Haagerup Tensor Product in Operator Spaces

• Haagerup tensor product is an operator-space tensor norm that linearizes completely bounded bilinear maps through matrix factorizations and a unique duality structure.
• It encompasses variants such as balanced, normal, extended, and module forms, with applications in C*-algebras, Hilbert modules, and quantum-group duality.
• Its duality properties and embedding theorems provide essential insights into operator module theory, ideal structure, and spectral synthesis.

Searching arXiv for additional relevant papers on the Haagerup tensor product. arXiv search query: "Haagerup tensor product operator spaces bidual embedding normal Haagerup tensor product". The Haagerup tensor product is an operator-space tensor product tailored to completely bounded bilinear maps. For operator spaces $X$ and $Y$, it equips $X\otimes Y$ with a norm defined by matrix factorizations, and its completion $X\otimes_h Y$ linearizes completely bounded bilinear forms through a completely isometric duality with $CB(X\times Y,\mathbb C)$. In the literature it appears in several closely related forms—balanced, normal, extended, module, and covariance-relative—and it functions both as a structural tensor product in operator space theory and as a technical device in $C^*$-algebras, Hilbert modules, Schur multiplier theory, multiple operator integrals, Fourier algebras, and quantum-group duality (Jain et al., 2011).

1. Definition and operator-space framework

An operator space is a Banach space equipped with compatible matrix norms on $M_n(X)$, typically coming from a concrete realization $X\subseteq B(H)$. A linear map $T:X\to Y$ is completely bounded when

$$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$

For bilinear forms, two notions are standard in the operator-space setting. A bilinear form $Y$0 is completely bounded in the Christensen–Sinclair sense if its amplifications

$Y$1

are uniformly bounded, while it is jointly completely bounded if the amplifications

$Y$2

are uniformly bounded as maps into $Y$3; every completely bounded bilinear form is jointly completely bounded and satisfies $Y$4 (Jain et al., 2011).

For $Y$5, the Haagerup norm is given by

$Y$6

where

$Y$7

The completion is the Haagerup tensor product $Y$8 (Jain et al., 2011). In the simpler two-factor presentation used for operator spaces and $Y$9-algebras, if $X\otimes Y$0, then

$X\otimes Y$1

and in the row-column form

$X\otimes Y$2

with $X\otimes Y$3 and $X\otimes Y$4 (He, 2020, Kumar et al., 2014).

The decisive structural feature is duality: $X\otimes Y$5 completely isometrically (Jain et al., 2011). This is the basic reason the tensor product is regarded as one of the fundamental tensor products in operator space theory, and why it repeatedly appears wherever completely bounded bilinear forms must be linearized or extended (Rajpal et al., 17 Aug 2025).

2. Variants: balanced, normal, extended, and covariance-relative forms

The balanced or relative Haagerup tensor product appears when $X\otimes Y$6 and $X\otimes Y$7 are operator modules over a common algebra $X\otimes Y$8. If $X\otimes Y$9 is an operator $X\otimes_h Y$0-$X\otimes_h Y$1 bimodule and $X\otimes_h Y$2 a left operator $X\otimes_h Y$3-module, the balanced tensor product

$X\otimes_h Y$4

is the completed quotient of the algebraic balanced tensor product $X\otimes_h Y$5 characterized by the universal property that every completely contractive $X\otimes_h Y$6-balanced bilinear map $X\otimes_h Y$7 extends uniquely to a completely contractive map on $X\otimes_h Y$8. It satisfies

$X\otimes_h Y$9

completely isometrically in the nondegenerate case, and

$CB(X\times Y,\mathbb C)$0

completely isometrically for a $CB(X\times Y,\mathbb C)$1-correspondence $CB(X\times Y,\mathbb C)$2 (Crisp, 2016).

In commutative base-change problems one uses the balanced product

$CB(X\times Y,\mathbb C)$3

obtained by completing the algebraic balanced tensor product in the relative Haagerup norm. On $CB(X\times Y,\mathbb C)$4 that norm is defined by factorizations $CB(X\times Y,\mathbb C)$5 with $CB(X\times Y,\mathbb C)$6, $CB(X\times Y,\mathbb C)$7, and

$CB(X\times Y,\mathbb C)$8

The resulting operator space is generally not symmetric under flipping the factors, even when the base algebra is commutative (Crisp, 2019).

The extended and weak-* Haagerup products are dual variants adapted to dual operator spaces. If $CB(X\times Y,\mathbb C)$9 and $C^*$0 are dual operator spaces, then

$C^*$1

completely isometrically (Alaghmandan et al., 2018). This weak-* variant is the natural receptacle for separately weak-* continuous completely bounded bilinear forms and plays a central role in biduality results.

A further generalization replaces module balancing over a subalgebra by balancing relative to a completely positive normal covariance map $C^*$2. In that setting the generalized product $C^*$3 is defined as a dual operator space determined by the image of a Hilbert-space predual map attached to a semicircular system with covariance $C^*$4, and $C^*$5 is defined as the image of the ordinary extended Haagerup product under the corresponding quotient map. When $C^*$6 is a conditional expectation $C^*$7, these constructions recover the module products $C^*$8 and $C^*$9; for general $M_n(X)$0 they encode covariance relations not reducible to module balancing over a subalgebra (Dabrowski, 2015).

The same module-Haagerup formalism also produces a duality theory. For a completely contractive Banach algebra $M_n(X)$1 and a left operator $M_n(X)$2-module $M_n(X)$3, the Haagerup dual is

$M_n(X)$4

and one has

$M_n(X)$5

For essential $M_n(X)$6-modules over algebras with bounded approximate identity, this recovers the ordinary operator-space dual, while in harmonic-analytic settings it yields genuinely different dual objects (Alaghmandan et al., 2018).

3. Biduality, normal Haagerup products, and extension theorems

A central structural theorem is the bidual embedding

$M_n(X)$7

for arbitrary operator spaces $M_n(X)$8 and $M_n(X)$9, and this embedding is a complete isometry (Jain et al., 2011). The proof rests on the canonical extension theory for completely bounded bilinear maps: if $X\subseteq B(H)$0 is completely bounded and $X\subseteq B(H)$1 is a dual operator space, then $X\subseteq B(H)$2 admits a unique separately weak-* continuous extension

$X\subseteq B(H)$3

with

$X\subseteq B(H)$4

This identifies completely bounded bilinear forms on $X\subseteq B(H)$5 with separately weak-* continuous completely bounded bilinear forms on $X\subseteq B(H)$6, and hence identifies the normal Haagerup tensor product on the biduals with $X\subseteq B(H)$7 (Jain et al., 2011).

The embedding is canonical but not asserted to be onto in general. The finite-dimensional case is exceptional: if one factor is finite dimensional, then

$X\subseteq B(H)$8

completely isometrically (Jain et al., 2011). The general theorem is therefore an inclusion theorem, not a full bidual commutation theorem.

The same paper develops a parallel extension theory for jointly completely bounded bilinear forms. If $X\subseteq B(H)$9 and $T:X\to Y$0 are exact operator spaces and $T:X\to Y$1 is jointly completely bounded, then $T:X\to Y$2 extends uniquely to a separately weak-* continuous jointly completely bounded form on $T:X\to Y$3 with quantitative norm control

$T:X\to Y$4

For $T:X\to Y$5-algebras $T:X\to Y$6 and $T:X\to Y$7, the extension is norm preserving: $T:X\to Y$8 for every jointly completely bounded $T:X\to Y$9, and the extension is separately normal on $$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$0 (Jain et al., 2011).

These weak-* extension results make the Haagerup tensor product the conceptual model for later embedding theorems for the operator space projective tensor product, and they identify the normal Haagerup tensor product as the appropriate weak-* completion of the ordinary Haagerup tensor product (Jain et al., 2011).

4. Commutativity, base change, adjunction, and Hilbert-module realizations

The Haagerup tensor product is generally not commutative as an operator-space tensor product, and the commutative-base case makes this visible in a sharp geometric form. For continuous maps $$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$1, the balanced flip

$$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$2

is completely bounded if and only if

$$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$3

and in that case

$$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$4

Equivalently, the canonical comparison map with the fiber-product algebra

$$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$5

is a completely bounded isomorphism exactly under the same fibrewise finiteness condition, yielding a Beck–Chevalley criterion for base change of operator modules (Crisp, 2019).

In operator-module category theory, the balanced Haagerup tensor product is the tensor product that turns a $$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$6-correspondence $$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$7 into a functor

$$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$8

A complete characterization is available: this functor has a left adjoint if and only if the left action of $$\|T\|_{cb}=\sup_{n\in\mathbb N}\|T_n:M_n(X)\to M_n(Y)\|<\infty.$$9 on $Y$00 lands in $Y$01, equivalently if and only if

$Y$02

captures the left action. Under that condition the left adjoint is $Y$03, and the adjunction isomorphisms are completely contractive. In the representation theory of locally compact groups, this yields a Frobenius reciprocity statement: unitary induction for a closed subgroup $Y$04 admits a left adjoint in the operator-module setting if and only if $Y$05 is cocompact in $Y$06 (Crisp, 2016).

For Hilbert modules over a commutative $Y$07-algebra $Y$08, the balanced Haagerup tensor product also identifies trace-class operators. If $Y$09 is a Hilbert $Y$10-module that is countably generated by multipliers, then the rank-one map

$Y$11

is a completely isometric isomorphism, generalizing the Hilbert-space fact

$Y$12

In the free-module case $Y$13, this identifies the balanced Haagerup tensor product with continuous fields of trace-class operators $Y$14 (Crisp et al., 2024).

5. Schur multipliers and multiple operator integrals

A classical analytic role of the Haagerup tensor product is the description of Schur multipliers. In the measure-theoretic and spectral-measure settings, the multiplier spaces are exactly Haagerup tensor products of $Y$15-spaces, with equality of norms: $Y$16 The corresponding factorization may be written either as

$Y$17

with square-function $Y$18-bounds, or as a Hilbert-space factorization

$Y$19

for $Y$20, $Y$21 (Aleksandrov et al., 2024).

In discrete operator-valued Schur multiplier theory, the Haagerup tensor product describes complete compactness. For a $Y$22-algebra $Y$23, the paper on compact Schur $Y$24-multipliers shows that

$Y$25

and, more generally,

$Y$26

At the level of formulas, a completely compact Schur $Y$27-multiplier has a symbol factorization

$Y$28

with the coefficient families satisfying Haagerup-type square-summability and $Y$29-vanishing conditions (He, 2020).

For multiple operator integrals, Haagerup tensor products of $Y$30-spaces provide the correct class of integrands for sharp Schatten-von Neumann estimates. If

$Y$31

with endpoint operators $Y$32, $Y$33 for $Y$34 and middle operators bounded, then the corresponding $Y$35-fold operator integral belongs to $Y$36, where

$Y$37

and satisfies the norm estimate

$Y$38

The threshold $Y$39 is sharp, and the paper introduces Haagerup-like tensor products to treat perturbation-theoretic integrands not lying in the ordinary Haagerup tensor product (Aleksandrov et al., 2016).

6. Harmonic analysis, Fourier algebras, and quantum-group duality

For the Fourier algebra $Y$40 of a compact group, the Haagerup tensor product governs convolution factorization in a way that is stronger than Banach-space projective factorization. The convolution maps

$Y$41

extend from $Y$42 as complete quotient maps. The twisted convolution range is the diagonal Fourier algebra

$Y$43

while ordinary convolution has full range

$Y$44

One consequence is that the convolution algebra $Y$45 is completely isomorphic to an operator algebra; another is that the anti-diagonal

$Y$46

is a set of spectral synthesis for $Y$47 (Rostami et al., 2014).

The module Haagerup tensor product also supports a duality theory with direct quantum-group content. For a left operator $Y$48-module $Y$49, the Haagerup dual $Y$50 recovers the ordinary operator-space dual for essential modules over $Y$51-algebras, but in convolution settings it yields the dual quantum object. In particular, for a compact Kac algebra $Y$52 whose dual $Y$53 is weakly amenable,

$Y$54

completely isometrically and weak-* homeomorphically. Corresponding group cases include

$Y$55

for compact groups and

$Y$56

for weakly amenable discrete groups (Alaghmandan et al., 2018).

These harmonic-analytic examples show that the Haagerup tensor product is not only a tensor norm but also a mechanism for converting algebraic operations such as convolution into completely bounded products and for recovering dual objects from operator-module structures (Rostami et al., 2014, Alaghmandan et al., 2018).

7. Ideal structure, regularity, and structural comparisons

The Haagerup tensor product has an extensive ideal theory. In the $Y$57-algebraic setting, one recorded consequence is that spectral synthesis for $Y$58 implies property $Y$59, meaning that product states $Y$60 coming from pure states of $Y$61 and $Y$62 separate closed ideals. Under spectral synthesis every closed ideal $Y$63 satisfies

$Y$64

and the paper states the implication

$Y$65

for $Y$66-algebras $Y$67 and $Y$68 (Kumar et al., 2011).

Arens regularity is another structural theme. For operator algebras $Y$69 and $Y$70,

$Y$71

For $Y$72-algebras, the Arens regularity of

$Y$73

is equivalent. The paper also gives concrete nonregularity results, such as non-Arens-regularity of $Y$74 for infinite-dimensional $Y$75 (Kumar et al., 2014).

More recent work extends ideal-space analysis beyond $Y$76-algebras to mixed Haagerup tensor products of ternary rings of operators and $Y$77-algebras. For a TRO $Y$78 and a $Y$79-algebra $Y$80, the map

$Y$81

controls the primal, factorial, and Glimm ideals of $Y$82. In particular, the factorial ideal space and the Glimm ideal space of $Y$83 are identified homeomorphically with the corresponding product spaces of the factors, and

$Y$84

(Rajpal et al., 17 Aug 2025).

The Haagerup tensor product also serves as a comparison standard for other tensor norms. In analyses of the Banach space projective tensor product $Y$85, the canonical map

$Y$86

is used repeatedly to transfer center and ideal information, and the known Haagerup identity

$Y$87

functions as the model for corresponding projective-tensor results (Gupta et al., 2018). In this sense the Haagerup tensor product occupies an intermediate position: it is finer than purely Banach-space tensor constructions, weaker than $Y$88-tensor symmetry, and often the most effective tensor product for combining operator-space duality, weak-* structure, and algebraic control.