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# A twisted inclusion between tensor products of operator spaces, with an
application to $2$-cocycles

Yemon Choi

(20th June 2016)

###### Abstract

Given operator spaces $V$ and $W$, let $\widetilde{W}$ denote the opposite
operator space structure on the same underlying Banach space. Although the
identity map $W\to\widetilde{W}$ is in general not completely bounded, we show
that the identity map on $V\mathbin{{\otimes}}W$ extends to a contractive
linear map $V\mathbin{\widehat{{\otimes}}}\widetilde{W}\to
V\mathbin{{\otimes}}_{\rm min}W$, where $\mathbin{\widehat{{\otimes}}}$ and
$\mathbin{{\otimes}}_{\rm min}$ denote the projective and injective tensor
products of operator spaces. We then sketch how this aids us in constructing
anti-symmetric $2$-cocycles on certain Fourier algebras.

Dedicated to John Rainwater, with thanks for his varied contributions and
generous spirit.

##  1 Introduction

We start by emphasising a convention that will be adhered to throughout this
note: all operator spaces are assumed, as part of the definition, to be
complete.

A core result in the theory of operator spaces is the following observation:
the operation on $K(\ell_{2})$ given by matrix transpose, $a\mapsto a^{\top}$,
fails to be completely bounded, even though it is an isometric involution of
Banach spaces. This fact serves to explain certain phenomena in non-
commutative harmonic analysis, and can be exploited to prove structural
results about Fourier and Fourier-Stieltjes algebras of locally compact
groups: see, for instance, [4, 12].

More generally: given an operator space $W$ — by which we mean a complex
vector space $W$, equipped with a sequence of complete norms
$\|{\cdot}\|_{(n)}$ on $M_{n}(W)={\mathbb{M}}_{n}\mathbin{{\otimes}}W$ that
satisfy Ruan’s axioms — one may define a new sequence of norms as follows:

$\|{\sum_{i}a_{i}\mathbin{{\otimes}}w_{i}}\|_{(n),{\rm
opp}}\mathbin{:=}\|{\sum_{i}a_{i}^{\top}\mathbin{{\otimes}}w_{i}}\|_{(n)}\qquad(a_{i}\in{\mathbb{M}}_{n},w_{i}\in
W).$

These matrix norms also define an operator space structure on $W$, which we
denote by $\widetilde{W}$ and call111A more natural name might be the
transposed operator space structure, but the terminology here appears to be
the standard one. the opposite operator space of $W$, or the opposite operator
space structure on the underlying Banach space of $W$. While $W$ and
$\widetilde{W}$ have the same underlying Banach space, in general they are not
isomorphic as operator spaces.

All this is well known. However, the following result appears to be new, or at
least not recorded in the literature. We write $\mathbin{\widehat{{\otimes}}}$
for the projective tensor product of operator spaces and
$\mathbin{{\otimes}}_{\rm min}$ for the injective tensor product (also known
as the minimal tensor product).

######  Theorem 1.1 (Main theorem).

Let $V$ and $W$ be operator spaces. The identity map on their algebraic tensor
product $V\mathbin{{\otimes}}W$ extends to a linear contraction

$\Psi_{V,W}:V\mathbin{\widehat{{\otimes}}}\widetilde{W}\to
V\mathbin{{\otimes}}_{\rm min}W.$

The proof is based on an interpolation argument suggested to the author by
John Rainwater (personal communication) and will be explained in Section 3.
Note that in general, $\Psi_{V,W}$ cannot be completely bounded (just take
$V={\mathbb{C}}$), and so it seems hard to obtain a more direct proof by
considering operator-space tensor norms on $V\mathbin{{\otimes}}W$. We also
note that in this theorem, we cannot replace the projective tensor product by
the Haagerup tensor product, even if we weaken “linear contraction” to
“bounded linear map” (Proposition 3.3).

Theorem 1.1 was originally motivated by a technical issue that arose in
studying the Hochschild cohomology of Fourier algebras, specifically the
problem of higher-dimensional weak amenability as defined in [7]. The main
issue is that when $A$ and $B$ are cb-versions of Banach algebras, and $V$ and
$W$ are cb-modules over $A$ and $B$ respectively, then
$V\mathbin{\widehat{{\otimes}}}\widetilde{W}$ might not be a Banach module
over $A\mathbin{\widehat{{\otimes}}}B$; however, $V\mathbin{{\otimes}}_{\rm
min}W$ is such a module, and then Theorem 1.1 allows us to replace the bad
space $V\mathbin{\widehat{{\otimes}}}\widetilde{W}$ with the better space
$V\mathbin{{\otimes}}_{\rm min}W$.

More precise statements can be found in Section 4, whose main result – on
existence of certain 2-cocycles – is stated as Theorem 4.8. The presentation
in Section 4 is more of a sketch than a comprehensive account: in future work
we intend to investigate these cohomology problems in much more detail, and
develop the appropriate theoretical framework in a more complete way.

#### Remarks on notation.

What we have written here as $\widetilde{W}$ is often denoted in the
literature by $W^{\rm op}$. We have chosen different notation because in some
of the intended applications, one is dealing with an operator space $A$ which
is also an algebra; and hence there is a potential conflict with the usage of
$A^{\rm op}$ to denote the “opposite algebra”, i.e. the algebra with the same
underlying vector space but with reversed product. In longer expressions, when
considering the opposite operator space, we use the notation
${(\dots)}^{\sim}$; for instance ${B(H)}^{\sim}$ denotes $B(H)$ equipped with
the opposite of its usual operator space structure.

##  2 Conventions and technical preliminaries

We assume the reader is familiar with the basic definitions of operator spaces
and completely bounded maps, as presented in [3] or [11]. We remind her that,
given operator spaces $E$ and $F$, the space $\operatorname{\it CB}(E,F)$ is
itself an operator space in a natural way. If $V$ is an operator space, then
the dual Banach space $V^{*}$ becomes an operator space under the
identification $V^{*}=\operatorname{\it CB}(V,{\mathbb{C}})$, while
$\operatorname{\it CB}({\mathbb{C}},V)$ is completely isometrically isomorphic
to $V$.

Although we will not use any category theory, it is convenient occasionally to
refer to the category of operator spaces and completely bounded maps, which we
denote by OpSp.

We shall abbreviate the phrase “operator space structure” to o.s.s. Whenever
$H$ is a Hilbert space and we refer to $B(H)$ as an operator space, we assume
(unless explicitly stated otherwise) that it is equipped with its usual,
canonical o.s.s.; note that if we do this, then there is a natural and
completely isometric identification of $B(H)$ with $\operatorname{\it
CB}(\textsf{COL}_{H})$, where $\textsf{COL}_{H}$ denotes $H$ equipped with the
column o.s.s.

#### Opposite operator spaces.

The opposite o.s.s. was already defined in the introduction; see also [11,
§2.10]. Let us collect some basic properties that do not seem to be mentioned
in [3] or [11]. It is easily checked that if $f:X\to Y$ is completely bounded,
then so is $f:\widetilde{X}\to\widetilde{Y}$, with the same cb-norm. For sake
of clarity, and to emphasise the functorial behaviour, we write this as
$\widetilde{f}:\widetilde{X}\to\widetilde{Y}$. The same calculation gives,
with some book-keeping, a more precise result: we omit the details.

######  Lemma 2.1.

Given operator spaces $E$ and $F$, we have a completely isometric isomorphism
${\operatorname{\it CB}(E,F)}^{\sim}\cong_{1}\operatorname{\it
CB}(\widetilde{E},\widetilde{F})$. In particular, we can identify
${(E^{*})}^{\sim}$ with $(\widetilde{E})^{*}$.

If $K$ is a Hilbert space, the transpose operator $\top:B(K)\to B(K)$,
$b\mapsto b^{\top}$, is defined by

$\langle b^{\top}\xi,\eta\rangle\mathbin{:=}\langle
b\eta,\xi\rangle\qquad(\xi,\eta\in K).$

More explicitly: if we fix a basis for $K$, then the matrix of $b^{\top}$ with
respect to this basis is the transpose of the matrix representing $b$ (with
respect to the same basis). It is clear that $\top$ provides a complete
isometry from $B(K)$ onto ${B(K)}^{\sim}$, and vice versa.

#### Tensor products and tensor norms.

For clarity, we repeat some notation. The algebraic tensor product of two
complex vector spaces $E$ and $F$ is denoted by $E\mathbin{{\otimes}}F$. The
projective and injective tensor products in the category OpSp are denoted by
$\mathbin{\widehat{{\otimes}}}$ and $\mathbin{{\otimes}}_{\rm min}$
respectively; this follows the notation of [11], rather than that of [3]. If
$f\in\operatorname{\it CB}(E,X)$ and $g\in\operatorname{\it CB}(F,Y)$ then by
tensoring we obtain completely bounded maps
$E\mathbin{\widehat{{\otimes}}}F\to X\mathbin{\widehat{{\otimes}}}Y$ and
$E\mathbin{{\otimes}}_{\rm min}F\to X\mathbin{{\otimes}}_{\rm min}Y$; for
extra emphasis, these maps will be denoted by
$f\mathbin{\widehat{{\otimes}}}g$ and $f\mathbin{{\otimes}}_{\rm min}g$
respectively. We have
$\|{f\mathbin{\widehat{{\otimes}}}g}\|_{cb}=\|{f\mathbin{{\otimes}}_{\rm
min}g}\|_{cb}=\|{f}\|_{cb}\|{g}\|_{cb}$.

In proving Theorem 1.1, we exploit the fact that the injective tensor norm on
$B(H)\mathbin{{\otimes}}B(K)$ can be calculated in terms of the action of this
algebra on $S_{2}(K,H)$, the space of Hilbert-Schmidt operators $K\to H$.

######  Lemma 2.2.

  1. (i)

The linear map $\theta_{0}:B(H)\mathbin{{\otimes}}B(K)\to B(S_{2}(K,H))$ that
is defined by $\theta_{0}(a\mathbin{{\otimes}}b)(c)\mathbin{:=}acb^{\top}$
extends to a complete isometry $\theta:B(H)\mathbin{{\otimes}}_{\rm
min}B(K)\to B(S_{2}(K,H))$.

  2. (ii)

There is a complete isometry $\Lambda:B(H)\mathbin{{\otimes}}_{\rm
min}{B(K)}^{\sim}\to B(S_{2}(K,H))$, which satisfies
$\Lambda(a\mathbin{{\otimes}}b)(c)=acb$ for all $a\in B(H)$, $b\in B(K)$ and
$c\in S_{2}(K,H))$.

###### Sketch of the proof.

Part (i) can be checked by considering the expression
$\langle(a\mathbin{{\otimes}}b^{\top})\xi,\eta\rangle$ for $\xi,\eta\in
H\otimes_{2}K$, expanding out $\xi$ and $\eta$ as linear combinations of
elementary tensors, and making direct calculations; see also Proposition 2.9.1
in [11], or the calculations in Section 3.5 of [3]. Part (ii) follows from
part (i) by composing $\theta$ with the completely isometric isomorphism
$\iota\mathbin{{\otimes}}\top:B(H)\mathbin{{\otimes}}_{\rm
min}{B(K)}^{\sim}\to B(H)\mathbin{{\otimes}}_{\rm min}B(K)$. ∎

Instead of the “concrete” definition of the matrix norms on the projective
tensor product of operator spaces, we prefer to use its characteristic
universal property: it linearizes those bilinear maps $h:E\times F\to G$ which
are “completely bounded” in the following sense

$\sup_{m,n\geq
1}\left\\{\|{h(x_{ij},y_{pq})}\|_{M_{mn}(E\mathbin{{\otimes}}F)}\colon[x_{ij}]\in{\sf
ball}_{1}M_{m}(E)\;,\;[y_{pq}]\in{\sf ball}_{1}M_{n}(F)\right\\}<\infty.$

(Here, our terminology is that of [3]; the reader should beware that often
such maps are instead called “jointly completely bounded”, and that the term
“completely bounded” is then used for what [3] call “multiplicatively
bounded”.) Denoting the space of such maps by $\operatorname{\it CB}_{\rm
bil}(E\times F;G)$, there are natural and completely isometric identifications

$\operatorname{\it
CB}(E\mathbin{\widehat{{\otimes}}}F,G)\cong_{1}\operatorname{\it CB}_{\rm
bil}(E\times F;G)\cong_{1}\operatorname{\it CB}(E,\operatorname{\it
CB}(F,G)).$

(See [3, Proposition 7.1.2].) We note, for future reference, that there is a
natural and completely isometric identification of
$\widetilde{E}\mathbin{\widehat{{\otimes}}}\widetilde{F}$ with
${(E\mathbin{\widehat{{\otimes}}}F)}^{\sim}$.

######  Remark 2.3.

If $H$ is a Hilbert space equipped with column o.s.s., the natural left action
of $B(H)$ on $H$ defines a completely contractive bilinear map $B(H)\times
H\to H$. This can be verified directly, but it is more illuminating to see it
as a special case of the following general fact: given operator spaces $E$,
$F$ and $G$, the composition operation

$\operatorname{\it CB}(F,G)\times\operatorname{\it
CB}(E,F)\to\operatorname{\it CB}(E,G)$

is completely contractive as a bilinear map. (To deduce the original
statement, take $E={\mathbb{C}}$ and $F=G=\textsf{COL}_{H}$.) Similarly, if we
denote the Banach-space adjoint of an operator $b:H\to H$ by $b^{\\#}:H^{*}\to
H^{*}$, then the adjoint action of $B(H)$ on $H^{*}$ defines a completely
contractive bilinear map $H^{*}\times B(H)\to H^{*}$. (Take
$E=F=\textsf{COL}_{H}$ and $G={\mathbb{C}}$.)

We also need a standard lemma on “interchanging tensor products”: see [3,
Theorem 8.1.10] for a proof.

######  Lemma 2.4.

Let $E$, $F$ and $G$ be operator spaces. There are complete contractions

$E\mathbin{\widehat{{\otimes}}}(F\mathbin{{\otimes}}_{\rm
min}G)\to(E\mathbin{\widehat{{\otimes}}}F)\mathbin{{\otimes}}_{\rm
min}G\quad,\quad(E\mathbin{{\otimes}}_{\rm
min}F)\mathbin{\widehat{{\otimes}}}G\to E\mathbin{{\otimes}}_{\rm
min}(F\mathbin{\widehat{{\otimes}}}G),$

both of which are the identity map when restricted to elementary tensors.

##  3 The main technical result

We will deduce the main result (Theorem 1.1) from the special case $V=B(H)$,
$W=B(K)$. It is convenient to reformulate this special case slightly, using
Lemma 2.2.

######  Theorem 3.1.

Let $H$ and $K$ be Hilbert spaces. Then the linear map
$\Phi_{0}:B(H)\mathbin{{\otimes}}B(K)\to B(S_{2}(K,H))$ that is defined by
$\Phi_{0}(a\mathbin{{\otimes}}b)(c)=acb$ extends to a contractive linear map
$\Phi_{2}:B(H)\mathbin{\widehat{{\otimes}}}B(K)\to B(S_{2}(K,H))$.

The following proof is based on suggestions of John Rainwater (see the
Acknowledgements for further details) and we thank him for his consent to
include the proof here.

###### Proof (Rainwater)..

Write $S_{0}(K,H)$ for the space of finite-rank operators $K\to H$,
$S_{1}(K,H)$ for the space of nuclear operators $K\to H$, and
$S_{\infty}(K,H)$ for the space of all compact operators $K\to H$. Then we
have three linear maps

$\Psi_{j}:B(H)\mathbin{{\otimes}}S_{j}(K,H)\mathbin{{\otimes}}B(K)\to
S_{j}(K,H)\qquad(j=0,1,\infty),$

each of which is defined on elementary tensors by
$a\mathbin{{\otimes}}c\mathbin{{\otimes}}b\mapsto acb$. The key observation is
that, if we equip $S_{1}(K,H)$ and $S_{\infty}(K,H)$ with appropriate o.s.
structures, then we can extend both $\Psi_{1}$ and $\Psi_{\infty}$ to
completely contractive linear maps on the threefold projective tensor product.

The details are as follows. Equip $H$ and $K$ with column o.s.s., and consider
the two operator spaces $H\mathbin{\widehat{{\otimes}}}K^{*}$ and
$H\mathbin{{\otimes}}_{\rm min}K^{*}$. The usual identification of
$H\mathbin{{\otimes}}K^{*}$ with $S(K,H)$ extends to give two isometric
isomorphisms of Banach spaces
$H\mathbin{\widehat{{\otimes}}}K^{*}\cong_{1}S_{1}(K,H)$ and
$H\mathbin{{\otimes}}_{\rm min}K^{*}\cong_{1}S_{\infty}(K,H)$. (See e.g. [3],
Proposition 8.2.1.) So we can identify $\Psi_{0}$, $\Psi_{1}$ and
$\Psi_{\infty}$ with the following linear maps:

$\begin{array}[]{rccl}\Psi_{0}:&B(H)\mathbin{{\otimes}}[H\mathbin{{\otimes}}K^{*}]\mathbin{{\otimes}}B(K)&\to&H\mathbin{{\otimes}}K^{*}\\\
\Psi_{1}:&B(H)\mathbin{{\otimes}}[H\mathbin{\widehat{{\otimes}}}K^{*}]\mathbin{{\otimes}}B(K)&\to&H\mathbin{\widehat{{\otimes}}}K^{*}\\\
\Psi_{\infty}:&B(H)\mathbin{{\otimes}}[H\mathbin{{\otimes}}_{\rm
min}K^{*}]\mathbin{{\otimes}}B(K)&\to&H\mathbin{{\otimes}}_{\rm
min}K^{*}\end{array}$

On elementary tensors, these maps satisfy

$\Psi_{j}(a\mathbin{{\otimes}}(v\mathbin{{\otimes}}\phi)\mathbin{{\otimes}}b)=av\mathbin{{\otimes}}b^{\\#}\phi\qquad\text{($a\in
B(H)$, $v\in H$, $\phi\in K^{*}$, $b\in B(K)$)}$

for $j=0,1,\infty$. (Recall that $b^{\\#}:K^{*}\to K^{*}$ is the adjoint in
the Banach space sense.)

As noted in Remark 2.3, there are completely contractive maps
$\lambda:B(H)\mathbin{\widehat{{\otimes}}}H\to H$ and
$\rho:K^{*}\mathbin{\widehat{{\otimes}}}B(K)\to K$, satisfying
$\lambda(a\mathbin{{\otimes}}v)=av$ ($a\in B(H)$, $v\in H$) and
$\rho(\phi\mathbin{{\otimes}}b)=b^{\\#}\phi$ ($b\in B(K)$, $\phi\in K^{*}$).
Hence, by associativity of $\mathbin{\widehat{{\otimes}}}$, we see that
$\lambda\mathbin{\widehat{{\otimes}}}\rho$ defines a complete contraction

$B(H)\mathbin{\widehat{{\otimes}}}H\mathbin{\widehat{{\otimes}}}K^{*}\mathbin{\widehat{{\otimes}}}B(K)\to
H\mathbin{\widehat{{\otimes}}}K^{*}$ (1)

which extends $\Psi_{1}$. Furthermore, by using associativity of
$\mathbin{\widehat{{\otimes}}}$, and using Lemma 2.4 twice, we have complete
contractions

$\displaystyle B(H)\mathbin{\widehat{{\otimes}}}[H\mathbin{{\otimes}}_{\rm
min}K^{*}]\mathbin{\widehat{{\otimes}}}B(K)$
$\displaystyle\cong_{1}\left(B(H)\mathbin{\widehat{{\otimes}}}[H\mathbin{{\otimes}}_{\rm
min}K^{*}]\right)\mathbin{\widehat{{\otimes}}}B(K)$
$\displaystyle\to\left([B(H)\mathbin{\widehat{{\otimes}}}H]\mathbin{{\otimes}}_{\rm
min}K^{*}\right)\mathbin{\widehat{{\otimes}}}B(K)$
$\displaystyle\to[B(H)\mathbin{\widehat{{\otimes}}}H]\mathbin{{\otimes}}_{\rm
min}[K^{*}\mathbin{\widehat{{\otimes}}}B(K)].$

Composing these maps with $\lambda\mathbin{{\otimes}}_{\rm min}\rho$, we
obtain a complete contraction

$B(H)\mathbin{\widehat{{\otimes}}}[H\mathbin{{\otimes}}_{\rm
min}K^{*}]\mathbin{\widehat{{\otimes}}}B(K)\to H\mathbin{{\otimes}}_{\rm
min}K^{*}$ (2)

which extends $\Psi_{\infty}$.

Fix $x\in B(H)\mathbin{{\otimes}}B(K)$ such that
$\|{x}\|_{B(H)\mathbin{\widehat{{\otimes}}}B(K)}\leq 1$, and consider the
corresponding linear map $\Phi_{0}(x):S_{0}(K,H)\to S_{0}(K,H)$. In view of
(1) and (2), we obtain contractive linear maps

$\Phi_{1}(x):S_{1}(K,H)\to
S_{1}(K,H)\quad,\quad\Phi_{\infty}(x):S_{\infty}(K,H)\to S_{\infty}(K,H),$

which both extend the map $\Phi_{0}(x)$.

Viewing $(S_{1}(K,H),S_{\infty}(K,H))$ as a compatible interpolation couple of
Banach spaces, we have
$(S_{1}(K,H),S_{\infty}(K,H))_{1/2}\cong_{1}S_{2}(K,H)$. By the Riesz–Thorin
interpolation theorem, $\Phi_{0}(x)$ extends to a contractive linear map
$\Phi_{2}(x):S_{2}(K,H)\to S_{2}(K,H)$. It is now routine to check that
$\Phi_{2}$ defines a linear contraction
$B(H)\mathbin{\widehat{{\otimes}}}B(K)\to B(S_{2}(K,H))$, which completes the
proof of Theorem 3.1. ∎

###### Proof of Theorem 1.1.

Let $V$ and $W$ be operator spaces, and fix two completely isometric
embeddings $j_{V}:V\hookrightarrow B(H)$ and $j_{W}:W\hookrightarrow B(K)$ for
some choices of Hilbert spaces $H$ and $K$. Consider the diagram

$\begin{diagram}$

where the top arrow $\Psi_{B(H),B(K)}$ restricts to the identity map on
elementary tensors; note that $\Psi_{B(H),B(K)}$ is well-defined and
contractive by combining Theorem 3.1 with Lemma 2.2. Now observe that the
left-hand vertical arrow in the diagram is a (complete) contraction, while the
right-hand vertical arrow is a (complete) isometry (using the “injective”
property of $\mathbin{{\otimes}}_{\rm min}$). Hence, for any elementary tensor
$x\in V\mathbin{{\otimes}}W$, we have

$\displaystyle\|{x}\|_{V\mathbin{{\otimes}}_{\rm min}W}$
$\displaystyle=\|{(j_{V}\mathbin{{\otimes}}j_{W})(x)}\|_{B(H)\mathbin{{\otimes}}_{\rm
min}B(K)}$
$\displaystyle\leq\|{\Psi_{B(H),B(K)}}\|\|{(j_{V}\mathbin{{\otimes}}\widetilde{j_{W}}(x)}\|_{B(H)\mathbin{\widehat{{\otimes}}}{B(K)}^{\sim}}\leq\|{x}\|_{V\mathbin{\widehat{{\otimes}}}\widetilde{W}}\;,$

which completes the proof. ∎

######  Remark 3.2.

If $K$ is infinite-dimensional, $\Phi_{2}$ cannot be completely bounded.
However, let $\textsf{OH}_{K,H}$ denote the Hilbert space $S_{2}(K,H)$
equipped not with column o.s.s., but with the self-dual o.s.s. of Pisier; then
by applying complex interpolation in the category OpSp, one can show that
$\Phi_{2}$ is completely contractive as a map
$B(H)\mathbin{\widehat{{\otimes}}}B(K)\to\operatorname{\it
CB}(\textsf{OH}_{K,H})$. We have chosen to omit the details from this note,
since this sharper form of Theorem 3.1 does not seem to be helpful for the
applications in Section 4.

We finish by briefly justifying the claim, made after the statement of Theorem
1.1, that one cannot replace the projective tensor product with the Haagerup
tensor product in that theorem. The result is probably well known but we
include a proof for sake of completeness.

######  Proposition 3.3.

There exist $\operatorname{C}^{*}$-algebras $A$ and $B$ such that the identity
map $A\mathbin{{\otimes}}B\to A\mathbin{{\otimes}}B$ does not extend to any
bounded linear map $A\mathbin{{\otimes}}_{h}\widetilde{B}\to
A\mathbin{{\otimes}}_{\rm min}B$.

###### Proof.

For convenience we take $A=B=B(K)$ for an infinite-dimensional Hilbert space
$K$: it will be seen that we actually get separable counterexamples in the
end.

Recall that $b\mapsto b^{\top}$ is a complete isometry on $B(K)$. Hence it
suffices to show that the map ${\rm
id}\mathbin{{\otimes}}\top:B(K)\mathbin{{\otimes}}B(K)\to
B(K)\mathbin{{\otimes}}B(K)$ has no bounded extension to a map
$B(K)\mathbin{{\otimes}}_{h}B(K)\to B(K)\mathbin{{\otimes}}_{\rm min}B(K)$.

Fix an infinite sequence of isometries $s_{1},s_{2},\dots$ in $B(K)$ with the
property that the range projections $s_{j}s_{j}^{*}$ are pairwise orthogonal.
In particular, $s_{j}^{*}s_{k}=0$ for $j\neq k$ and
$P_{n}\mathbin{:=}\sum_{j=1}^{n}s_{j}s_{j}^{*}$ is an orthogonal projection
for each $n$.

Let $x_{n}=\sum_{j=1}^{n}s_{j}\mathbin{{\otimes}}s_{j}^{\top}\in
B(H)\mathbin{{\otimes}}B(K)$: then using the standard formula for the Haagerup
tensor norm (see [11, Chapter 5]),

$\|{x_{n}}\|_{B(K)\mathbin{{\otimes}}_{h}B(K)}\leq\bigl{\|}{\sum\nolimits_{j=1}^{n}s_{j}s_{j}^{*}}\bigr{\|}^{1/2}\bigl{\|}{\sum\nolimits_{k=1}^{n}(s_{k}^{\top})^{*}s_{k}^{\top}}\bigr{\|}^{1/2}=\|{P_{n}}\|^{1/2}\|{P_{n}^{\top}}\|_{1/2}\leq
1\;;$

On the other hand, let $y_{n}=({\rm
id}\mathbin{{\otimes}}\top)(x_{n})=\sum_{j=1}^{n}s_{j}\mathbin{{\otimes}}s_{j}\in
B(K)\mathbin{{\otimes}}_{\rm min}B(K)$. By the $\operatorname{C}^{*}$-identity
$\|{y_{n}}\|=\|{y_{n}^{*}y_{n}}\|^{1/2}$, with both norms taken in
$B(K)\mathbin{{\otimes}}_{\rm min}B(K)$. But since the $s_{j}$ are isometries
with pairwise orthogonal ranges,

$y_{n}^{*}y_{n}=\sum\nolimits_{j,k=1}^{n}s_{j}^{*}s_{k}\mathbin{{\otimes}}s_{j}^{*}s_{k}=nI\mathbin{{\otimes}}I\;.$

Hence $\|{({\rm
id}\mathbin{{\otimes}}\top)(x_{n})}\|_{B(K)\mathbin{{\otimes}}_{\rm
min}B(K)}\geq n^{1/2}\|{x_{n}}\|_{B(K)\mathbin{{\otimes}}_{h}B(K)}$, and since
$n$ is arbitrary the result follows. ∎

######  Remark 3.4.

Since $\mathbin{{\otimes}}_{h}$ and $\mathbin{{\otimes}}_{\rm min}$ are both
injective tensor norms, the proof of Proposition 3.3 shows that we could take
$A$ to be the $\operatorname{C}^{*}$-algebra generated by the isometries
$(s_{j})_{j\geq 1}$, i.e. the Cuntz algebra ${\mathcal{O}}_{\infty}$, and
$B=A^{\rm op}$, the $\operatorname{C}^{*}$-algebra obtained from $A$ by
reversing the product. In particular, we get nuclear examples.

##  4 Constructing $2$-cocycles on certain Fourier algebras

In this longer section, we give an outline of a problem arising in the study
of Fourier algebras, and how Theorem 1.1 supplies a missing piece of the
puzzle. This may be of interest to some specialists in abstract harmonic
analysis. Since this will not be a systematic or comprehensive exposition of
the Hochschild cohomology groups of Fourier algebras, we shall omit many
definitions and details. The reader is referred to the articles cited in this
section, and their bibliographies, for the relevant definitions and
background.

###  4.1 Obtaining cocycles from derivations

Consider commutative algebras $A$ and $B$ (not necessarily unital), a
symmetric $A$-bimodule $X$, and a symmetric $B$-bimodule $Y$. Given
derivations $D_{A}:A\to X$, $D_{B}:B\to Y$, we can “wedge
$D_{A}\mathbin{{\otimes}}\iota_{B}$ and $\iota_{A}\mathbin{{\otimes}}D_{B}$”
to obtain an antisymmetric $2$-cocycle $F_{0}\in
Z^{2}(A\mathbin{{\otimes}}B,X\mathbin{{\otimes}}Y)$:

$F_{0}(a_{1}\mathbin{{\otimes}}b_{1},a_{2}\mathbin{{\otimes}}b_{2})\mathbin{:=}[D_{A}(a_{1})\cdot
a_{2}]\mathbin{{\otimes}}[b_{1}\cdot D_{B}(b_{2})]-[a_{1}\cdot
D_{A}(a_{2})\mathbin{{\otimes}}D_{B}(b_{1})\cdot b_{2}]$ (3)

for $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B$. One can verify the $2$-cocycle
identity by hand: this makes it clear where we require $D_{A}$ and $D_{B}$ to
be derivations, and also why we want the bimodules $X$ and $Y$ to be
symmetric.

Furthermore, under various mild conditions (for instance, if $A$ and $B$ are
unital), $F_{0}$ is non-zero provided that $D_{A}$ and $D_{B}$ are non-zero.
Since degree-$2$ coboundaries are symmetric as functions of two variables,
while $F$ is antisymmetric, it follows that $F_{0}$ defines a non-zero element
of the Hochschild cohomology group
$H^{2}(A\mathbin{{\otimes}}B,X\mathbin{{\otimes}}Y)$.

The following example should be kept in mind as a motivation for the general
construction.

######  Example 4.1.

Let $A=X={\mathbb{C}}[w]$, $B=Y={\mathbb{C}}[z]$, $D_{A}g=dg/dw$ and
$D_{B}h=dh/dz$. Identifying $A\mathbin{{\otimes}}B$ with ${\mathbb{C}}[z,w]$,
the $2$-cocycle $F$ is given by

$F(f_{1},f_{2})=\frac{\partial f_{1}}{\partial w}\frac{\partial
f_{2}}{\partial z}-\frac{\partial f_{2}}{\partial w}\frac{\partial
f_{1}}{\partial z}\qquad(f_{1},f_{2}\in{\mathbb{C}}[z,w]).$

With minor modifications, the same procedure still works in the setting of
commutative Banach algebras and symmetric Banach bimodules. Let
$\mathbin{\widehat{{\otimes}}}_{\gamma}$ denote the projective tensor product
of Banach spaces. Then, given commutative Banach algebras $A$ and $B$,
symmetric Banach bimodules $X$ and $Y$ over $A$ and $B$ respectively, and
continuous derivations $D_{A}:A\to X$, $D_{B}:B\to Y$, the formula (3) defines
an antisymmetric, continuous $2$-cocycle

$F:A\mathbin{\widehat{{\otimes}}}_{\gamma}B\times
A\mathbin{\widehat{{\otimes}}}_{\gamma}B\to
X\mathbin{\widehat{{\otimes}}}_{\gamma}Y\;.$ (4)

In particular, this gives a method for finding commutative Banach algebras
whose degree-$2$ continuous Hochschild cohomology groups are non-zero. For
instance, since
$\ell^{1}({\mathbb{Z}}_{+}^{2})\cong_{1}\ell^{1}({\mathbb{Z}}_{+})\mathbin{\widehat{{\otimes}}}_{\gamma}\ell^{1}({\mathbb{Z}}_{+})$,
a modified version of Example 4.1 shows222This is essentially the same as the
example given in [5, §5]. that
${\mathcal{H}}^{2}(\ell^{1}({\mathbb{Z}}_{+}^{2}),{\mathbb{C}}_{\varphi})\neq
0$ for suitable choices of a character
$\varphi:\ell^{1}({\mathbb{Z}}_{+}^{2})\to{\mathbb{C}}$.

Can we do something similar for Fourier algebras of locally compact groups? We
now know many examples of locally compact groups $G$ for which there exist
non-zero continuous derivations from $\operatorname{A}(G)$ to suitable
symmetric bimodules (see [8] for some of the history and a guide to the
relevant literature). However, attempting to use (4) runs into a problem:

  * •

the natural map
$\operatorname{A}(G_{1})\mathbin{\widehat{{\otimes}}}_{\gamma}\operatorname{A}(G_{2})\to\operatorname{A}(G_{1}\times
G_{2})$ is surjective if and only if either $G_{1}$ or $G_{2}$ has an abelian
subgroup of finite index [9];

  * •

if $G$ has an abelian subgroup of finite index, then all continuous
derivations from $\operatorname{A}(G)$ into symmetric Banach
$\operatorname{A}(G)$-bimodules must vanish [4, Theorem 3.3].

Since there is a completely isometric algebra isomorphism
$\operatorname{A}(G_{1})\mathbin{\widehat{{\otimes}}}\operatorname{A}(G_{2})\cong_{1}\operatorname{A}(G_{1}\times
G_{2})$ (see e.g. [3, Theorem 7.2.4]), we might try to mimic (4) in the
operator-space category OpSp. However, if we try to do this directly, we run
into a new problem: all algebras, module actions and derivations would now
live in OpSp; and in this setting, all derivations of the appropriate form are
zero (see [13, Theorem 5.2] or [14]).

Nevertheless, not all is lost. For many non-abelian, connected groups $G$, one
can actually find derivations that are completely bounded as maps
$\operatorname{A}(G)\to{(\operatorname{A}(G)^{*})}^{\sim}$. This is not
mentioned explicitly in the original articles, although it is lurking
implicitly (see e.g. Remark 3.2 in [1] and [8, Theorem 2.1]). The next
subsection shows how we may exploit this.

###  4.2 The needle returns to the start of the song

We use some slightly non-standard terminology. By a cb-Banach algebra, we mean
an operator space $A$ equipped with a bilinear, completely bounded and
associative map $A\times A\to A$. Given such an $A$, we define a cb-Banach
$A$-bimodule to be an operator space $X$, equipped with an $A$-bimodule
structure such that the left action $A\times X\to X$ and the right action
$X\times A\to X$ are both completely bounded.

These notions interact well with the “opposite o.s.s. functor”, but care is
needed. If $A$ is a cb-Banach algebra then so is $\widetilde{A}$; and if $X$
is a cb-Banach $A$-bimodule, $\widetilde{X}$ is a cb-Banach
$\widetilde{A}$-bimodule. (Note that when passing to the opposite o.s.s. we
are not reversing the algebra product or the module actions in any way; we are
merely changing the way we put norms on certain maps between matricial
spaces.) However, there is no reason to suppose $\widetilde{X}$ will be a cb-
Banach $A$-bimodule! In particular, although $A$ and $A^{*}$ are cb-Banach
$A$-bimodules, $\widetilde{A}$ and ${(A^{*})}^{\sim}$ need not be.

######  Proposition 4.2.

Let $A$ and $B$ be cb-Banach algebras; let $X$ be a cb-Banach $A$-bimodule and
$Y$ a cb-Banach $B$-bimodule. Let $T_{A}\in\operatorname{\it
CB}(A,\widetilde{X})$ and $T_{B}\in\operatorname{\it CB}(B,\widetilde{Y})$.
Then, if we define
$F_{1},F_{2}:(A\mathbin{{\otimes}}B)\times(A\mathbin{{\otimes}}B)\to
X\mathbin{{\otimes}}Y$ by

$\displaystyle
F_{1}(a_{1}\mathbin{{\otimes}}b_{1},a_{2}\mathbin{{\otimes}}b_{2})$
$\displaystyle=[T_{A}(a_{1})\cdot a_{2}]\mathbin{{\otimes}}[b_{1}\cdot
T_{B}(b_{2})]\;,$ $\displaystyle
F_{2}(a_{1}\mathbin{{\otimes}}b_{1},a_{2}\mathbin{{\otimes}}b_{2})$
$\displaystyle=[a_{1}\cdot T_{A}(a_{2})]\mathbin{{\otimes}}[T_{B}(b_{1})\cdot
b_{2}]\;,$

both $F_{1}$ and $F_{2}$ extend to bounded bilinear maps
$(A\mathbin{\widehat{{\otimes}}}B)\times(A\mathbin{\widehat{{\otimes}}}B)\to
X\mathbin{\widehat{{\otimes}}}\widetilde{Y}$.

###### Proof.

We will only give the proof for $F_{1}$; the proof for $F_{2}$ is very
similar.

Since $T_{A}:A\to\widetilde{X}$ is completely bounded, so is
$\widetilde{T_{A}}:\widetilde{A}\to X$. Therefore,
$S=\widetilde{T_{A}}\mathbin{\widehat{{\otimes}}}\iota_{\widetilde{B}}\mathbin{\widehat{{\otimes}}}\iota_{A}\mathbin{\widehat{{\otimes}}}T_{B}$
is a complete contraction from
$\widetilde{A}\mathbin{\widehat{{\otimes}}}\widetilde{B}\mathbin{\widehat{{\otimes}}}A\mathbin{\widehat{{\otimes}}}B$
to
$X\mathbin{\widehat{{\otimes}}}\widetilde{B}\mathbin{\widehat{{\otimes}}}A\mathbin{\widehat{{\otimes}}}\widetilde{Y}$.
Then, since $X$ is a cb-Banach $A$-bimodule, and $\widetilde{Y}$ is a cb-
Banach $\widetilde{B}$-bimodule, we have a complete contraction
$R:X\mathbin{\widehat{{\otimes}}}\widetilde{B}\mathbin{\widehat{{\otimes}}}A\mathbin{\widehat{{\otimes}}}\widetilde{Y}\to
X\mathbin{\widehat{{\otimes}}}\widetilde{Y}$, which satisfies
$R(x\mathbin{{\otimes}}b_{1}\mathbin{{\otimes}}a_{2}\mathbin{{\otimes}}y)=(x\cdot
a_{2})\mathbin{{\otimes}}(b_{1}\mathbin{{\otimes}}y)$.

Since
$\widetilde{A}\mathbin{\widehat{{\otimes}}}\widetilde{B}=(A\mathbin{\widehat{{\otimes}}}B)^{\sim}$,
the composite map $RS$ defines a completely bounded bilinear map from
$(A\mathbin{\widehat{{\otimes}}}B)^{\sim}\times(A\mathbin{\widehat{{\otimes}}}B)$
to $X\mathbin{\widehat{{\otimes}}}\widetilde{Y}$, which agrees with $F_{1}$ on
$(A\mathbin{{\otimes}}B)\times(A\mathbin{{\otimes}}B)$. In particular, $F_{1}$
extends to a bounded bilinear map (no longer completely bounded!) from
$(A\mathbin{\widehat{{\otimes}}}B)\times(A\mathbin{\widehat{{\otimes}}}B)$ to
$X\mathbin{\widehat{{\otimes}}}\widetilde{Y}$. ∎

######  Remark 4.3.

The proof of Proposition 4.2 might seem like overkill. Things would be much
easier if
$T_{A}\mathbin{{\otimes}}\iota_{B}\mathbin{{\otimes}}\iota_{A}\mathbin{{\otimes}}T_{B}$
extended to a continuous linear map from
$(A\mathbin{\widehat{{\otimes}}}B)\mathbin{\widehat{{\otimes}}}_{\gamma}(A\mathbin{\widehat{{\otimes}}}B)$
to
$(X\mathbin{\widehat{{\otimes}}}A)\mathbin{\widehat{{\otimes}}}_{\gamma}(B\mathbin{\widehat{{\otimes}}}Y)$;
however, we see no reason why this should always hold.

Why does Proposition 4.2 not give us the version of Equation (3) that we need?
Ultimately, we want to consider $F_{1}-F_{2}$ as our candidate for a
$2$-cocycle on the Banach algebra $A\mathbin{\widehat{{\otimes}}}B$, and so we
need it to take values in a Banach
$(A\mathbin{\widehat{{\otimes}}}B)$-bimodule. But there seems to be no reason
that the action
$a\mathbin{{\otimes}}b\mathbin{{\otimes}}x\mathbin{{\otimes}}y\mapsto
ax\mathbin{{\otimes}}by$ should extend to a linear map
$(A\mathbin{\widehat{{\otimes}}}B)\mathbin{\widehat{{\otimes}}}_{\gamma}(X\mathbin{\widehat{{\otimes}}}\widetilde{Y})\to
X\mathbin{\widehat{{\otimes}}}\widetilde{Y}$; in general
$X\mathbin{\widehat{{\otimes}}}\widetilde{Y}$ and
$X\mathbin{\widehat{{\otimes}}}Y$ need not have the same underlying Banach
space! On the other hand, there is no problem with $X\mathbin{{\otimes}}_{\rm
min}Y$.

######  Lemma 4.4.

Let $A$ and $B$ be cb-Banach algebras; let $X$ be a cb-Banach $A$-bimodule and
$Y$ a cb-Banach $B$-bimodule. Then $X\mathbin{{\otimes}}_{\rm min}Y$ is a
(cb-)Banach $A\mathbin{\widehat{{\otimes}}}B$-bimodule.

###### Proof.

Consider the map
$(A\mathbin{\widehat{{\otimes}}}B)\mathbin{\widehat{{\otimes}}}(X\mathbin{{\otimes}}_{\rm
min}Y)\cong_{1}A\mathbin{\widehat{{\otimes}}}(X\mathbin{{\otimes}}_{\rm
min}Y)\mathbin{\widehat{{\otimes}}}B\longrightarrow(A\mathbin{\widehat{{\otimes}}}X)\mathbin{{\otimes}}_{\rm
min}(Y\mathbin{\widehat{{\otimes}}}B)$; this is well-defined and completely
contractive by repeated use of Lemma 2.4. Then the left and right actions of
$A$ and $B$ on $X$ and $Y$ give two completely bounded bilinear maps
$A\mathbin{\widehat{{\otimes}}}X\to X$ and $Y\mathbin{\widehat{{\otimes}}}B$;
min-tensoring these and composing with the previous map, we obtain the desired
cb-actions of $A\mathbin{\widehat{{\otimes}}}B$ on $X\mathbin{{\otimes}}_{\rm
min}Y$. ∎

###  4.3 The application to Fourier algebras

The next definition is purely for convenience of presentation.

######  Definition 4.5.

$G\in{\mathcal{S}}$ if there exists a non-zero bounded derivation
$\operatorname{A}(G)\to\operatorname{A}(G)^{*}$ that is completely bounded as
a map $\operatorname{A}(G)\to{(\operatorname{A}(G)^{*})}^{\sim}$.

Note that by Herz’s restriction theorem, if $G$ contains a closed subgroup
belongin to ${\mathcal{S}}$, then $G\in{\mathcal{S}}$.

We take a diversion to record some easy calculations, which would otherwise
clutter up the proof of Theorem 4.8.

######  Lemma 4.6.

Let $R$ be a commutative algebra. Consider the following subsets of $R$:

$X_{1,1}=\\{ab\colon a,b\in R\\}\quad,\quad X_{2}=\\{a^{2}\colon a\in
R\\}\quad,$ $X_{1,1,1,1}=\\{abcd\colon a,b,c,d\in R\\}\quad,\quad
X_{2,2}=\\{a^{2}b^{2}\colon a,b\in R\\}\quad,\quad X_{4}=\\{a^{4}\colon a\in
R\\}.$

Then $\operatorname{lin}X_{1,1}=\operatorname{lin}X_{2}$ and
$\operatorname{lin}X_{1,1,1,1}=\operatorname{lin}X_{2,2}=\operatorname{lin}X_{4}$.

###### Idea of the proof..

These follow from the polarization identities
$ab=\frac{1}{4}\left[(a+b)^{2}-(a-b)^{2}\right]$ and
$x^{2}y^{2}=\frac{1}{24}\left[(x+y)^{4}+(x-y)^{4}-(x+iy)^{4}-(x-iy)^{4}\right]$.
We leave it to the reader to fill in the details. ∎

######  Corollary 4.7.

Let $G$ be a locally compact group. Then $\\{a^{4}\colon
a\in\operatorname{A}(G)\\}$ and $\\{b^{2}\colon b\in\operatorname{A}(G)\\}$
both have dense linear span in $\operatorname{A}(G)$.

###### Proof.

Let $A_{0}=\operatorname{A}(G)\cap C_{c}(G)$. Standard results on Fourier
algebras tell us that $A_{0}$ is dense in $\operatorname{A}(G)$; moreover, for
each $f\in A_{0}$ there exists $g\in A_{0}$ such that $fg=f$. It then follows
from Lemma 4.6 that $A_{0}=\operatorname{lin}\\{a^{4}\colon a\in A_{0}\\}$ and
$A_{0}=\operatorname{lin}\\{a^{2}\colon a\in A_{0}\\}$. The rest is clear. ∎

Putting everything together, we arrive at our main theorem concerning Fourier
algebras.

######  Theorem 4.8 (Constructing antisymmetric $2$-cocycles on Fourier
algebras).

Let $H,L\in{\mathcal{S}}$ with corresponding non-zero derivations
$D_{H}:\operatorname{A}(H)\to\operatorname{A}(H)^{*}$ and
$D_{L}:\operatorname{A}(L)\to\operatorname{A}(L)^{*}$ that are “completely
bounded into the opposite o.s.s.” Then the bilinear map

$F_{0}:(\operatorname{A}(H)\mathbin{{\otimes}}\operatorname{A}(L))\times(\operatorname{A}(H)\mathbin{{\otimes}}\operatorname{A}(L))\to\operatorname{A}(H)^{*}\mathbin{{\otimes}}\operatorname{A}(L)^{*}\;,$

defined as in Equation (3), extends to a bounded bilinear map
$F:\operatorname{A}(H\times L)\times\operatorname{A}(H\times
L)\to\operatorname{A}(H\times L)^{*}$. Moreover, $F$ is a non-zero,
antisymmetric $2$-cocycle.

###### Proof.

For this proof, just to ease notation slightly, we denote the Fourier algebras
of $H$, $L$ and $H\times L$ by $A_{H}$, $A_{L}$ and $A_{H\times L}$
respectively.

By Proposition 4.2, $F_{0}$ extends to a bounded bilinear map from
$(A_{H}\mathbin{\widehat{{\otimes}}}A_{L})\times(A_{H}\mathbin{\widehat{{\otimes}}}A_{L})$
to $A_{H}^{*}\mathbin{\widehat{{\otimes}}}{(A_{L}^{*})}^{\sim}$. Applying
Theorem 1.1 with $X=A_{H}^{*}$ and $Y=A_{L}^{*}$, we obtain a bounded bilinear
map

$F:(A_{H}\mathbin{\widehat{{\otimes}}}A_{L})\times(A_{H}\mathbin{\widehat{{\otimes}}}A_{L})\to
A_{H}^{*}\mathbin{{\otimes}}_{\rm min}A_{L}^{*}$

that extends $F_{0}$. Clearly $F$ is antisymmetric, by construction. As
mentioned after Equation (3), one can check that $F$ satisfies the $2$-cocycle
identity on the dense subalgebra $A_{H}\mathbin{{\otimes}}A_{L}$. A routine
continuity argument shows that it satisfies the identity on all of
$A_{H}\mathbin{\widehat{{\otimes}}}A_{L}$.

Recall that $A_{H}\mathbin{\widehat{{\otimes}}}A_{L}\cong_{1}A_{H\times L}$ as
algebras, while $A_{H}^{*}\mathbin{{\otimes}}_{\rm
min}A_{L}^{*}\cong_{1}\operatorname{VN}(H)\mathbin{{\otimes}}_{\rm
min}\operatorname{VN}(L)$ embeds isometrically in $\operatorname{VN}(H\times
L)=(A_{H\times L})^{*}$ (and this is an embedding of $A_{H\times L}$
bimodules). So $F$ can be viewed as a bilinear map $A_{H\times L}\times
A_{H\times L}\to(A_{H\times L})^{*}$.

Finally, we must show that $F$ is not identically zero. Since $F_{0}$ takes
values in $\operatorname{VN}(H)\mathbin{{\otimes}}\operatorname{VN}(L)$ and
the natural map
$\operatorname{VN}(H)\mathbin{{\otimes}}\operatorname{VN}(L)\to\operatorname{VN}(H\times
L)$ is injective, it suffices to show that $F_{0}$ is not identically zero.
Observe that if $a\in\operatorname{A}_{H}$ and $b\in\operatorname{A}_{L}$ we
have

$\displaystyle F_{0}(a^{3}\mathbin{{\otimes}}b,a\mathbin{{\otimes}}b)$
$\displaystyle=[D_{H}(a^{3})\cdot a]\mathbin{{\otimes}}[b\cdot
D_{L}(b)]-[a^{3}\cdot D_{H}(a)]\mathbin{{\otimes}}[D_{L}(b)\cdot b]$
$\displaystyle=2a^{3}\cdot D_{H}(a)\mathbin{{\otimes}}b\cdot D_{L}(b)$
$\displaystyle=\frac{1}{4}D_{H}(a^{4})\mathbin{{\otimes}}D_{L}(b^{2}).$

By Corollary 4.7, elements of the form $a^{4}$ span a dense subspace of
$A_{H}$, and elements of the form $b^{2}$ span a dense subspace of $A_{L}$.
Therefore, since $D_{H}$ is continuous and non-zero, there exists $a\in A_{H}$
such that $D_{H}(a^{4})\neq 0$; similarly, there exists $b\in A_{L}$ such that
$D_{L}(b^{2})\neq 0$. We conclude that
$F_{0}(a^{3}\mathbin{{\otimes}}b,a\mathbin{{\otimes}}b)\neq 0$, as required. ∎

######  Remark 4.9.

The last part of the proof of Theorem 4.8 was somewhat fiddly because we
wished to formulate the theorem in a very general setting. For many of the
known examples of groups in ${\mathcal{S}}$, one can find derivations where it
is “obvious” that the resulting $2$-cocycle $F_{0}$ cannot vanish identically
on
$(\operatorname{A}(H)\mathbin{{\otimes}}\operatorname{A}(L))\times(\operatorname{A}(H)\mathbin{{\otimes}}\operatorname{A}(L))$.

It is tricky to give precise references for the fact that certain groups
belong to the class ${\mathcal{S}}$, since the extra “completely bounded into
the opposite o.s.s.” property is merely implicit rather than explicit in the
relevant papers [1, 2, 6, 8]. We shall merely indicate some places where
specific cases can be analyzed: in all cases, the key point is that the check
map on $\operatorname{A}(G)$, defined by $\check{f}(x)=f(x^{-1})$, defines a
complete isometry from $\operatorname{A}(G)$ onto
${\operatorname{A}(G)}^{\sim}$.

######  Example 4.10.

By examining the calculations in [1, §3] or [8, Proposition 2.4], it is quite
easy to check that ${\rm SO}(3,{\mathbb{R}})$ and ${\rm SU}(2,{\mathbb{C}})$
belong to ${\mathcal{S}}$. With a little more work, one sees that the same is
true for the affine group of the real line: see Theorem 2.9 in [8]. Therefore,
by the remark after Definition 4.5, and structure theory for Lie groups, it
follows that every non-abelian connected semisimple Lie group belongs to
${\mathcal{S}}$.

It seems plausible, given the results of [8], that every connected non-abelian
Lie group belongs to ${\mathcal{S}}$; we intend to investigate this in future
work.

Finally: suppose $H\in{\mathcal{S}}$ and $L\in{\mathcal{S}}$, and let $G$ be
any locally compact group that containins a copy of $H\times L$ as a closed
subgroup. For instance: in view of Example 4.10, we could take $H=L={\rm
SU}(2,{\mathbb{C}})$ and then $G={\rm SU}(n,{\mathbb{C}})$ for any $n\geq 4$.
Then Herz’s restriction theorem provides a quotient homomorphism from
$\operatorname{A}(G)$ onto $\operatorname{A}(H\times L)$, and so by routine
arguments (see e.g. [7]) we can “lift” the non-trivial antisymmetric
$2$-cocycle on $\operatorname{A}(H\times L)$ to get the same kind of
$2$-cocycle on $\operatorname{A}(G)$. In the language of [7],
$\operatorname{A}(G)$ is not $2$-dimensionally weakly amenable; these are the
first confirmed examples of Fourier algebras with this property.

### Acknowledgements

The author thanks Matthew Daws for various helpful exchanges on operator-space
tensor products, and Adam Skalski for comments and corrections on an earlier
draft of this note.

He also wishes to thank John Rainwater for supplying the key ideas in the
proof of Theorem 1.1. Rainwater has set an admirable if idiosyncratic example
over the years (see [10] for further details) but, in these more sober times,
he has declined an invitation to be a co-author of this note. Nevertheless, in
view of Rainwater’s generous habits in acknowledging the input of colleagues
and correspondents, it seems only fitting to return the favour.

## References

  * [1] Y. Choi and M. Ghandehari, Weak and cyclic amenability for Fourier algebras of connected Lie groups, J. Funct. Anal., 266 (2014), pp. 6501–6530.
  * [2] , Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups, J. Funct. Anal., 268 (2015), pp. 2440–2463.
  * [3] E. G. Effros and Z.-J. Ruan, Operator spaces, vol. 23 of London Mathematical Society Monographs. New Series, Oxford University Press, New York, 2000.
  * [4] B. E. Forrest and V. Runde, Amenability and weak amenability of the Fourier algebra, Math. Z., 250 (2005), pp. 731–744.
  * [5] F. Gourdeau and M. C. White, Vanishing of the third simplicial cohomology group of $l^{1}({\bf Z}_{+})$, Trans. Amer. Math. Soc., 353 (2001), pp. 2003–2017.
  * [6] B. E. Johnson, Non-amenability of the Fourier algebra of a compact group, J. London Math. Soc. (2), 50 (1994), pp. 361–374.
  * [7] , Higher-dimensional weak amenability, Studia Math., 123 (1997), pp. 117–134.
  * [8] H. H. Lee, J. Ludwig, E. Samei, and N. Spronk, Weak amenability of Fourier algebras and local synthesis of the anti-diagonal, Adv. Math., 292 (2016), pp. 11–41.
  * [9] V. Losert, On tensor products of Fourier algebras, Arch. Math. (Basel), 43 (1984), pp. 370–372.
  * [10] R. Phelps, Biography of John Rainwater, Topological Commentary, 7 (2002).  Available online at http://at.yorku.ca/t/o/p/d/47.htm, see also arXiv math.HO/0312462.
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Department of Mathematics and Statistics
Fylde College, Lancaster University
Lancaster, United Kingdom LA1 4YF Email<EMAIL_ADDRESS>


# Prompt-Engineering and Transformer-based Question Generation and Evaluation

Rubaba Amyeen

###### Abstract

Question generation has numerous applications in the educational context.
Question generation can prove helpful for students when reviewing content and
testing themselves. Furthermore, a question-generation model can aid teachers
by lessening the burden of creating assessments and other practice material.
This paper aims to find the best method to generate questions from textual
data through a transformer model and prompt engineering. In this research, we
finetuned a pre-trained distilBERT model on the SQuAD question-answering
dataset to generate questions. In addition to training a transformer model,
prompt engineering was applied to generate questions effectively using Meta’s
LLaMa model. The generated questions were compared against the baseline
questions in the SQuAD dataset to evaluate the effectiveness of four different
prompts. All four prompts demonstrated over 60% similarity on average. Of the
prompt-generated questions, 30% achieved a high similarity score greater than
70%.

##  1 Introduction

Question generation is an essential NLP task. One of its primary applications
is that it can be used as a learning tool. Active recall is the practice of
retrieving information by answering questions. According to the Association
for Psychological Science (APS), active recall is a high-utility learning
technique since learners demonstrate higher academic performance after
studying with active recall [DRM+13]. A model that can generate questions
would help students use active recall to review material and lessen the time
it takes to create questions. According to a study from CBE Life Science
Education, students who used active learning strategies scored 5.5% and 10.2%
higher than students who did not use active learning strategies on exams 1 and
2, respectively [WSRF21]. Question generation systems can also motivate
students to engage in educational activities because learners will know they
will be tested on the material [Hei11]. Online courses and materials need many
questions, so this model would reduce question generation costs [PLCK19].
Furthermore, it would allow students to test their knowledge on materials that
do not include guiding questions. On the other hand, question generation is
also valuable for teachers by allowing educators to spend their time elsewhere
from formulating questions. Assessments need a constant flow of new questions
as the value of questions decreases after multiple usages since test takers
may share them.

Question generation is a generative task that involves supervised learning
with language data. The task concerns taking the input data (natural language
texts of the context and answer) and transforming it into the output data,
which is the question. We tested two models: The transformer model distilBERT
and Meta’s LLaMA large language model (LLM) through prompt engineering. We
evaluated the transformer model using the F1 score and the LLaMA using spaCy
vector similarity.

##  2 Background

The task of question generation has rapidly developed over the past few years.
Ali and Chali train their model using the TREC 2007 Question Answering Track.
The dataset has a series of factoids, lists, and other questions under targets
[ACH10]. They simplified complex sentences to elementary structures with
syntactic parsing. However, they were unable to address word disambiguation
with semantic information. Yuan and Wang implemented a recurrent model to
generate questions using the SQuAD dataset [YWG+17]. In their implementation,
the encoder processes the answer and document while the decoder generates the
question. They faced challenges with similar entities, related verbs being
swapped, and their model needing common sense. Similarly, Duan and Tang used a
convolution neural network (CNN) and recurrent neural network (RNN) [DTCZ17].
Chen and Wu recently implemented a reinforcement learning-based Graph2Sequence
model [CWZ19].

Most prior works often generate poor-quality questions. We propose using large
language models (LLMs) with prompt engineering to combat these challenges. The
prior papers have not addressed a prompt engineering-based approach for the
question generation task. Furthermore, as LLMs have been trained on large
amounts of data, they have common sense knowledge that prior models lack.

##  3 Dataset

In this paper, Stanford’s SQuAD dataset was used to generate questions.
[RZLL16], [RJL18]. The transformer model was evaluated and trained on the
SQuAD dataset. Since the SQuAD version 2 dataset has unanswerable questions,
we used the SQuAD 1.1 dataset [G+22]. For the question generation task,
unanswerable questions would not be useful as we wanted to generate questions
that test reading comprehension. This dataset contains 100,000 questions from
over 500 articles. While this dataset is commonly used for answering
questions, we reversed it for question generation. The SQuAD dataset was split
approximately 90/10 between the training set (87599 samples) and the
validation set (10570 samples). For data preprocessing, the inputs were
tokenized. Long contexts are split; however, in case the answer is where the
context is split, the hyperparameter, doc_strides, allows overlap between the
two parts of the split context. We also used the SQuAD dataset for prompt
engineering. We compared the SQuAD questions to the prompt-generated questions
by LLaMA to evaluate the quality of the generated questions.

We visualized the SQuAD dataset with simple techniques, such as a histogram of
the length of questions with outliers accounted for and the most frequent
keywords with stop words removed.

Figure 1: Length of the Questions in SQuAD. Figure 2: Most Frequent Words in
SQuAD Questions.

##  4 Methodology/Models

###  4.1 Transformer DistilBERT

Transformers are deep learning models that consist of a unique architecture
that retains the relationships between all the words in a sentence.
Transformers are not implemented with recurrent neural networks (RNNs) and
only use attention mechanisms [VSP+17]. They consist of an encoder and a
decoder. An encoder extracts features from the input, and a decoder produces a
prediction for the task using the features. BERT (Bi-directional Encoder
Representation from Transformers) is a widely used pre-trained language model
in deep neural networks [DCLT18]. Figure 3 illustrates the distilBERT model
architecture, a smaller version of the BERT model.

Figure 3: DistillBERT model architecture.

DistilBERT is a distilled version of BERT with 40% fewer hyperparameters than
BERT-base-uncased [SDCW19]. We finetuned the DistilBERT model on the SQuAD
dataset for question generation. First, the SQuAD dataset was reversed, that
is, we transformed the dataset into Pandas Dataframe and swapped the
“question” and “answer” columns.

The parameters to finetune the pre-trained model included a learning rate of
.00005, weight decay of 0.01, and training epochs set to 3. Figure 4 displays
the procedure for the distilBERT question generation task.

Figure 4:  Transformer question generation flow

###  4.2 Prompt Engineering

Prompt engineering is the practice of formulating specific prompts for LLMs to
generate desired outputs. Andrew Ng and Isa Fulford outlined two main
principles for prompt engineering: writing clear and specific instructions and
allowing the model to think [NF]. We developed four prompts for the question
generation task using these guidelines for the Meta’s LLaMA model [TLI+23].
The temperature, the prompts’ variability, was set to 0.5 to ensure varied
responses. Each of the prompts generated five questions. Table 1 shows the
four prompts.

Prompt |  Description
---|---
Prompt-A |  “Generate 5 questions from the text;”
Prompt-B |  “Generate 5 complex questions from the text.”
Prompt-C |  “Generate 5 questions from the text; make sure the questions can be answered.”
Prompt-D |  “Generate 5 questions from the text; answer the question in the text; if the question is answered in the context, output 5 questions.”
Table 1:  Written Prompts.

We randomly selected a sample of 50 contexts from the SQuAD dataset. We
evaluated the four distinct prompts against the baseline questions under each
context from SQuAD. For each prompt, 250 questions were generated. We used the
spaCy metric that used word embeddings to measure the sentence similarities
between the generated questions and baseline questions. We recorded the max
similarity score for each generated question and the max similarity score for
each prompt type under every context. Figure 5 shows the process of prompt
engineering and how the prompts were evaluated.

Figure 5:  Flow Diagram of Data Collection Process

Table 2 is a sample of the data collection for the context: “Beyoncé Giselle
Knowles-Carter (born September 4, 1981) is an American singer, songwriter,
record producer and actress. Born and raised in Houston, Texas, she performed
in various singing and dancing competitions as a child, and rose to fame in
the late 1990s as lead singer of R&B girl-group Destinyś Child. Managed by her
father, Mathew Knowles, the group became one of the worldś best-selling girl
groups of all time. Their hiatus saw the release of Beyoncéś debut album,
Dangerously in Love (2003), which established her as a solo artist worldwide,
earned five Grammy Awards and featured the Billboard Hot 100 number-one
singles ‘Crazy in Love’ and ‘Baby Boy’.” The generated questions from the four
prompts are displayed. The “Prompt Max” is the max similarity score for all
the generated questions under the specific prompt.

Context | Prompt |  Generated Question |  Question Max |  Prompt Max
---|---|---|---|---
Beyoncé … | A |  What is Beyoncé’s full name? |  0.833 |  0.935
| A |  When was Beyoncé born? |  0.534 |
| A |  Where did Beyoncé grow up? |  0.749 |
| A |  Who managed Destiny’s Child? |  0.857 |
| A |  What was the name of Beyoncé’s debut album? |  0.935 |
| B |  What is Beyoncé’s full name? |  0.833 |  0.833
| B |  Where was she born and raised? |  0.472 |
| B |  In what decade did she rise to fame as the lead singer of Destiny’s Child? |  0.722 |
| B |  Who managed her career, her father Mathew Knowles or someone else? |  0.433 |
| B |  Which album marked her debut as a solo artist worldwide and earned five Grammy Awards? |  0.639 |
| C |  What is Beyoncé’s first name? |  0.865 |  0.865
| C |  When was Beyoncé born? |  0.534 |
| C |  Where did Beyoncé grow up? |  0.749 |
| C |  Who managed Destiny’s Child? |  0.857 |
| C |  Which album established Beyoncé as a solo artist worldwide? |  0.719 |
| D |  What is Beyoncé’s birth name? |  0.879 |  0.956
| D |  When was Beyoncé born? |  0.535 |
| D |  Where did Beyoncé grow up? |  0.749 |
| D |  How did Beyoncé rise to fame? |  0.956 |
| D |  What was the name of Beyoncé’s debut solo album? |  0.934 |
Table 2:  Data Collection Table for the context $1$

##  5 Results and Discussion

###  5.1 Transformer Model

Using the evaluation metric under the Trainer class from PyTorch, the F1 score
for the transformer model was 15.88. A key reason for a low F1 score was found
to be that the transformer model could not generate questions and instead
generated sentences similar to the answers in SQuAD. These predicted outputs
may be because the model was already familiar with question-answering tasks
and generating answers, so it could not generate questions. The distilBERT
model was trained on 3.3 billion words from the Wikipedia and Toronto Book
Corpus, so finetuning on a much smaller dataset like SQuAD was not effective
in reversing distilBERT’s inherent behavior to answer questions. As the
transformer model was not able to generate questions, we did not use a metric
to evaluate the generated outputs.

###  5.2 Prompt Engineering

Figure 6:  Score distribution across diffent prompts

We used the spaCy metric for similarity scoring. Figure 6 is a boxplot of the
similarity scores for all the generated questions under each prompt. The
boxplot reveals that the median similarity score for Prompt D’s generated
questions was the highest at 0.6444. Furthermore, Prompt D also had the
highest outliers, with one being a perfect match from the similarity score of
1. Prompts A and C had an average similarity score of 0.6387 and 0.6321
respectively. Prompt B performed the worst, with the lowest average of 0.6227.

Figure 7:  High Threshold Score Distribution for the Prompts

Another method we used to establish which prompt performed the best was the
prompt that had the most matches with the baseline questions. We established a
threshold of 0.7 similarity score to be considered a match. After comparing
generated questions with a similarity score above 0.7 with the baseline
questions, we discovered that most of the generated questions had the same
meaning as the baseline question; while generated questions with a similarity
score below 0.7 were inherently different. We defined a match as when the
baseline and generated questions were nearly identical. Figure 7 reveals that
Prompts A, B, C, and D had 79, 64, 76, and 81 matches, respectively, out of
the 250 questions each prompt generated in total. Prompt D had the highest
number of matches, so it could generate the highest quality questions,
assuming the SQuAD dataset has the highest quality questions. Prompt B had the
lowest number of matches. The word “complex” in Prompt B may have generated
unanswerable questions from the context and required more critical thinking,
so it performed poorly under the similarity metric. Table 3 displays an
unanswerable question generated by Prompt B.

Context |  Prompt-B Question
---|---
The Hanover Zoo is one of the most spectacular and best zoos in Europe. The zoo received the Park Scout Award for the fourth year running in 2009/10, placing it among the best zoos in Germany. |  Where is the Hanover Zoo located in Germany?
Table 3:  Sample Unanswerable Question from Prompt B.

Lastly, Figure 8 is a distribution graph of the max scores of every prompt
under each of the 50 contexts in total. Generally, the red (Prompt D) and blue
(Prompt A) lines are more significant than the orange (Prompt B) and green
(Prompt C) lines. Prompt B had a particularly low max similarity score for
context 45 of 0.42 compared to Prompt A, C, and D, which had max similarity
scores of 0.73, 0.70, and 0.77, respectively. This is most likely because
Prompt B generated compound questions, where two questions were combined into
one. For example, one of the questions was, “In what way can Nirvana be seen
as a liberation from the cycle of suffering, and how is this achieved?”

Figure 8:  Max score distribution across contexts

##  6 Conclusion

In this paper, we finetuned a distilBERT model on the question generation
task. This transformer model could not generate questions with an F1 score of
15.88. On the other hand, Meta’s LLaMA model successfully generated questions
using the prompts we created with the prompt engineering guidelines. As the
transformer model was not able to generate questions and Meta’s LLaMA model
was, we decided to only evaluate the generated questions from LLaMA. We used a
similarity metric to determine which of the four prompts we created was the
most effective. We found that the most complex prompt (Prompt D) resulted in
the most matches with the baseline questions from SQuAD at 81 matches. The
most simple prompt (Prompt A) performed the second best with 79 matches.
Prompt C had 76 matches. Lastly, Prompt B had a significantly lower number of
matches at 64 matches. It should be noted that Prompt B was the same as Prompt
A except for the added word “complex,” suggesting that the similarity score
metric rewards simple factual questions rather than compound questions.

We plan to extend our research to include other LLMs, such as the Dolly model,
GPT-4, and PaLM 2. Furthermore, we plan to combine the question generation
model with a classification model that classifies the contexts under specific
educational subjects to make the questions generated more appropriate for each
subject beyond reading comprehension.

##  7 Acknowledgements

We would like to acknowledge the technical contributions and mentorship of
Tanish Jain from Stanford University.

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