimes} \mathcal{H}_c \Big)^*.\end{align*} $$
2.2 Schatten classes
Let
$\mathcal {H}$
and
$\mathcal {K}$
be separable Hilbert spaces. For any
$1\leq p < +\infty $
, let
$\mathcal {S}^p(\mathcal {H}, \mathcal {K})$
be the space of compact operators
$T \colon \mathcal {H} \rightarrow \mathcal {K}$
such that
$$ \begin{align*} \|T\|_p \colon= \text{tr}(|T|^p)^{\frac{1}{p}} < \infty. \end{align*} $$
Then
$\|\cdot \|_p$
is a norm on
$\mathcal {S}^p(\mathcal {H}, \mathcal {K})$
and
$(\mathcal {S}^p(\mathcal {H}, \mathcal {K}), \|\cdot \|_p)$
is called the Schatten class of order p. When
$p=\infty $
, the space
$\mathcal {S}^{\infty }(\mathcal {H}, \mathcal {K})$
will denote the space of compact operators equipped with the operator norm.
Recall that
$\left ( \mathcal {S}^1(\mathcal {H}, \mathcal {K}) \right )^* = \mathcal {B}(\mathcal {K}, \mathcal {H})$
and that for
$1<p\leq +\infty $
,
$\left ( \mathcal {S}^p(\mathcal {H}, \mathcal {K}) \right )^* = \mathcal {S}^{p'}(\mathcal {K}, \mathcal {H})$
where
$p'$
is the conjugate exponent of p, for the duality pairing
$$ \begin{align*} \left\langle S,T \right\rangle = \text{tr}(ST), \ \ S \in \mathcal{S}^p(\mathcal{H}, \mathcal{K}), T\in \mathcal{S}^{p'}(\mathcal{K}, \mathcal{H}). \end{align*} $$
Using the Haagerup tensor product introduced in Subsection 2.1, we have, by [Reference Effros and Ruan20, Proposition 9.3.4], a complete isometry
$$ \begin{align} (\mathcal{H}_c)^* \overset{h}{\otimes} \mathcal{K}_c = \mathcal{S}^1(\mathcal{H}, \mathcal{K}). \end{align} $$
where
$\mathcal {S}^1(\mathcal {H}, \mathcal {K})$
is equipped with its operator space structure as the predual of
$\mathcal {B}(\mathcal {K}, \mathcal {H})$
.
Similarly, we have a complete isometry
$$ \begin{align} \mathcal{K}_c \overset{h}{\otimes} (\mathcal{H}_c)^* = \mathcal{S}^{\infty}(\mathcal{H}, \mathcal{K}). \end{align} $$
Finally, if
$(\Omega _1, \mu _1)$
and
$(\Omega _2, \mu _2)$
are two
$\sigma $
-finite measure spaces, we will identify
$L^2(\Omega _1 \times \Omega _2)$
with the space
$\mathcal {S}^2(L^2(\Omega _1), L^2(\Omega _2))$
of Hilbert-Schmidt operators as follows. If
$K\in L^2(\Omega _1 \times \Omega _2)$
, the operator
$$ \begin{align} \begin{array}[t]{l@{\ }c@{\ }c@{\ }c} X_K \colon & L^2(\Omega_1) & \longrightarrow & L^2(\Omega_2) \\ & f & \longmapsto & \displaystyle \int_{\Omega_1} K(t,\cdot)f(t) \mathrm{d}\mu_1(t) \end{array} \end{align} $$
is a Hilbert-Schmidt operator and
$\|X_K\|_2=\|K\|_{L^2}$
. Moreover, any element of
$\mathcal {S}^2(L^2(\Omega _1), L^2(\Omega _2))$
has this form. Hence, the space
$L^2(\Omega _1 \times \Omega _2)$
is isometrically isomorphic to
$\mathcal {S}^2(L^2(\Omega _1), L^2(\Omega _2))$
through the mapping
$K \mapsto X_K$
.
2.3
$L_{\sigma }^p$
-spaces and Duality
Let
$(\Omega , \mu )$
be a
$\sigma $
-finite measure space and let F be a Banach space. For any
$1\leq p \leq +\infty $
, we let
$L^p(\Omega ;F)$
denote the classical Bochner space of measurable functions
$f \colon \Omega \to F$
.
Assume that E is a separable Banach space. A function
$f \colon \Omega \rightarrow E^*$
is said to be
$w^*$
-measurable if for all
$ e\in E$
, the function
$t \in \Omega \mapsto \langle \phi (t), e \rangle $
is measurable. We denote by
$L^p_{\sigma }(\Omega ;E^*)$
the space of all
$w^*$
-measurable
$f \colon \Omega \rightarrow E^*$
such that
$\|f(\cdot )\| \in L^p(\Omega )$
, after taking quotient by the functions which are equal to
$0$
almost everywhere. Equipped with the norm
$$ \begin{align*} \| f\|_p = \| \|f(.)\| \|_{L^p(\Omega)}, \end{align*} $$
$(L^p_{\sigma }(\Omega ; E^*), \|.\|_p)$
is a Banach space.
Let
$1\leq p' < +\infty $
be the conjugate exponent of p. Then we have an isometric isomorphism
$$ \begin{align*} L^p(\Omega;E)^* = L^{p'}_{\sigma}(\Omega; E^*) \end{align*} $$
through the duality pairing
$$ \begin{align*} \langle f, g \rangle \colon= \int_{\Omega} \langle f(t), g(t) \rangle \,\text{d}\mu(t)\,. \end{align*} $$
See [Reference Coine, Le Merdy and Sukochev5, Section 4] and the references therein for a proof of that result and more information about
$L_{\sigma }^p$
-spaces.
Note that by [Reference Diestel and Uhl9, Chapter IV], the equality
$L^p_\sigma (\Omega ; E^*)=L^p(\Omega ; E^*)$
is equivalent to
$E^*$
having the Radon-Nikodym property. It is for instance the case for Hilbert spaces.
The important identification we will need in this paper is the following. For any
$f\in L^\infty _\sigma (\Omega ; E^*)$
, define
$$ \begin{align} u_f \colon \psi \in L^1(\Omega) \mapsto \left[ e\in E \mapsto \int_{\Omega} \left\langle f(t),e \right\rangle \psi(t) \,\text{d}t \right] \in E^*. \end{align} $$
Then
$f \mapsto u_f$
yields an isometric identification (see [Reference Dunford and Pettis10, Theorem 2.1.6])
$$ \begin{align} L^\infty_\sigma(\Omega; E^*) = \mathcal{B}(L^1(\Omega),E^*). \end{align} $$
In particular, for a Hilbert space
$\mathcal {H}$
we have the equality
$$ \begin{align} L^{\infty}(\Omega; \mathcal{H}) = \mathcal{B}(L^1(\Omega), \mathcal{H}). \end{align} $$
3 Multiple Operator Integrals
3.1 Multiple Operator Integrals Associated with Operators
Let
$\mathcal {H}$
be a separable Hilbert space and let A be a (possibly unbounded) normal operator on
$\mathcal {H}$
. We denote by
$\sigma (A)$
the spectrum of A and by
$E^A$
its spectral measure. A scalar-valued spectral measure for A is a positive measure
$\lambda _A$
on the Borel subsets of
$\sigma (A)$
such that
$\lambda _A$
and
$E^A$
have the same sets of measure zero. Let e be a separating vector of the von Neumann algebra
$W^*(A)$
generated by A (see [Reference Conway8, Corollary 14.6]).
Then, by [Reference Conway8, Proposition 15.3], the measure
$\lambda _A$
defined by
$$ \begin{align*} \lambda_A = \|E^A(.)e\|^2 \end{align*} $$
is a scalar-valued spectral measure for A. We refer the reader to [Reference Conway8, Section 15] and [Reference Coine, Le Merdy and Sukochev5, Section 2.1] for more details.
For any bounded Borel function
$f \colon \sigma (A) \to \mathbb {C}$
, we define
$f(A) \in \mathcal {B}(\mathcal {H})$
by
$$ \begin{align*} f(A):=\int_{\sigma(A)} f(t) \ \text{d}E^A(t), \end{align*} $$
and this operator only depends on the class of f in
$L^{\infty }(\lambda _A)$
. According to [Reference Conway8, Theorem 15.10], we obtain a
$w^*$
-continuous
$*$
-representation
$$ \begin{align*} f \in L^{\infty}(\lambda_A) \mapsto f(A) \in \mathcal{B}(\mathcal{H}). \end{align*} $$
Moreover, the space
$L^{\infty }(\lambda _A)$
does not depend on the choice of the scalar-valued spectral measure.
Let
$n\in \mathbb {N}, n\geq 1$
and let
$E_1, \ldots , E_n, E$
be Banach spaces. We denote by
$\mathcal {B}_n(E_1 \times \cdots \times E_n, E)$
the space of n-linear continuous mappings from
$E_1 \times \cdots \times E_n$
into E equipped with the norm
$$ \begin{align*} \|T\|_{\mathcal{B}_n(E_1 \times \cdots \times E_n, E)} := \mathop{\mathrm{sup}}\limits_{\|e_i\| \leq 1, 1\leq i \leq n} \ \|T(e_1, \ldots, e_n)\|. \end{align*} $$
When
$E_1 = \cdots = E_n = E$
, we will simply write
$\mathcal {B}_n(E)$
.
Let
$n\in \mathbb {N}, n\geq 2$
and let
$A_1, A_2, \ldots , A_n$
be normal operators in
$\mathcal {H}$
with scalar-valued spectral measures
$\lambda _{A_1}, \ldots , \lambda _{A_n}$
. We let
$$ \begin{align*} \Gamma^{A_1,A_2, \ldots, A_n} \colon L^{\infty}(\lambda_{A_1}) \otimes \cdots \otimes L^{\infty}(\lambda_{A_n}) \rightarrow \mathcal{B}_{n-1}(\mathcal{S}^2(\mathcal{H})) \end{align*} $$
be the unique linear map such that for any
$f_i \in L^{\infty }(\lambda _{A_i}), i=1, \ldots , n$
and for any
$X_1, \ldots , X_{n-1} \in \mathcal {S}^2(\mathcal {H})$
,
$$ \begin{align*} &\left[\Gamma^{A_1,A_2, \ldots, A_n}(f_1\otimes\cdots\otimes f_n)\right] (X_1,\ldots, X_{n-1})\\ \nonumber &\quad =f_1(A_1)X_1f_2(A_2) \cdots f_{n-1}(A_{n-1})X_{n-1}f_n(A_n). \end{align*} $$
We have a natural inclusion
$L^{\infty }(\lambda _{A_1}) \otimes \cdots \otimes L^{\infty }(\lambda _{A_n}) \subset L^{\infty }\left (\prod _{i=1}^n \lambda _{A_i}\right )$
which is
$w^*$
-dense. The following shows that
$\Gamma ^{A_1,A_2, \ldots , A_n}$
extends to
$L^{\infty }\left (\prod _{i=1}^n \lambda _{A_i}\right )$
. It was proved in [Reference Coine, Le Merdy and Sukochev5, Theorem 4 and Proposition 5].
Theorem 3.1
$\Gamma ^{A_1,A_2, \ldots , A_n}$
extends to a unique
$w^*$
-continuous isometry still denoted by
$$ \begin{align*} \Gamma^{A_1,A_2, \ldots, A_n} \colon L^{\infty}\left(\prod\limits_{i=1}^n \lambda_{A_i}\right) \longrightarrow \mathcal{B}_{n-1}(\mathcal{S}^2(\mathcal{H})). \end{align*} $$
Definition 3.1 For
$\phi \in L^{\infty }\left (\prod _{i=1}^n\lambda _{A_i}\right )$
, the transformation
$\Gamma ^{A_1,A_2, \ldots , A_n}(\phi )$
is called a multiple operator integral associated with
$A_1, A_2, \ldots , A_n$
and
$\phi $
.
The
$w^*$
-continuity of
$\Gamma ^{A_1,A_2, \ldots , A_n}$
means that if a net
$(\phi _i)_{i\in I}$
in
$L^{\infty }\left (\prod _{i=1}^n \lambda _{A_i}\right )$
converges to
$\phi \in L^{\infty }\left (\prod _{i=1}^n \lambda _{A_i}\right )$
in the
$w^*$
-topology, then for any
$X_1, \ldots , X_{n-1} \in \mathcal {S}^2(\mathcal {H})$
, the net
$$ \begin{align*} \bigl(\left[\Gamma^{A_1,A_2, \ldots, A_n}(\phi_i)\right](X_1,\ldots, X_{n-1})\bigr)_{i\in I} \end{align*} $$
converges to
$\left [\Gamma ^{A_1,A_2, \ldots , A_n}(\phi )\right ](X_1,\ldots , X_{n-1})$
weakly in
$\mathcal {S}^2(\mathcal {H})$
. We refer the reader to [Reference Coine, Le Merdy and Sukochev5, Section 3.1] for more details.
3.2 Measurable Multilinear Schur Multipliers
Let
$n\in \mathbb {N}$
. Let
$(\Omega _1, \mu _1), \ldots , (\Omega _n, \mu _n)$
be
$\sigma $
-finite measure spaces, and let
$\phi \in L^{\infty }(\Omega _1 \times \cdots \times \Omega _n)$
. Let
$\Omega = \Omega _2 \times \cdots \times \Omega _{n-1}$
. For any
$K_i \in L^2(\Omega _i\times \Omega _{i+1})$
,
$1\leq i \leq n-1$
, we let
$\Lambda (\phi )(K_1,\ldots , K_{n-1})$
be the function
$$ \begin{align*} (t_1,t_n) \mapsto \int_{\Omega} \phi(t_1,\ldots,t_n)K_1(t_1,t_2)\ldots K_{n-1}(t_{n-1},t_n) \,\text{d}\mu_2(t_2) \ldots \text{d}\mu_{n-1}(t_{n-1}) \end{align*} $$
By Cauchy-Schwarz inequality,
$\Lambda (\phi )(K_1,\ldots , K_{n-1}) \in L^2(\Omega _1 \times \Omega _n)$
and
$$ \begin{align} \| \Lambda(\phi)(K_1,\ldots, K_{n-1}) \|_2 \leq \|\phi\|_{\infty}\|K_1\|_2 \ldots \|K_{n_1}\|_2. \end{align} $$
Thus,
$\Lambda (\phi )$
defines a bounded
$(n-1)$
-linear map
$$ \begin{align*}\Lambda(\phi) \colon L^2(\Omega_1 \times \Omega_2) \times L^2(\Omega_2\times \Omega_3) \times \cdots \times L^2(\Omega_{n-1} \times \Omega_n) \longrightarrow L^2(\Omega_1 \times \Omega_n),\end{align*} $$
or, equivalently, by (2.3) and the obvious equality
$L^2(\Omega _i \times \Omega _j) = L^2(\Omega _j \times \Omega _i), 1\leq i,j \leq n,$
a bounded
$(n-1)$
-linear map
$$ \begin{align*} \Lambda(\phi) \colon \mathcal{S}^2(L^2(\Omega_2), L^2(\Omega_1)) \times \cdots \times \mathcal{S}^2(L^2(\Omega_n), L^2(\Omega_{n-1})) \rightarrow \mathcal{S}^2(L^2(\Omega_n), L^2(\Omega_1)). \end{align*} $$
For simplicity, write
$E_i = L^2(\Omega _i), 1\leq i \leq n$
. Then, the map
$\Lambda \colon \phi \mapsto \Lambda (\phi )$
is a linear isometry
$$ \begin{align*} \Lambda \colon L^{\infty}(\Omega_1 \times\cdots \times \Omega_n) \longrightarrow B_{n-1}(\mathcal{S}^2(E_2, E_1) \times \cdots \times \mathcal{S}^2(E_n, E_{n-1}), \mathcal{S}^2(E_n, E_1)). \end{align*} $$
This follow, for example, from similar computations as those in the proof of [Reference Coine, Le Merdy and Sukochev5, Proposition 8] or from [Reference Juschenko, Todorov and Turowska13, Theorem 3.1].
Let
$\mathcal {H}$
be a separable Hilbert space and let
$A_1, \ldots , A_n$
be normal operators on
$\mathcal {H}$
. For any
$1\leq i \leq n$
, let
$e_i \in \mathcal {H}$
be such that
$$ \begin{align*} \lambda_{A_i}(\cdot)=\|E^{A^i}(\cdot)e_i\|^2. \end{align*} $$
By [Reference Coine, Le Merdy and Sukochev5, Subsection 4.2], the linear mappings
$\rho _i \colon L^2(\sigma (A_i), \lambda _{A_i}) \to \mathcal {H}$
defined for any measurable subset
$F\subset \sigma (A_i)$
by
$$ \begin{align*} \rho_i(\chi_F) = E^{A_i}(F)e_i \end{align*} $$
extends uniquely to an isometry
$\rho _i \colon L^2(\sigma (A_i), \lambda _{A_i}) \to \mathcal {H}$
. Hence, denoting by
$\mathcal {H}_i$
the range of
$\rho _i$
, we get that
$\rho _i \colon L^2(\sigma (A_i), \lambda _{A_i}) \equiv \mathcal {H}_i$
is a unitary.
In the next result, we will consider the map
$\Lambda $
introduced before and associated with the measure spaces
$(\Omega _i,\mu _i)=(\sigma (A_i),\lambda _{A_i})$
. We see any operator
$T\in \mathcal {S}^2(\mathcal {H}_i, \mathcal {H}_j)$
as an element of
$\mathcal {S}^2(\mathcal {H})$
by identifying T with the matrix
$ \begin {pmatrix} T & 0 \\ 0 & 0\end {pmatrix}\ \in \,\mathcal {S}^2\bigl (\mathcal {H}_i\overset {2}\oplus \mathcal {H}_i^\perp , \mathcal {H}_j\overset {2}\oplus \mathcal {H}_j^\perp \bigr ). $
The following makes the connection between the multiple operator integrals associated with operators and the map
$\Lambda $
defined above. In particular, when one restricts the Hilbert space
$\mathcal {H}$
to the subspaces
$\mathcal {H}_i$
, then the associated multiple operator integral coincides with
$\Lambda $
. It is the analogue of [Reference Coine, Le Merdy and Sukochev5, Proposition 9] for n operators. The proof is similar and we leave it to the reader.
Proposition 3.2 For any
$1\leq i \leq n-1, K_i \in \mathcal {S}^2(L^2(\lambda _{A_{i+1}}), L^2(\lambda _{A_i}))$
and set
$$ \begin{align*} \widetilde{K}_i=\rho_i \circ K_i \circ \rho_{i+1}^{-1} \in \mathcal{S}^2(\mathcal{H}_{i+1}, \mathcal{H}_i). \end{align*} $$
For any
$\phi \in L^\infty (\lambda _{A_1} \times \cdots \times \lambda _{A_n})$
,
$\Gamma ^{A_1, \ldots , A_n}(\phi )(\widetilde {K}_1, \ldots , \widetilde {K}_{n-1})$
belongs to
$\mathcal {S}^2(\mathcal {H}_n,\mathcal {H}_1)$
and
$$ \begin{align} \Lambda(\phi)(K_1, \ldots,K_{n-1}) = \rho_1^{-1} \circ \Gamma^{A_1, \ldots, A_n}(\phi)(\widetilde{K}_1, \ldots, \widetilde{K}_{n-1}) \circ \rho_n. \end{align} $$
4 Characterization of the Complete Boundedness of Multiple Operator Integrals
Let
$A_1, \ldots , A_n$
be n normal operators on a separable Hilbert space
$\mathcal {H}$
associated with scalar-valued spectral measures
$\lambda _{A_1}, \ldots , \lambda _{A_n}$
. For
$\phi \in L^{\infty }(\lambda _{A_1} \times \cdots \times \lambda _{A_n})$
,
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
belongs to
$\mathcal {B}_{n-1}(\mathcal {S}^2(\mathcal {H}))$
, which is equivalent, by [Reference Coine, Le Merdy and Sukochev5, Section 3.1], to having a continuous mapping defined on the projective tensor product of
$n-1$
copies
$\mathcal {S}^2(\mathcal {H})$
and still denoted by
$$ \begin{align*}\Gamma^{A_1, \ldots, A_n}(\phi) \colon \mathcal{S}^2(\mathcal{H}) \overset{\wedge}{\otimes}\cdots \overset{\wedge}{\otimes} \mathcal{S}^2(\mathcal{H}) \rightarrow \mathcal{S}^2(\mathcal{H}).\end{align*} $$
We will make this identification for the rest of the paper.
The purpose of this section is to characterize the functions
$\phi \in L^{\infty }(\lambda _{A_1} \times \cdots \times \lambda _{A_n})$
such that
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
extends to a (completely) bounded map
$$ \begin{align*}\Gamma^{A_1, \ldots, A_n}(\phi) \colon \underbrace{\mathcal{S}^{\infty}(\mathcal{H}) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(\mathcal{H})}_{n-1 \ \text{times}} \longrightarrow \mathcal{S}^{\infty}(\mathcal{H}).\end{align*} $$
We will also consider the measurable multilinear Schur multipliers
$\Lambda (\phi )$
. In [Reference Juschenko, Todorov and Turowska13], the authors studied and characterized the boundedness of measurable multilinear Schur multipliers
$$ \begin{align*}\mathcal{S}^{\infty}(L^2(\lambda_{A_{n-1}}), L^2(\lambda_{A_n})) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(L^2(\lambda_{A_1}),L^2(\lambda_{A_2})) \rightarrow \mathcal{S}^{\infty}(L^2(\lambda_{A_1}), L^2(\lambda_{A_n})).\end{align*} $$
They proved that we have such extension if and only if
$\phi $
has a certain factorization that will be given in the theorem below. They also proved that the boundedness for the Haagerup norm in this setting implies the complete boundedness.
The proof of Theorem 4.1 below provides another proof of [Reference Juschenko, Todorov and Turowska13, Theorem 3.4]. We show that for multiple operator integrals, boundedness and complete boundedness are also equivalent and that the same characterization holds.
Theorem 4.1 Let
$n\in \mathbb {N}, n\geq 2$
, let
$A_1, \ldots , A_n$
be normal operators on a separable Hilbert space
$\mathcal {H}$
and let
$\phi \in L^{\infty }(\lambda _{A_1} \times \cdots \times \lambda _{A_n})$
. For any
$1\leq i \leq n$
, let
$E_i = L^2(\lambda _{A_i})$
. The following are equivalent:
-
(i)
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
extends to a bounded mapping
$$ \begin{align*}\Gamma^{A_1, \ldots, A_n}(\phi) \colon \mathcal{S}^{\infty}(\mathcal{H}) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(\mathcal{H}) \rightarrow \mathcal{S}^{\infty}(\mathcal{H}).\end{align*} $$
-
(ii)
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
extends to a completely bounded mapping
$$ \begin{align*}\Gamma^{A_1, \ldots, A_n}(\phi) \colon \mathcal{S}^{\infty}(\mathcal{H}) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(\mathcal{H}) \rightarrow \mathcal{S}^{\infty}(\mathcal{H}).\end{align*} $$
-
(iii)
$\Lambda (\phi )$
extends to a completely bounded mapping
$$ \begin{align*}\Lambda(\phi) \colon \mathcal{S}^{\infty}(E_2, E_1) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(E_n, E_{n-1}) \rightarrow \mathcal{S}^{\infty}(E_n, E_1).\end{align*} $$
-
(iv) There exist separable Hilbert spaces
$H_1, \ldots , H_{n-1}$
,
$a_1\in L^{\infty }(\lambda _{A_1} ; H_1), a_n \in L^{\infty }(\lambda _{A_n} ; H_{n-1})$
and
$a_i\in L_{\sigma }^{\infty }(\lambda _{A_i} ; \mathcal {B}(H_i, H_{i-1})), 2\leq i \leq n-1,$
such that
$$\begin{align} \phi(t_1,\ldots,t_n)= \left\langle a_1(t_1), [a_2(t_2)\ldots a_{n-1}(t_{n-1})](a_n(t_n)) \right\rangle \end{align} $$
for a.-e.
$(t_1,\ldots ,t_n) \in \sigma (A_1) \times \cdots \times \sigma (A_n).$
In this case,
$$ \begin{align} \left\|\Gamma^{A_1, \ldots, A_n}(\phi) \right\| &= \left\|\Gamma^{A_1, \ldots, A_n}(\phi) \right\|_{\text{cb}} \\&= \left\| \Lambda(\phi) \right\|_{\text{cb}} = \inf \left\lbrace \|a_1\|_{\infty} \cdots \|a_n\|_{\infty} \ | \ \phi \ \text{as in} \ (4.1) \right\rbrace. \nonumber\end{align} $$
Remark 4.2 Using the normal Haagerup tensor product
$\otimes _{\sigma h}$
of operator spaces, for which we refer the reader to [Reference Blecher and Le Merdy4], one can prove, by simply considering the bi-adjoint of
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
, that the above four statements are also equivalent to:
-
(v)
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
extends to a
$w^*$
-continuous and completely bounded mapping
$$ \begin{align*}\Gamma^{A_1, \ldots, A_n}(\phi) \colon \mathcal{B}^{\infty}(\mathcal{H}) \otimes_{\sigma h} \cdots \otimes_{\sigma h} \mathcal{B}^{\infty}(\mathcal{H}) \rightarrow \mathcal{B}^{\infty}(\mathcal{H}).\end{align*} $$
Proof
$\underline {\text {Proof of (i) } \Leftrightarrow \text { (ii)}}$
Clearly (ii)
$\Rightarrow $
(i) so we only prove (i)
$\Rightarrow $
(ii). We keep the notation
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
for the associated multilinear map defined on
$\mathcal {S}^{\infty }(\mathcal {H}) \times \cdots \times \mathcal {S}^{\infty }(\mathcal {H})$
. Let
$\mathcal {D} = W^*(A_1)'$
and
$\mathcal {C} = W^*(A_n)'$
be the commutant of
$W^*(A_1)$
and
$W^*(A_n)$
, respectively, where the von Neumann algebra
$W^*(A)$
was defined in Section 3.1. Then
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
is a multilinear
$(\mathcal {D}, \mathcal {C})$
-module map, that is, for any
$d\in \mathcal {D}, c\in \mathcal {C}$
, and any
$X_1, \ldots , X_{n-1} \in \mathcal {S}^{\infty }(\mathcal {H})$
,
$$ \begin{align} \left[\Gamma^{A_1, \ldots, A_n}(\phi)\right](dX_1, \ldots, X_{n-1}c) = d\left[\Gamma^{A_1, \ldots, A_n}(\phi)\right](X_1, \ldots, X_{n-1})c. \end{align} $$
By density, it is sufficient to check this equality when
$X_i \in \mathcal {S}^2(\mathcal {H})$
. But in this case, by linearity and
$w^*$
-continuity of
$\Gamma ^{A_1, \ldots , A_n}$
, we can further assume that
$\phi $
is an elementary tensor
$\phi = f_1 \otimes \cdots \otimes f_n$
, where
$f_i \in L^{\infty }(\lambda _{A_i})$
. Then, since
$f_1(A_1) \in W^*(A_1)$
and
$f_n(A_n) \in W^*(A_n)$
we have
$$ \begin{align*} & \left[\Gamma^{A_1, \ldots, A_n}(\phi)\right](dX_1, \ldots, X_{n-1}c) \\ & \ \ \ \ \ = f_1(A_1) d X_1 f_2(A_2) \ldots f_{n-1}(A_{n-1})X_{n-1}c f_n(A_n)\\ & \ \ \ \ \ = d f_1(A_1) X_1 f_2(A_2) \ldots f_{n-1}(A_{n-1})X_{n-1} f_n(A_n) c\\ & \ \ \ \ \ = d\left[\Gamma^{A_1, \ldots, A_n}(\phi)\right](X_1, \ldots, X_{n-1})c. \end{align*} $$
Note that
$W^*(A_1)$
has a separating vector and hence, by [Reference Conway8, Proposition 14.3], this vector is cyclic for
$\mathcal {D}$
. Similarly,
$\mathcal {C}$
has a cyclic vector. It remains to apply [Reference Juschenko, Todorov and Turowska13, Lemma 3.3] to obtain the complete boundedness of
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
and the equality of the norms.
$\underline {\text {Proof of (ii) } \Rightarrow \text { (iii)}}$
We use the same notation as in Subsection 3.2 where we introduced the subspaces
$\mathcal {H}_i$
of
$\mathcal {H}, 1\leq i \leq n$
, with
$\mathcal {H}_i \equiv L^2(\sigma (A_i), \lambda _{A_i})$
. For any
$1\leq i \leq n-1$
,
$\mathcal {S}^{\infty }(\mathcal {H}_{i+1}, \mathcal {H}_i)$
is a closed subspace of
$\mathcal {S}^{\infty }(\mathcal {H})$
and by injectivity of the Haagerup tensor product (see Proposition 2.3), we have a closed subspace
$$ \begin{align*} \mathcal{S}^{\infty}(\mathcal{H}_2, \mathcal{H}_1) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(\mathcal{H}_n, \mathcal{H}_{n-1}) \subset \mathcal{S}^{\infty}(\mathcal{H}) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(\mathcal{H}). \end{align*} $$
By Proposition 3.2, the restriction of
$\Gamma ^{A_1,\ldots ,A_n}(\phi )$
to
$\mathcal {S}^{\infty }(\mathcal {H}_2, \mathcal {H}_1) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(\mathcal {H}_n, \mathcal {H}_{n-1})$
is valued in
$\mathcal {S}^{\infty }(\mathcal {H}_n, \mathcal {H}_1)$
. Moreover, this restriction is completely bounded and by the same proposition, we obtain the inequality
$$ \begin{align*}\left\| \Lambda(\phi) \right\|_{\text{cb}} \leq \left\| \Gamma^{A_1,\ldots,A_n}(\phi) \right\|_{\text{cb}}.\end{align*} $$
$\underline {\text {Proof of (iii) } \Rightarrow \text { (iv)}}$
In this part, the
$L^1-$
spaces will be equipped with their maximal operator space structure (Max) for which we refer the reader to [Reference Pisier17, Chapter 3]. If
$(\Omega , \mu )$
is a measure space, the mapping
$(f,g) \in L^2(\Omega )^2 \mapsto fg \in L^1(\Omega )$
induces a quotient map
$$ \begin{align*} f\otimes g \in L^2(\Omega) \overset{\wedge}{\otimes} L^2(\Omega) \mapsto fg \in L^1(\Omega).\end{align*} $$
We can identify
$L^2(\Omega )$
with its conjugate space so that by (2.1) we get a quotient map
$$ \begin{align*}q \colon \mathcal{S}^1(L^2(\Omega)) \rightarrow L^1(\Omega)\end{align*} $$
which turns out to be a complete metric surjection.
Let
$q_i \colon \mathcal {S}^1(L^2(\lambda _{A_i})) \rightarrow L^1(\lambda _{A_i}), i=1, \ldots , n$
be defined as above. Recall the notation
$E_i = L^2(\lambda _{A_i})$
. Using Proposition 2.3 together with the associativity of the Haagerup tensor product, we get a complete metric surjection
$$ \begin{align*} Q = q_1 \otimes \cdots \otimes q_n \colon \mathcal{S}^1(E_1) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^1(E_n) \rightarrow L^1(\lambda_{A_1}) \overset{h}{\otimes} \cdots \overset{h}{\otimes} L^1(\lambda_{A_n}). \end{align*} $$
Let
$N = \ker Q$
and let, for
$1\leq i \leq n, N_i = \ker q_i$
. For any
$1\leq j \leq n$
, let
$$ \begin{align*} F_j = \mathcal{S}^1(E_1) \otimes \cdots \otimes \mathcal{S}^1(E_{j-1}) \otimes N_j \otimes \mathcal{S}^1(E_j) \otimes \cdots \otimes \mathcal{S}^1(E_n). \end{align*} $$
By Proposition 2.3
$(iv)$
, we obtain that
$$ \begin{align*}N = \overline{F_1 + F_2 + \cdots + F_n}.\end{align*} $$
Assume that
$\Lambda (\phi )$
extends to a completely bounded mapping
$$ \begin{align*}\Lambda(\phi) \colon \mathcal{S}^{\infty}(E_2, E_1) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^{\infty}(E_n, E_{n-1}) \rightarrow \mathcal{S}^{\infty}(E_n, E_1).\end{align*} $$
Let
$E = \mathcal {S}^{\infty }(E_2, E_1) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(E_n, E_{n-1})$
. By Proposition 2.4, we have a complete isometry
$$ \begin{align*}CB(E, \mathcal{B}(E_n, E_1)) = \Big( ((E_1)_c)^* \overset{h}{\otimes} E \overset{h}{\otimes} (E_n)_c \Big)^*.\end{align*} $$
By (2.2) we have
$$ \begin{align*}E = (E_1)_c \overset{h}{\otimes} ((E_2)_c)^* \overset{h}{\otimes} (E_2)_c \overset{h}{\otimes} ((E_3)_c)^* \overset{h}{\otimes} \cdots \overset{h}{\otimes} (E_{n-1})_c \overset{h}{\otimes} ((E_n)_c)^*.\end{align*} $$
Thus, using (2.1) and the associativity of the Haagerup tensor product, we get that
$$ \begin{align*}CB(E, \mathcal{B}(E_n, E_1)) = \Big(\mathcal{S}^1(E_1) \overset{h}{\otimes} \cdots \overset{h}{\otimes} \mathcal{S}^1(E_n)\Big)^*.\end{align*} $$
Let
$u \colon \mathcal {S}^1(E_1) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^1(E_n) \rightarrow \mathbb {C}$
induced by
$\Lambda (\phi )$
. For any
$x_i \in \mathcal {S}^1(H_i), 1\leq i \leq n$
, we have
$$ \begin{align*}&u(x_1 \otimes \cdots \otimes x_n) \\&\quad= \int_{\Omega_1 \times \cdots \times \Omega_n} \phi(t_1,\ldots,t_n) [q_1(x_1)](t_1) \ldots [q_n(x_n)](t_n) \ \text{d}\mu_1(t_1) \ldots \text{d}\mu_n(t_n).\end{align*} $$
To see this, it is enough to check when the
$x_i$
are rank one operators and in that case, one can use the identifications above. In particular, the latter implies that u vanishes on
$N = \ker Q$
. Since Q is a complete metric surjection, we get a mapping
$$ \begin{align*}v \colon L^1(\lambda_{A_1}) \overset{h}{\otimes} \cdots \overset{h}{\otimes} L^1(\lambda_{A_n}) \rightarrow \mathbb{C}\end{align*} $$
such that
$u = v \circ Q$
. An application of Theorem 2.1 with suitable restrictions using the separability of the spaces
$L^1(\lambda _{A_i})$
gives the existence of separable Hilbert spaces
$H_1, \ldots , H_{n-1}$
and completely bounded maps
$$ \begin{align*} \alpha_1 \colon L^1(\lambda_{A_1}) &\rightarrow \mathcal{B}(H_1,\mathbb{C}) = (H_1)_c^*, \end{align*} $$
$$ \begin{align*} \alpha_i \colon L^1(\lambda_{A_i}) &\rightarrow \mathcal{B}(H_i, H_{i-1}), 2\leq i\leq n-1, \end{align*} $$
$$ \begin{align*} \alpha_n \colon L^1(\lambda_{A_n}) \rightarrow \mathcal{B}(\mathbb{C}, H_{n-1})= (H_{n-1})_c \end{align*} $$
such that for any
$f_j \in L^1(\lambda _{A_j}), 1\leq j \leq n$
,
$$ \begin{align*}v(f_1 \otimes \cdots \otimes f_n) = \left\langle \alpha_1(f_1), [\alpha_2(f_2) \ldots \alpha_{n-1}(f_{n-1})](\alpha_n(f_n)) \right\rangle.\end{align*} $$
Since
$L^1(\Omega _2)$
is equipped with the Max operator space structure, we have
$$ \begin{align*}CB(L^1(\lambda_{A_i}), \mathcal{B}(H_i, H_{i-1})) = \mathcal{B}(L^1(\lambda_{A_i}), \mathcal{B}(H_i, H_{i-1})).\end{align*} $$
Moreover, by (2.5), we have
$$ \begin{align*} \mathcal{B}(L^1(\lambda_{A_i}), \mathcal{B}(H_i, H_{i-1})) = L^{\infty}_{\sigma}(\lambda_{A_i}; \mathcal{B}(H_i, H_{i-1})).\end{align*} $$
Thus, for any
$2\leq i \leq n-1$
, we associate with
$\alpha _i$
an element
$a_i \in L^{\infty }_{\sigma }\kern-0.5pt(\lambda _{A_i}; \mathcal {B}(H_i, H_{i-1})\kern-1pt)$
. Similarly, we associate with
$\alpha _1$
an element
$a_1 \in L^{\infty }(\lambda _{A_1}; H_1)$
and with
$\alpha _n$
an element
$a_n \in L^{\infty }(\lambda _{A_n}; H_{n-1})$
. Using the identification (2.4), we obtain that
$$ \begin{align*} \phi(t_1,\ldots,t_n)= \left\langle a_1(t_1), [a_2(t_2)\ldots a_{n-1}(t_{n-1})](a_n(t_n)) \right\rangle \end{align*} $$
for a.-e.
$(t_1,\ldots ,t_n) \in \sigma (A_1) \times \cdots \times \sigma (A_n)$
, and one can choose
$a_1, \ldots , a_n$
such that we have the equality
$$ \begin{align*} \left\| \Lambda(\phi) \right\|_{\text{cb}} = \|a_1\|_{\infty} \cdots \|a_n\|_{\infty}. \end{align*} $$
$\underline {\text {Proof of (iv) }\Rightarrow \text { (ii)}}$
Assume that there exist separable Hilbert space
$H_1 , \ldots , H_{n-1}$
,
$a_1\kern1.2pt{\in}\kern1.2pt L^{\infty }(\lambda _{A_1} ; H_1), a_i \kern1.2pt{\in} L_{\sigma }^{\infty }(\lambda _{A_i} ; \mathcal {B}(H_i, H_{i-1})), 2\leq i\leq n-1$
and
$a_n \in L^{\infty }(\lambda _{A_n} ; H_{n-1})$
such that
$$ \begin{align*} \phi(t_1,\ldots,t_n)= \left\langle a_1(t_1), [a_2(t_2)\ldots a_{n-1}(t_{n-1})](a_n(t_n)) \right\rangle \end{align*} $$
for a.-e.
$(t_1,\ldots ,t_n) \in \sigma (A_1) \times \cdots \times \sigma (A_n)$
. For any
$1\leq i\leq n-1$
, let
$(\epsilon ^i_k)_{k\geq 1}$
be a Hilbertian basis of
$H_i$
. for
$k,l \geq 1$
, define
$$ \begin{align*}a^1_k = \left\langle a_1, \epsilon^1_k \right\rangle, a^i_{kl} = \left\langle \epsilon^{i-1}_k, a_i \epsilon^i_l \right\rangle \quad \text{and} \ \ a^n_l = \left\langle \epsilon^{n-1}_l, a_n \right\rangle.\end{align*} $$
Then
$a^1_k \in L^{\infty }(\lambda _{A_1}), a^i_{kl} \in L^{\infty }(\lambda _{A_i}), 2\leq i \leq n-1$
, and
$a^n_l \in L^{\infty }(\lambda _{A_n})$
. To see this, simply note that for
$2\leq i \leq n-1$
,
$$ \begin{align*}a^i_{kl} = \text{tr} (a_i(\cdot) \circ (\epsilon^{i-1}_k \otimes \epsilon^i_l)).\end{align*} $$
For
$N \geq 1$
and
$1\leq i \leq n-1$
, let
$P^i_N$
be the orthogonal projection onto
$\text {Span}(\kern-0.3pt\epsilon ^i_1, \ldots , \epsilon ^i_N\kern-0.3pt)$
. Then define
$$ \begin{align*}\phi_N = \left\langle P^1_N(a(t_1))), [a_2(t_2)P^2_N a_3(t_3)P^3_N \ldots a_{n-1}(t_{n-1})P^{n-1}_N](a_n(t_n)) \right\rangle.\end{align*} $$
It is clear that
$(\phi _N)_{N\geq 1}$
is bounded in
$L^{\infty }(\lambda _{A_1} \times \cdots \times \lambda _{A_n})$
and that
$\phi _N \to \phi $
pointwise when
$N \to \infty $
. Therefore, by the Dominated Convergence Theorem, we have that
$\phi _N \to \phi $
for the
$w^*-$
topology. This implies, by
$w^*-$
continuity of
$\Gamma ^{A_1,\ldots , A_n}$
, that for any
$X_j$
in
$\mathcal {S}^2(\mathcal {H}), 1\leq j \leq n-1$
,
$$ \begin{align*}\left[\Gamma^{A_1, \ldots, A_n}(\phi_N)\right](X_1 \otimes \cdots \otimes X_{n-1}) \to \left[\Gamma^{A_1, \ldots, A_n}(\phi)\right](X_1 \otimes \cdots \otimes X_{n-1})\end{align*} $$
weakly in
$\mathcal {S}^2(\mathcal {H})$
.
Assume that
$(\Gamma ^{A_1, \ldots , A_n}(\phi _N))_N$
is uniformly bounded in
$CB(\mathcal {S}^{\infty }(\mathcal {H}) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(\mathcal {H}), \mathcal {S}^{\infty }(\mathcal {H})).$
Then, the above approximation property together with the density of
$\mathcal {S}^2$
into
$\mathcal {S}^{\infty }$
imply that
$\Gamma ^{A_1, \ldots , A_n}(\phi )$
is completely bounded as well with
$\| \Gamma ^{A_1, \ldots , A_n}(\phi ) \|_{\text {cb}} \leq \mathop {\mathrm {sup}} _N \| \Gamma ^{A_1, \ldots , A_n}(\phi _N) \|_{\text {cb}}$
.
We will show now that for any
$N\geq 1$
,
$\| \Gamma ^{A_1, \ldots , A_n}(\phi _N) \|_{\text {cb}} \leq \|a_1\|_{\infty } \ldots \|a_n\|_{\infty }.$
For any
$N\geq 1$
and a.-e.
$(t_1,\ldots ,t_n) \in \sigma (A_1) \times \cdots \times \sigma (A_n)$
, we have
$$ \begin{align*}\phi_N(t_1, \ldots, t_n) = \sum\limits_{k_1, \ldots, k_{n-1} = 1}^N a^1_{k_1}(t_1) a^2_{k_1 k_2}(t_2) \ldots a^{n-1}_{k_{n-2} k_{n-1}}(t_{n-1}) a^n_{k_n}(t_n),\end{align*} $$
so that for any
$X_1, \ldots , X_{n-1} \in \mathcal {S}^2(\mathcal {H})$
,
$$ \begin{align*} & \left[\Gamma^{A_1, \ldots, A_n}(\phi_N)\right](X_1 \otimes \cdots \otimes X_{n-1}) \\ & = \sum\limits_{k_1, \ldots, k_{n-1} = 1}^N a^1_{k_1}(A_1) X_1 a^2_{k_1 k_2}(A_2) X_2 \ldots X_{n-2} a^{n-1}_{k_{n-2} k_{n-1}}(A_{n-1}) X_{n-1} a^n_{k_n}(A_n). \end{align*} $$
Note that the latter can be written as
$$ \begin{align*}\left[\Gamma^{A_1, \ldots, A_n}(\phi_N)\right](X_1 \otimes \cdots \otimes X_{n-1})= A^1_N (X_1 \otimes I_N) A^2_N (X_2 \otimes I_N) \cdots (X_{n-1} \otimes I_N) A^n_N,\end{align*} $$
where
$$ \begin{align*}A^1_N = [a^1_1(A_1) \ a^1_2(A_1) \ldots a^1_N(A_1)] \colon \ell_2^N(\mathcal{H}) \rightarrow \mathcal{H},\end{align*} $$
$$ \begin{align*}A^i_N = [a^i_{kl}(A_i)]_{\begin{subarray}{l} 1 \leq k \leq N\\ 1 \leq l \leq N \end{subarray}} \colon \ell_2^N(\mathcal{H}) \rightarrow \ell_2^N(\mathcal{H}), \ 2 \leq i \leq n-1\end{align*} $$
and
$$ \begin{align*}A^n_N = [a^n_1(A_n) \ a^n_2(A_n) \ldots A^n_N(A_n)]^t \colon \mathcal{H} \rightarrow \ell_2^N(\mathcal{H}).\end{align*} $$
The notation
$X \otimes I_N$
stands for the element of
$\mathcal {B}(\ell _2^N(\mathcal {H}))$
whose matrix is the
$N\times N$
diagonal matrix
$\text {diag}(X, \ldots , X)$
.
For any
$N\geq 1$
and any
$1\leq i \leq n$
, let
$\pi _N$
and
$\pi _i$
be the
$*-$
representations defined by
$$ \begin{align*} \begin{array}[t]{l@{\ }r@{\ }c@{\ }l} \pi_N \colon &\mathcal{B}(\mathcal{H}) & \longrightarrow & \mathcal{B}(\ell_2^N(\mathcal{H}))\\ & X & \longmapsto & X\otimes I_N \end{array}\quad\text{and}\quad\begin{array}[t]{l@{\ }r@{\ }c@{\ }l} \pi_{A_i} \colon &L^{\infty}(\lambda_{A_i}) & \longrightarrow & \mathcal{B}(\mathcal{H})\\ & f & \longmapsto & f(A_i) \end{array}. \end{align*} $$
By [Reference Pisier17, Proposition 1.5],
$\pi _N$
and
$\pi _{A_i}$
are completely bounded with cb-norm less than
$1$
. Note that the element
$[a^i_{kl}]_{1 \leq k,l \leq N} \in M_N(L^{\infty }(\lambda _B))$
has a norm less than
$\|a_i\|_{\infty }$
. Thus, the latter implies that
$A^i_N = [\pi _{A_i}(a^i_{kl})]_{1 \leq k,l \leq N}$
has an operator norm less than
$\|a_i\|_{\infty }$
. Similarly (using column and row matrices), we show that
$A^1_N$
and
$A^n_N$
have a norm less than
$\|a_1\|_{\infty }$
and
$\|a_n\|_{\infty }$
, respectively. Finally, write
$$ \begin{align*}\left[\Gamma^{A_1, \ldots, A_n}(\phi_N)\right](X_1 \otimes \cdots \otimes X_{n-1}) = \sigma^1_N(X_1) \sigma^2_N(X_2) \ldots \sigma^{n-1}_N(X_{n-1}),\end{align*} $$
where for any
$1 \leq i \leq n-2$
,
$\sigma ^i_N(X_1) = A^i_N \pi _N(X_i)$
and
$\sigma _{n-1}^N(X_{n-1}) = A^{n-1}_N \pi _N(X_{n-1}) A^n_N$
. By the easy part of Wittstock theorem (see e.g. [Reference Pisier17, Theorem 1.6]),
$\sigma ^i_N$
and
$\sigma _{n-1}^N$
are completely bounded with cb-norm less than
$\|a_i\|_{\infty }$
and
$\|a_{n-1}\|_{\infty } \|a_n\|_{\infty }$
, respectively. Hence, by Theorem 2.1, we get that
$\Gamma ^{A_1, \ldots , A_n}(\phi _N)$
is completely bounded with cb-norm less than
$\|a_1\|_{\infty } \ldots \|a_n\|_{\infty }$
. ▪
We now give an example of a class of functions for which the multiple operator integrals will be completely bounded (in the sense of the paper) for any normal operators. We will identify a bounded Borel function
$\psi : \mathbb {C}^n \rightarrow \mathbb {C}$
with the class of the restriction
$\tilde {\psi } = \psi _{|\sigma (A_1) \times \sigma (A_2) \times \cdots \times \sigma (A_n)}$
in
$L^{\infty }\left (\prod _{i=1}^n \lambda _{A_i}\right )$
. Then we will denote by
$\Gamma ^{A_1,A_2, \ldots , A_n}(\psi )$
the multiple operator integral
$\Gamma ^{A_1,A_2, \ldots , A_n}(\tilde {\psi })$
.
Example 4.3 Let
$C_b$
be the space of bounded and continuous functions
$f : \mathbb {C} \to \mathbb {C}$
. Let
$n\geq 1$
be an integer. We define the integral tensor product of
$C_b$
, denoted by
$C_b \, \widehat {\otimes }_i \cdots \widehat {\otimes }_i \, C_b$
, as the space of functions
$\phi : \mathbb {C}^n \to \mathbb {C}$
such that there exist a
$\sigma $
-finite measure space
$(\Sigma , \mu )$
and functions
$h_i : \mathbb {C} \times \Sigma \to \mathbb {C}$
,
$1\leq i \leq n$
, such that for a.e.
$w\in \Sigma , t \mapsto h_i(t, w) \in C_b$
,
$$ \begin{align} \int_{\Sigma} \| h_1(\cdot, w)\|_{\infty} \cdots \| h_n(\cdot,w)\|_{\infty} \ \text{d}\mu(w) < +\infty \end{align} $$
and for every
$t_1, \ldots , t_n \in \mathbb {C}$
,
$$ \begin{align} \phi(t_1,\ldots, t_n) = \int_{\Sigma} h_1(t_1, w) \cdots h_n(t_n,w) \ \text{d}\mu(w). \end{align} $$
The integral projective norm
$\|\phi \|_i$
of
$\phi $
is the infimum of the quantities (4.4) over all representations of
$\phi $
as above.
Let
$A_1, \ldots , A_n$
be normal operators on
$\mathcal {H}$
and let
$\phi \in C_b \, \widehat {\otimes }_i \cdots \widehat {\otimes }_i \, C_b$
. Then we have a completely bounded multiple operator integral
$\Gamma ^{A_1, \ldots , A_n}(\phi ) \colon \mathcal {S}^{\infty }(\mathcal {H}) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(\mathcal {H}) \rightarrow \mathcal {S}^{\infty }(\mathcal {H})$
with
$\left \|\Gamma ^{A_1, \ldots , A_n}(\phi ) \right \|_{\text {cb}} \leq \| \phi \|_i$
.
Proof To show this, we will find another factorization of
$\phi $
as in (4.5) that satisfies of Theorem 4.1(iv). First, note that by changing
$\Sigma $
if necessary, we can assume that for almost every
$w\in \Sigma , \| h_1(\cdot , w)\|_{\infty } \cdots \| h_n(\cdot ,w)\|_{\infty }> 0$
. We define
$g_i : \mathbb {C} \times \Sigma \to \mathbb {C}$
,
$1\leq i \leq n$
, for almost every
$(t_i, \sigma )$
by
$$ \begin{align*} g_1(t_1,w) = \dfrac{h_1(t_1, w)}{\sqrt{\| h_1(\cdot, w)\|_{\infty}}} \sqrt{\| h_2(\cdot, w)\|_{\infty} \cdots \| h_n(\cdot,w)\|_{\infty}}, \end{align*} $$
$$ \begin{align*} g_i(t_i,w) = \dfrac{\overline{h_i}(t_i, w)}{\| h_i(\cdot, w)\|_{\infty}} \ \ \text{if} \ 2\leq i \leq n-1, \end{align*} $$
and
$$ \begin{align*} g_n(t_n,w) = \dfrac{\overline{h_n}(t_n, w)}{\sqrt{\| h_n(\cdot, w)\|_{\infty}}} \sqrt{\| h_1(\cdot, w)\|_{\infty} \cdots \| h_{n-1}(\cdot,w)\|_{\infty}}. \end{align*} $$
It is straightforward to check that for every
$t_1, t_n \in \mathbb {C}$
,
$g_1(t_1, \cdot )$
and
$g_n(t_n, \cdot )$
belong to
$L^2(\Sigma , \mu )$
and we have, setting
$\alpha = \int _{\Sigma } \| h_1(\cdot , w)\|_{\infty } \cdots \| h_n(\cdot ,w)\|_{\infty } \ \text {d}\mu (w)$
,
$$ \begin{align*} \| g_1(t_1, \cdot) \|_{L^2(\Sigma, \mu)} \leq \sqrt{\alpha} \ \ \ \text{and} \ \ \ \|g_n(t_n, \cdot)\|_{L^2(\Sigma, \mu)} \leq \sqrt{\alpha}. \end{align*} $$
Moreover, the proof of [Reference Coine, Le Merdy, Sukochev and Skripka6, Proposition 5.4] shows that the associated mappings (after taking their restriction to the spectrum of
$A_1$
and
$A_n$
)
$a_1 : t_1 \in \sigma (A_1) \mapsto g_1(t_1, \cdot ) \in L^2(\Sigma , \mu )$
and
$a_n : t_n \in \sigma (A_n) \mapsto g_n(t_n, \cdot ) \in L^2(\Sigma , \mu )$
are continuous, hence measurable, so that, after taking the classes of these functions in
$L^{\infty }$
,
$a_1 \in L^{\infty }(\lambda _{A_1}, L^2(\Sigma , \mu ))$
and
$a_n \in L^{\infty }(\lambda _{A_n}, L^2(\Sigma , \mu ))$
.
Now, notice that for all
$2\leq i \leq n-1, g_i$
is bounded on
$\sigma (A_i) \times \Sigma $
and for almost every
$t_i \in \sigma (A_i), a_i(t_i) \in \mathcal {B}(L^2(\Sigma , \mu ))$
be the multiplication map by
$g_i(t_i, \cdot )$
. This defines a mapping
$a_i : \sigma (A_i) \to \mathcal {B}(L^2(\Sigma , \mu ))$
that is bounded by
$\| g_i \|_{\infty } =1$
. To prove that this map is
$w^*$
-measurable, it is sufficient, by linearity and density, to show that for any rank-one operator
$T \in \mathcal {S}^1(L^2(\Sigma , \mu ))$
,
$t_i \in \sigma (A_i) \mapsto \text {tr}(a_i(t_i)T)$
is measurable. Such an operator T can be written as
$T = b_1\otimes b_2$
with
$b_1,b_2 \in L^2(\Sigma , \mu )$
and
$T(f) = \left \langle f,b_1 \right \rangle b_2$
. Now, one easily checks that
$a_i(t_i)T = b_1 \otimes a_i(t_i)b_2 = b_1 \otimes g_i(t_i, \cdot )b_2$
. Hence,
$$ \begin{align} \text{tr}(a_i(t_i)T) &= \int_{\Sigma} \overline{b_1}(w)g_i(t_i, w)b_2(w) \text{d}w \\&= \int_{\Sigma} g_i(t_i, w)b(w) \text{d}w = \left\langle g_i(t_i, \cdot), b \right\rangle_{L^{\infty}, L^1}, \nonumber\end{align} $$
where
$b = \overline {b_1}b_2 \in L^1(\Sigma , \mu )$
. Since
$g_i \in L^{\infty }(\lambda _{A_i} \times \mu )$
, we get that
$t_i \mapsto g(t_i, \cdot )$
is a
$w^*$
-measurable map from
$\sigma (A_i)$
into
$L^{\infty }(\mu )$
. Together with equality (4.6), this implies that
$a_i$
is
$w^*$
-measurable and hence
$a_i \in L^{\infty }_{\sigma }(\lambda _{A_i}, \mathcal {B}(L^2(\Sigma , \mu )))$
.
Finally, we check that
$$ \begin{align*} \phi(t_1,\ldots,t_n)= \left\langle a_1(t_1), [a_2(t_2)\ldots a_{n-1}(t_{n-1})](a_n(t_n)) \right\rangle \end{align*} $$
for a.-e.
$(t_1,\ldots ,t_n) \in \sigma (A_1) \times \cdots \times \sigma (A_n)$
and that
$\|a_1\|_{\infty } \cdots \|a_n\|_{\infty } \leq \alpha $
. Taking the infimum over all representations of
$\phi $
gives
$\left \|\Gamma ^{A_1, \ldots , A_n}(\phi )\right \|_{\text {cb}} \leq \| \phi \|_i$
. ▪
Example 4.4 For fixed normal operator
$A_1, \ldots , A_n$
on
$\mathcal {H}$
, one can define in a similar way the space
$L^{\infty }(\lambda _{A_1}) \, \widehat {\otimes }_i \cdots \widehat {\otimes }_i \, L^{\infty }(\lambda _{A_n})$
. Then, any
$\phi $
in this space induces a completely bounded multiple operator integral
$\Gamma ^{A_1, \ldots , A_n}(\phi ) \colon \mathcal {S}^{\infty }(\mathcal {H}) \overset {h}{\otimes } \cdots \overset {h}{\otimes } \mathcal {S}^{\infty }(\mathcal {H}) \rightarrow \mathcal {S}^{\infty }(\mathcal {H}).$
This can be proved using the same ideas as in Example 4.3. We refer the reader to [Reference Juschenko, Todorov and Turowska13] for another proof.
