# Homological and cohomological properties of Banach algebras and their second
duals

M. J. Mehdipour and A. Rejali111Corresponding author

Abstract. In this paper, we investigate homological properties of Banach
algebras. We show that retractions Banach algebras preserve biprojectivity,
contractibility and biflatness. We also prove that contractibility of second
dual of a Banach algebra implies contractibility of the Banach algebra. For a
Banach algebra $A$ with $\Delta(A)\neq\emptyset$, let ${F}(X,A)$ be one of the
Banach algebras $C_{b}(X,A)$, $C_{0}(X,A)$, $\hbox{Lip}_{\alpha}(X,A)$ or
$\hbox{lip}_{\alpha}(X,A)$. In the following, we study homological properties
of Banach algebra ${F}(X,A)$, especially contractibility of it. We prove that
contractibility of ${F}(X,A)$ is equivalent to finiteness of $X$ and
contractibility of $A$. In the case where, $A$ is commutative, we show that
${F}(X,A)$ is contractible if and only if $A$ is a $C^{*}-$algebra and both
$X$ and $\Delta(A)$ are finite. In particular, $\hbox{lip}_{\alpha}^{0}(X,A)$
is contractible if and only if $X$ is finite. We also investigate
contractibility of $L^{1}(G,A)$ and establish $L^{1}(G,A)$ is contractible if
and only if $G$ finite and $A$ is contractible. Finally, we show that
biprojectivity of the Beurling algebra $L^{1}(G,\omega)$ is equivalent to
compactness of $G$, however, biprojectivity of the Banach algebras
$L^{1}(G,\omega)^{**}$ is equivalent to finiteness of $G$. This result holds
for the Banach algebra $M(G,\omega)^{**}$ instead of $L^{1}(G,\omega)^{**}$.
11footnotetext:  2020 Mathematics Subject Classification: 46H20, 46M20, 46M10,
46H25, 46E40, 43A20. Keywords: Biprojectivity, contractibility, biflatness,
Lipschitz algebra, vector-valued functions, weighted group algebras.

##  1 Introduction

Let $A$ be a Banach algebra and $E$ be a Banach $A-$bimodule. A bounded linear
operator $D:A\rightarrow E$ is called a _derivation_ if

$D(ab)=D(a)b+aD(b)$

for all $a,b\in A$. Let us recall that $A$ is called _contractible_ if every
continuous derivation $D$ from $A$ into Banach $A-$bimodule $E$ is inner, that
is, there exists $z\in E$ such that $D(a)=za-az$ for all $a\in A$. Also, $A$
is said to be _weakly amenable_ if every continuous derivation $D$ from $A$
into $A^{*}$ is inner.

For Banach spaces $E$ and $F$, a bounded linear operator $T:E\rightarrow F$
with closed range is said to be _admissible_ if $\hbox{ker}T$ and $T(E)$ are
complemented subspaces of $E$ and $F$, respectively. Now, let $A$ be a Banach
algebra and let $E$ and $F$ be Banach left $A-$modules. We reserve the symbol
${}_{A}{\cal B}(E,F)$ for the set of all bounded left $A-$module homomorphisms
from $E$ into $F$. A Banach left $A-$module $P$ is called _projective_
(respectively, _flat_) if for every Banach left $A-$module $E$ and $F$, for
each admissible epimorphism (respectively, monomorphism) $T\in_{A}{\cal
B}(E,F)$ and each $S_{1}\in_{A}{\cal B}(P,F)$ (respectively,
$S_{2}\in_{A}{\cal B}(E,P^{*})$), there exists $R_{1}\in_{A}{\cal B}(P,E)$
(respectively, $R_{2}\in_{A}{\cal B}(F,P^{*})$ ) such that

$T\circ R_{1}=S_{1}\quad(\hbox{respectively},R_{2}\circ T=S_{2}).$

A Banach algebra $A$ is called _biprojective_ (respectively, _biflat_) if $A$
is projective (respectively, flat) as a Banach $A-$bimodule. These definitions
due to Helemskii [14]. It is proved that $A$ is biprojective if and only if
the corresponding diagonal operator

$\pi_{A}:A\hat{\otimes}A\rightarrow A$

has a bounded bimodule right inverse. Also, $A$ is biflat if and only if
$\pi_{A}^{*}$ has a bounded bimodule left inverse; for more details see [4, 6,
12, 13, 14, 16, 30, 38].

Johnson [17] studied the space ${\cal H}^{1}(L^{1}(G),E)$, the first
Hochschild cohomology group of $L^{1}(G)$ with coefficients in Banach
$L^{1}(G)-$bimodule $E$. He showed that ${\cal H}^{1}(L^{1}(G),E^{*})$
vanishes for all Banach $L^{1}(G)-$bimodules $E$ if and only if $G$ is
amenable. This result leads to Banach algebras with the property ${\cal
H}^{1}(A,E^{*})$ vanishes for all Banach $A-$bimodule $E$ are called
_amenable_. Some authors gave equivalent definitions of amenable Banach
algebras. For example, Helemskii [12, 13, 14, 15, 16] introduced the
homological properties Banach algebras and found equivalent definitions of
amenability. For instance, he showed that $A$ is an amenable Banach algebra if
and only if the unitization of $A$ is biflat. For other equivalent definitions
see [18]; see also [15, 27], see Selivanov [33] for characterization the
biprojective and biflat Banach algebra through the first Hochschild cohomology
groups, [32] for biprojective semisimple Banach algebras, for biprojectivity
and biflatness of Banach algebra related to locally compact groups and semi
groups see [2, 8, 11, 25, 28, 31].

Let $X$ be a locally compact space and $A$ be a Banach algebra. Let $X^{A}$ be
the set of all vector-valued functions from $X$ into $A$. Let us recall that
$f\in X^{A}$ is called _bounded_ if

$\|f\|_{\infty}:=\sup_{x\in X}\|f(x)\|$

is finite. We denote by $C_{b}(X,A)$ the space of all bounded continuous
functions from $X$ into $A$. We also denote by $C_{0}(X,A)$ the set of all
continuous functions from $X$ into $A$ that vanish at infinity. If $A={C}$,
then we use from the symbols $C_{b}(X)$ and $C_{0}(X)$ instead of $C_{b}(X,A)$
and $C_{0}(X,A)$, respectively.

Let $f$ be a bounded vector-valued function from metric space $(X,d)$ into
Banach algebra $(A,\|.\|)$. Then $f$ is called _vector-valued Lipschitz
function of order_ $\alpha$ if there exists $M>0$ such that for every $x,y\in
X$

$\|f(x)-f(y)\|\leq Md^{\alpha}(x,y),$

where $\alpha$ is a positive real number. The space of all vector-valued
Lipschitz functions of order $\alpha$ from $(X,d)$ into $(A,\|.\|)$ is denoted
by $\hbox{Lip}_{\alpha}(X,A)$. Clearly, every vector-valued Lipschitz function
of order $\alpha$ is continuous. Hence $\hbox{Lip}_{\alpha}(X,A)$ is a
subspace of $C_{b}(X,A)$. It is proved that $\hbox{Lip}_{\alpha}(X,A)$ with
the pointwise multiplication and the norm

$\|f\|_{\infty,\alpha}:=\|f\|_{\infty}+p_{\alpha}(f)$

is a Banach algebra, where

$p_{\alpha}(f)=\sup_{x,y\in X,x\neq y}\frac{\|f(x)-f(y)\|}{d^{\alpha}(x,y)}.$

Let $\hbox{lip}_{\alpha}(X,A)$ be the subalgebra of $\hbox{Lip}_{\alpha}(X,A)$
consisting of $f$ such that

$\lim_{d(x,y)\rightarrow 0}\frac{\|f(x)-f(y)\|}{d(x,y)^{\alpha}}=0.$

If $X$ is a locally compact metric space, then we set

$\hbox{lip}^{\circ}_{\alpha}(X,A):=\hbox{lip}_{\alpha}(X,A)\cap C_{0}(X,A).$

In the case where, $A={C}$, we write
$\hbox{Lip}_{\alpha}(X,{C})=\hbox{Lip}_{\alpha}X$,
$\hbox{lip}_{\alpha}(X,{C})=\hbox{lip}_{\alpha}X$ and
$\hbox{lip}^{\circ}_{\alpha}(X,{C})=\hbox{lip}^{\circ}_{\alpha}X$. For an
extensive study of Lipschitz spaces see [1, 4, 35, 34, 36].

Let $G$ be a locally compact group with identity element $e$ and weight
function $\omega$, i.e., a continuous function $\omega:G\rightarrow[1,\infty)$
such that $\omega(e)=1$ and $\omega(xy)\leq\omega(x)\omega(y)$ for all $x,y\in
G$. Let $L^{1}(G,\omega)$ and $M(G,\omega)$ be the Beurling and weighted
measure algebra of $G$, respectively; for more details see [5, 29]. We put

$L^{1}(G,1)=L^{1}(G)\quad\hbox{and}\quad M(G,1)=M(G).$

For a Banach algebra $A$, let $L^{1}(G,A)$ be the Banach algebra of $A-$valued
integrable functions on $G$. It is well-known that $L^{1}(G,A)$ isometrically
isomorphic to the projective tensor product $L^{1}(G)\hat{\otimes}A$; see for
example Proposition 1.5.4 in [19].

This paper is organized as follow. In Section 2, we give some results on
homological properties of Banach algebras. For example, we prove that
biprojectivity, contractibility and biflatness preserve under retractions.
Also, if $A^{**}$ is contractible, then $A$ is contractible. Then, we show
that the character space of a commutative contractible Banach algebra is
finite. Section 3 is devoted to study homological properties of vector-valued
Banach algebras. To this end, let $A$ be a Banach algebra such that
$\Delta(A)$, the character space of $A$, is non-empty. Let also ${F}(X,A)$ be
one of the Banach algebras $C_{b}(X,A)$, $C_{0}(X,A)$,
$\hbox{Lip}_{\alpha}(X,A)$ or $\hbox{lip}_{\alpha}(X,A)$. We show that
biprojectivity of ${F}(X,A)$ implies discreteness of $X$ and $\Delta(A)$. We
also prove that ${F}(X,A)$ is contractible if and only if $X$ is finite and
$A$ is contractible. Then, for commutative Banach algebra $A$, we prove that
${F}(X,A)$ is contractible if and only if $A$ is a $C^{*}-$algebra and both
$X$ and $\Delta(A)$ are finite. A special case of this result shows that
$\hbox{lip}_{\alpha}^{0}X$ is contractible if and only if $X$ is finite. In
Section 4, we investigate homological properties weighted Banach algebras and
show that if ${X}$ is a Banach algebra containing $C_{0}(G,1/\omega)$, then
contractibility and biprojectivity of ${X}^{*}$ are equivalent to finitness of
$G$, however, biflatness of ${X}$ implies discreteness of $G$ and boundedness
of the weight function $\omega^{*}$ defined by
$\omega^{*}(x)=\omega(x)\omega(x^{-1})$ for all $x\in G$. We also prove that
biprojectivity, contractibility and biflatness of the Banach algebras
$M(G,\omega)^{**}$ and $L^{1}(G,\omega)^{**}$ are equivalent and occur only if
$G$ is finite. Finally, we prove that $L^{1}(G,A)$ is contractible if and only
if $M(G)\hat{\otimes}A$ is contractible; or equivalently, $G$ is finite and
$A$ is contractible.

##  2 Homological and cohomological properties of Banach algebras

Curtis and Loy [3] and Helemskii [12, 13, 14, 15, 16] investigated the
relation between the homological properties and amenability and obtained the
following result.

###### Theorem 2.1

Let $A$ be a non-zero Banach algebra. Then the following statements hold.

_(i)_ Every $C^{*}-$ algebra is biflat.

_(ii)_ If $A$ is biflat, then $A$ is weakly amenable.

_(iii)_ If $A$ is biprojective, then $A$ is biflat.

_(iv)_ $A$ is contractible if and only if $A$ is unital and biprojective.

_(v)_ If $A$ is contractible, then $A$ is amenable.

_(vi)_ Commutative Banach algebra $A$ is contractible if and only if $A$ is
finite dimensional.

_(vii)_ $A$ is amenable if and only if $A$ has a bounded approximate identity
and $A$ is biflat.

_(viii)_ If $A$ is a commutative biprojective Banach algebra, then $\Delta(A)$
is discrete. Furthermore, if $A$ is a commutative $C^{*}-$algebra, then the
converse remain valid.

###### Example 2.2

Let $E$ be a Banach space with $\hbox{dim}(E)>1$ and let $\varphi\in E^{*}$ be
non-zero. Then $E$ with the following multiplication is a Banach algebra.

$a\cdot b:=\varphi(b)a.$

It is clear that $A:=(E,\cdot)$ is biprojective and so $A$ is biflat. Also,
$\Delta(A)=\\{\varphi\\}$. Note that $A$ does not have bounded approximate
identity. Hence $A$ is neither contractible nor amenable. Thus the assumption
that $A$ has a bounded approximate identity in Theorem 2.1 (iv) and (vii) can
not drop.

Let $A$ and $B$ be Banach algebras. Then continuous homomorphism
$\phi:A\rightarrow B$ is called a _retraction_ if there exists a continuous
homomorphism $\psi:B\rightarrow A$ such that $\phi\circ\psi=id_{B}$, the
identity map on $B$. Also, a linear map $L$ from $A$ into Banach left
$A-$module $E$ is called a _left multiplier_ if $L(ab)=L(a)b$ for all $a,b\in
A$. Similarly, one can define the notion right multiplier. Pirkovskii and
Selivanov [26] proved that $A$ is biprojective (respectively, biflat) if and
only if every continuous derivation $D$ from $A$ into Banach $A-$bimodule $E$
(respectively, $E^{*}$), there exist left multiplier $L$ and right multiplier
$R$ from $A$ into $E$ (respectively, $E^{*}$) such that $D=R-L$.

###### Theorem 2.3

Let $A$ and $B$ be Banach algebras and $\phi:A\rightarrow B$ be a retraction.
Then the following statements hold.

_(i)_ If $A$ is biprojective, then $B$ is biprojective.

_(ii)_ If $A$ is contractible, then $B$ is contractible.

_(iii)_ If $A$ is biflat, then $B$ is biflat.

Proof. First note that if $E$ is a Banach $B-$bimodule, then $E$ is a Banach
$A-$bimodule with the following actions.

$a\cdot x=\phi(a)x\quad\hbox{and}\quad x\cdot a=x\phi(a)$

for all $a\in A$ and $x\in E$. Now, let $D:B\rightarrow E$ be a continuous
derivation. Then the linear operator $\bar{D}:A\rightarrow E$ defined by
$\bar{D}=D\circ\phi$ is a continuous derivation; indeed, for every
$a_{1},a_{2}\in A$, we have

$\displaystyle\bar{D}(a_{1}a_{2})$ $\displaystyle=$ $\displaystyle
D(\phi(a_{1})\phi(a_{2}))$ $\displaystyle=$ $\displaystyle
D(\phi(a_{1}))\phi(a_{2})+\phi(a_{1})D(\phi(a_{2}))$ $\displaystyle=$
$\displaystyle\bar{D}(a_{1})\cdot a_{2}+a_{1}\cdot\bar{D}(a_{2}).$

Let $A$ be biprojective. Then there exist left multiplier $L$ and right
multiplier $R$ from $A$ into $E$ such that $\bar{D}=R-L$. Since $\phi$ is a
retraction, there exists a bounded homomorphism $\psi:B\rightarrow A$ with
$\phi\circ\psi=id_{B}$. Define left multiplier and right multiplier
$\tilde{L}$ and $\tilde{R}$ by

$\tilde{L}=L\circ\psi\quad\hbox{and}\quad\tilde{R}=R\circ\psi.$

Then for every $b\in B$ we have

$\displaystyle D(b)$ $\displaystyle=$ $\displaystyle
D(\phi\psi(b))=\bar{D}(\psi(b))$ $\displaystyle=$ $\displaystyle
R\psi(b)-L\psi(b)$ $\displaystyle=$ $\displaystyle\tilde{L}(b)-\tilde{R}(b).$

Therefore, $B$ is biprojective. That is, (i) holds. For (ii), note that if $A$
is unital, then $B$ is unital. This together with (i) proves (ii). The
statement (iii) proves similar to (i). $\hfill\mathchar 1027\relax$

###### Corollary 2.4

Let $A$ and $B$ be Banach algebras and $\phi:A\rightarrow B$ be an isomorphism
and a homeomorphism. Then $\phi$ preserves biprojectivity, contractibility and
biflatness.

It is natural to ask whether continuous homomorphisms with dense range
preserve biprojectivity, contractibility and biflatness. Mewomo [23] proved
that the answer is affirmative for contractibility. But, the following example
shows that the question has a negative answer for biprojectivity and
biflatness.

###### Example 2.5

Let $E$ be a Banach space. Then $E\hat{\otimes}E^{*}$ with the following
multiplication is a Banach algebra.

$(x_{1}\otimes f_{1})\cdot(x_{2}\otimes f_{2})=f_{1}(x_{2})x_{1}\otimes
f_{2}\quad\quad(x_{1},x_{2}\in E,f_{1},f_{2}\in E^{*})$

Set $A:=(E\hat{\otimes}E^{*},\cdot)$. Define the mapping
$Q:A\hat{\otimes}A\rightarrow A$ by

$Q(x\otimes f)\otimes(y\otimes g)=f(y)x\otimes g.$

Choose $y\in G$ such that $f(y)\neq 0$. Define the function $S:A\rightarrow
A\hat{\otimes}A$ by

$S(x\otimes f)=(x\otimes f)\otimes(y\otimes 1/f(y)f).$

Then for every $x\in E$ and $f\in E^{*}$, we have

$\displaystyle Q\circ S(x\otimes f)$ $\displaystyle=$ $\displaystyle
Q(x\otimes f)\otimes(y\otimes 1/f(y)f)$ $\displaystyle=$ $\displaystyle
f(y)x\otimes 1/f(y)f$ $\displaystyle=$ $\displaystyle x\otimes f.$

This shows that $Q\circ S=id_{A}$. It follows that $A$ is biprojective and
hence it is biflat. We define the function
$\phi:E\hat{\otimes}E^{*}\rightarrow B(E)$ by

$\phi(a\otimes f)(x)=f(x)a.$

If $X_{i}=x_{i}\otimes f_{i}$, for $i=1,2$, then for every $x\in E$, we have

$\displaystyle\phi(X_{1})\circ\phi(X_{2})(x)$ $\displaystyle=$
$\displaystyle\phi(X_{1})[(f_{2}(x)x_{2})]$ $\displaystyle=$ $\displaystyle
f_{1}(f_{2}(x)x_{2})x_{1}$ $\displaystyle=$ $\displaystyle
f_{2}(x)f_{1}(x_{2})x_{1}=\phi(X_{1}\cdot X_{2})(x)$

Hence $\phi$ is a continuous homomorphism. Usually, the range of $\phi$ is
called the _space of_ _nuclear operators_ on $E$ and is denoted by ${\cal
N}(E)$. So $\phi$ is a continuous epimorphism from $A$ onto ${\cal N}(E)$. It
is well-known from [10] that ${\cal N}(E)$ is weakly amenable if and only if

$\hbox{dim}(\hbox{ker}\phi)\leq 1.$

So if $\hbox{dim}(\hbox{ker}\phi)\geq 2$, then ${\cal N}(E)$ is not weakly
amenable and thus it is neither biprojective nor biflat. Therefore, continuous
epimorphisms don’t preserve biprojectivity and biflatness.

On can prove that $(\hbox{ker}\phi)^{2}=\\{0\\}$ and so it is not essential.
This shows that $\hbox{ker}\phi$ is not weakly amenable. Note that
$\hbox{ker}\phi$ is an ideal of Banach algebra $E\hat{\otimes}E^{*}$. So
closed ideals of a biprojective (respectively, biflat) algebras is not
biprojective (respectively, biflat), in general.

###### Proposition 2.6

Let $A_{1}$ and $A_{2}$ be Banach algebras. Then the following statements
hold.

_(i)_ $A_{1}\oplus A_{2}$ is biprojective if and only if $A_{1}$ and $A_{2}$
are biprojective.

_(ii)_ $A_{1}\oplus A_{2}$ is contractible if and only if $A_{1}$ and $A_{2}$
are contractible.

_(iii)_ $A_{1}\oplus A_{2}$ is biflat if and only if $A_{1}$ and $A_{2}$ are
biflat.

Proof. (i) For $i=1,2$, let $\pi_{i}:A_{1}\oplus A_{2}\rightarrow A_{i}$ and
$\iota_{i}:A_{i}\rightarrow A_{1}\oplus A_{2}$ be canonical projection and
injection, respectively. Then $\pi_{i}\circ\iota_{i}=id_{A_{i}}$, the identity
map on $A_{i}$. Hence $\pi_{i}$ is a retraction of $A_{1}\oplus A_{2}$. So, if
$A_{1}\oplus A_{2}$ is biprojective, then $A_{i}$ is biprojective. The
converse is clear.

(ii) This follows from (i) and Theorem 2.1 (iv).

(iii) This is proved by the argument used in (i).$\hfill\mathchar 1027\relax$

Ramsden [28] showed that if $A_{1}$ and $A_{2}$ are biprojective (biflat),
then $A_{1}\hat{\otimes}A_{2}$ is biprojective (biflat). He also proved that
if $A_{1}$ is unital and $A_{2}$ containing a non-zero idempotent element,
then biprojectivity (biflatness) of $A_{1}\hat{\otimes}A_{2}$ implies
biprojectivity (biflatness) of $A_{1}$.

###### Theorem 2.7

Let $A_{1}$ and $A_{2}$ be Banach algebras. Then the following statements
hold.

_(i)_ If $A_{1}$ and $A_{2}$ are contractible, then $A_{1}\hat{\otimes}A_{2}$
is contractible. The converse is true if $\Delta(A_{i})$ is a nonempty set,
for $i=1,2$.

_(ii)_ If $A_{2}$ is unital and $\Delta(A_{2})$ is a nonempty set, then
biprojectivity _(_ biflatness _)_ of $A_{1}\hat{\otimes}A_{2}$ implies
biprojectivity _(_ biflatness _)_ of $A_{1}$.

_(iii)_ If $A_{1}$ is unital and $\Delta(A_{1})$ is a nonempty set, then
biprojectivity _(_ biflatness _)_ of $A_{1}\hat{\otimes}A_{2}$ implies
biprojectivity _(_ biflatness _)_ of $A_{2}$.

Proof. Let $A_{1}$ and $A_{2}$ be contractible. Then $A_{1}$ and $A_{2}$ are
unital and biprojective. Thus $A_{1}\hat{\otimes}A_{2}$ is unital and
biprojective. So $A_{1}\hat{\otimes}A_{2}$ is contractible.

Let $\varphi_{2}\in\Delta(A_{2})$. Define the mapping
$F:A_{1}\hat{\otimes}A_{2}\rightarrow A_{1}$ by $F(a_{1}\otimes
a_{2})=\varphi_{2}(a_{2})a_{1}$. Then for every $a_{1},x_{1}\in A_{1}$ and
$a_{2},x_{2}\in A_{2}$ we have

$\displaystyle F((a_{1}\otimes a_{2})(x_{1}\otimes x_{2}))$ $\displaystyle=$
$\displaystyle F(a_{1}x_{1}\otimes a_{2}x_{2})$ $\displaystyle=$
$\displaystyle\varphi_{2}(a_{2}x_{2})a_{1}x_{1}$ $\displaystyle=$
$\displaystyle\varphi_{2}(a_{2})\varphi_{2}(x_{2})a_{1}x_{1}$ $\displaystyle=$
$\displaystyle F(a_{1}\otimes a_{2})F(x_{1}\otimes x_{2}).$

Choose $a_{2}\in A_{2}$ with $\varphi_{2}(a_{2})=1$. If $a_{1}\in A_{1}$, then
$F(a_{1}\otimes a_{2})=a_{1}$. Thus $F$ is a continuous epimorphism. Hence
contractibility of $A_{1}$ follows from the fact that every continuous
epimorphism of Banach algebras is contractibility preserving. Similarly,
$A_{2}$ is contractible. So (i) holds.

Assume that $A_{2}$ is unital and $\Delta(A_{2})$ is a nonempty set. Define
the function $\psi:A_{1}\rightarrow A_{1}\hat{\otimes}A_{2}$ by
$\psi(a_{1})=a_{1}\otimes 1_{A_{2}}$. Then $\psi$ is a homomorphism and
$F\circ\psi=id_{A_{1}}$. Thus $F$ is a retraction. Therefore, (ii) holds.
Similarly, (iii) is proved.$\hfill\mathchar 1027\relax$

For a Banach algebra $A$, Moslehian and Niknam [24] showed that biflatness of
$A^{**}$ implies biflatness of $A$. Essmaili et al. [7] established that if
$A$ is an ideal of $A^{**}$, then biprojectivity of $A^{**}$ forces $A$ to be
biprojective. In the next result, we give relation between contractibility of
$A^{**}$ and $A$. To this end, let us recall that a derivation $D$ from Banach
algebra $A$ into Banach $A-$bimodule $E$ is called _weakly approximately
inner_ if there exists a bounded net $(x_{\alpha})$ in $E$ such that for every
$a\in A$

$D(a)=w-\lim_{\alpha}\;x_{\alpha}.a-a.x_{\alpha}.$

###### Theorem 2.8

Let $A$ be a Banach algebra. If $A^{**}$ is contractible, then $A$ is
contractible.

Proof. Let $D$ be a continuous derivation from $A$ into Banach $A-$bimodule
$E$. Then ${B}=A\oplus E$ is a Banach algebra with the product

$(a,x)(b,y)=(ab,ay+xb)$

and the norm $\|(a,x)\|=\|a\|+\|x\|$. Also, the mapping $\theta:A\rightarrow
A\oplus E$ defined by $\theta(a)=(a,D(a))$ is a continuous homomorphism and
$\theta^{**}(n)=(n,D^{**}(n))$ for all $n\in A^{**}$. Theses facts show that
$D^{**}$ is a continuous derivation; see Lemma 2.2 in [9]. Since $A^{**}$ is
contractible, $A^{**}$ is amenable and so $D^{**}$ is inner. Hence there
exists $\Phi\in E^{**}$ such that for every $m\in A^{**}$ and $f\in E^{*}$, we
have

$\langle D^{**}(m),f\rangle=\langle\Phi\cdot
m-m\cdot\Phi,f\rangle=\langle\Phi,m\cdot f-f\cdot m\rangle.$

Assume that $(x_{\alpha})$ be a bounded net in $E$ such that
$x_{\alpha}\rightarrow\Phi$ in the weak∗ topology of $E^{**}$. For every $a\in
A$, we have

$\displaystyle\langle D(a),f\rangle$ $\displaystyle=$ $\displaystyle\langle
D^{**}(a),f\rangle$ $\displaystyle=$ $\displaystyle\Phi(a.f)-\Phi(f.a)$
$\displaystyle=$ $\displaystyle\lim_{\alpha}(a.f(x_{\alpha})-f.a(x_{\alpha}))$
$\displaystyle=$ $\displaystyle\lim_{\alpha}\langle
f,x_{\alpha}.a-a.x_{\alpha}\rangle.$

This implies that

$D(a)=w-\lim_{\alpha}\;x_{\alpha}.a-a.x_{\alpha}.$

Hence $D$ is weakly approximately inner and therefore it is inner. That is,
$A$ is contractible.$\hfill\mathchar 1027\relax$

Essmaili et al. [7] raised the question of whether there exists a Banach
algebra $A$ such that $A^{**}$ is biprojective, but $A$ is not biprojective.
The next example gives a positive answer to the question. This example also
shows that the converse of Theorem 2.8 is not true, in general.

###### Example 2.9

Let $A=c_{0}$. Then $A$ is biprojective. If $A^{**}=\ell^{\infty}$ is
biprojective, then it is commutative and contractible. So $A^{**}$ is finite
dimensional, a contradiction. Thus $A^{**}$ is not biprojective. Note that
$A^{\sharp}$, the unitization of $A$, is contractible, however,
$(A^{\sharp})^{**}$ is a commutative infinite dimensional Banach algebra. So
$(A^{\sharp})^{**}$ is not contractible.

It is well-known that the character space of unital, commutative Banach
algebras is weak∗ compact; see for example Theorem 2.2.3 in [19]. Here, we
give a simple proof for it. Note that an application of Shilov’s idempotent
theorem shows that if the character space of a semisimple commutative Banach
algebra is compact, then the Banach algebra is unital; see Theorems 3.5.1 and
3.5.5 in [19].

###### Lemma 2.10

Let $A$ be a unital, commutative Banach algebra. Then $\Delta(A)$ is weak∗
compact.

Proof. Let $(\varphi_{\alpha})$ be a net in $\Delta(A)$ such that
$\varphi_{\alpha}\rightarrow\varphi\in A^{*}$ in the weak∗ topology of
$A^{*}$. Then

$\varphi(1_{A})=\lim_{\alpha}\varphi_{\alpha}(1_{A})=1.$

So $\varphi$ is non-zero. If $f,g\in A$, then

$\displaystyle\varphi(fg)$ $\displaystyle=$
$\displaystyle\lim_{\alpha}\varphi_{\alpha}(fg)=\varphi_{\alpha}(fg)$
$\displaystyle=$
$\displaystyle\varphi_{\alpha}(f)\varphi_{\alpha}(g)=\varphi(f)\varphi(g).$

Hence $\varphi$ is multiplicative. Since $A$ is commutative, by [19],
$\varphi$ is continuous. It follows that $\varphi\in\Delta(A)$. Thus
$\Delta(A)$ is a weak∗ closed subspace of the compact unit ball of $A^{*}$.
Therefore, $\Delta(A)$ is weak∗ compact.$\hfill\mathchar 1027\relax$

We finish this section with the following result.

###### Theorem 2.11

Let $A$ be a commutative Banach algebra. Then the following assertions are
equivalent

_(a)_ $A$ is contractible.

_(b)_ $A$ is finite dimensional.

_(c)_ $A$ is a $C^{*}-$algebra and $\Delta(A)$ is finite.

Proof. By Theorem 2.1 the statements (a) and (b) are equivalent. Let $A$ be
contractible. Then $A$ is unital and biprojective. By Theorem 2.1 (viii) and
Lemma 2.10, $\Delta(A)$ is finite. Since $A$ is commutative, $A$ is a
$C^{*}-$algebra; see Corollary 4.1.3 in [31]. Thus (a)$\Rightarrow$(c).
Finally, the implication (c)$\Rightarrow$(a) follows from Theorem 2.1 (iv) and
(vii).$\hfill\mathchar 1027\relax$

##  3 Homological and cohomological properties of vector-valued function
algebras

Let $X$ be a locally compact space and $A$ be a Banach algebra. For $x\in X$
and $\varphi\in\Delta(A)$, we define the multiplicative linear functional
$x\otimes\varphi:X^{A}\rightarrow{{C}}$ by

$x\otimes\varphi(f):=\varphi(f(x))$

for all $f\in X^{A}$. We set

$X\otimes\Delta(A)=\\{x\otimes\varphi:x\in X,\varphi\in\Delta(A)\\}.$

###### Proposition 3.1

Let $(X,d)$ be a locally compact Hausdorff space and $A$ be a Banach algebra
with $\Delta(A)\neq\emptyset$. Then the function
$\tilde{\Gamma}:(x,\varphi)\mapsto x\otimes\varphi$ from $X\times\Delta(A)$
onto $X\otimes\Delta(A)$ is separately continuous and bijective.

Proof. Let $x_{0}$ and $x_{1}$ be distinct points of $X$. Then there exists a
function $f\in C_{c}(X)$ such that $f(x_{0})=1$ and $f(x_{1})=0$. Now, let
$\varphi$ and $\psi$ be distinct elements of $\Delta(A)$. Choose $a\in A$ such
that $\varphi(a)$ is non-zero. Define the function $g:X\rightarrow A$ by
$g(x)=af(x)$. Then $g(x_{0})=a$ and $g(x_{1})=0$. So
$\varphi(g(x_{0}))\neq\psi(g(x_{1}))$ . Hence, $x_{0}\otimes\varphi\neq
x_{1}\otimes\psi$. Thus $\tilde{\Gamma}$ is injective. It is clear that
$\tilde{\Gamma}$ is surjective and separately continuous.$\hfill\mathchar
1027\relax$

Let ${A}(X)$ be one of the Banach algebras $C_{b}(X)$, $\hbox{Lip}_{\alpha}X$
or $\hbox{lip}_{\alpha}X$, where $\alpha>0$. Let also ${F}(X)$ be ${A}(X)$ or
$C_{0}(X)$. Note that in the case where ${F}(X)$ is $C_{b}(X)$ or $C_{0}(X)$,
the set $X$ is a locally compact space and the other cases, $X$ is a metric
space. For a Banach algebra $A$, let ${A}(X,A)$ and ${F}(X,A)$ be the vector-
valued Banach algebras corresponding to ${A}(X)$ and ${F}(X)$, respectively.

###### Lemma 3.2

Let $A$ be a Banach algebra. Then the following statements hold.

_(i)_ If $\varphi\in\Delta(A)$, then ${F}(X)=\varphi\otimes{F}(X,A)$, where
$\varphi\otimes f(x)=\varphi(f(x))$ for all $f\in{F}(X)$ and $x\in X$.

_(ii)_ If $x\in X$, then $x\otimes{A}(X,A)=A$, where $x\otimes g=g(x)$ for all
$g\in{A}(X,A)$.

_(iii)_ For every $\varphi\in\Delta(A)$, the mapping $f\mapsto\varphi\otimes
f$ is a continuous epimorphism from ${F}(X,A)$ onto ${F}(X)$.

_(iv)_ For every $x\in X$, the mapping $f\mapsto x\otimes f$ is a continuous
epimorphism from ${A}(X,A)$ onto $A$.

_(v)_ ${A}(X,A)$ is unital if and only if $A$ is unital.

_(vi)_ ${A}(X,A)$ is commutative if and only if $A$ is commutative.

Proof. Let $\varphi\in\Delta(A)$ and $g\in{F}(X)$. Choose $a\in A$ such that
$\varphi(a)=1$. Then $ag\in{F}(X,A)$ and

$\varphi\otimes(ag)(x)=\varphi(ag(x))=g(x)\varphi(a)=g(x)$

for all $x\in X$. So $g=\varphi\otimes(ag)$. Thus ${F}(X)$ contains in
$\varphi\otimes{F}(X,A)$. It is easy to see that $\varphi\otimes f\in{F}(X)$
for all $f\in{F}(X,A)$. So (i) holds.

Let $a\in A$. Define the vector-valued function $f\in{A}(X,A)$ by $f(x)=a$ for
all $x\in X$. Then $a=x\otimes f$. This shows that (ii) holds.

The statements (iii) and (iv) follow from (i) and (ii), respectively. For (v),
let $u$ be an identity element of ${A}(X,A)$. Fix $x_{0}\in X$. Then for every
$a\in A$, we have

$au(x_{0})=x_{0}\otimes(au)=x_{0}\otimes a=a.$

Hence $u(x_{0})$ is an identity of $A$. Thus (v) holds. The proof of statement
(vi) is rutin.$\hfill\mathchar 1027\relax$

###### Proposition 3.3

Let $A$ be a commutative Banach algebra with $\Delta(A)\neq\emptyset$ and
$\alpha>0$. If ${F}(X,A)$ is biprojective, then $X$ and $\Delta(A)$ are
discrete.

Proof. Let ${F}(X,A)$ be biprojective. In view of Theorem 2.10,
$\Delta({F}(X,A))$ is discrete. Since $X\otimes\Delta(A)$ is a subset of
$\Delta({F}(X,A))$, it follows that $X$ and $\Delta(A)$ are
discrete.$\hfill\mathchar 1027\relax$

It is well-known that if $X$ is a locally compact space, then
$\Delta(C_{0}(X,A))$ and $X\otimes\Delta(A)$ are homeomorphic.

###### Theorem 3.4

Let $A$ be a commutative Banach algebra with $\Delta(A)\neq\emptyset$. Then
the following assertions are equivalent.

_(a)_ $C_{b}(X,A)^{**}$ is contractible.

_(b)_ $C_{b}(X,A)$ is contractible.

_(c)_ $C_{0}(X,A)^{**}$ is contractible.

_(d)_ $C_{0}(X,A)$ is contractible.

_(e)_ $X$ is finite and $A$ is finite dimensional.

_(f)_ $A$ is a $C^{*}-$algebra and both $X$ and $\Delta(A)$ are finite.

Proof. It follows from Theorem 2.8 that (a)$\Rightarrow$(b) and
(c)$\Rightarrow$(d). Let ${F}(X,A)$ be contractible. Then $\Delta({F}(X,A))$
is finite. Since $X\otimes\Delta(A)$ is a subset of $\Delta({F}(X,A))$, it
follows that $X\otimes\Delta(A)$ is finite. So $X$ is finite. In view of Lemma
3.2 and Proposition 2.4.2 in [23], $A$ is unital, commutative and
contractible. Hence $A$ is finite dimensional. Thus (b)$\Rightarrow$ (e) and
(d)$\Rightarrow$(e).

If $A$ is finite dimensional, then there exists $n\in{{N}}$ such that $A$ and
${{C}}^{n}$ are isomorphism. Thus $A$ is a contractible $C^{*}-$algebra and so
$\Delta(A)$ is finite. Hence (e)$\Rightarrow$(f).

Let $X$ be finite. Then ${F}(X,A)=C_{0}(X,A)$ and so

$\Delta({F}(X,A))=X\otimes\Delta(A).$

If $A$ is a $C^{*}-$algebra and $\Delta(A)$ is finite, then ${F}(X,A)$ is a
commutative $C^{*}-$algebra and its character space is finite. By Theorem
2.10, ${F}(X,A)$ is contractible. That is, (f)$\Rightarrow$(b) and
(f)$\Rightarrow$(d).

Finally, assume that $X$ is finite and $A$ is finite dimensional. Then there
exists $n\in{{N}}$ such that ${B}(X,A)^{**}$ is a finite number of copies
${{C}}\hat{\otimes}{{C}}^{n}$. Hence ${F}(X,A)^{**}$ is contractible. That is,
(e)$\Rightarrow$(a) and (e)$\Rightarrow$(c). $\hfill\mathchar 1027\relax$

The argument used in the proof of Theorem 3.4 shows the next result holds.

###### Theorem 3.5

Let $A$ be a commutative Banach algebra with $\Delta(A)\neq\emptyset$. Then
the following assertions are equivalent.

_(a)_ $\emph{Lip}_{\alpha}(X,A)^{**}$ is contractible.

_(b)_ $\emph{Lip}_{\alpha}(X,A)$ is contractible.

_(c)_ $\emph{lip}_{\alpha}(X,A)^{**}$ is contractible.

_(d)_ $\emph{lip}_{\alpha}(X,A)$ is contractible.

_(e)_ $X$ is finite and $A$ is finite dimensional.

_(f)_ $A$ is a $C^{*}-$algebra and both $X$ and $\Delta(A)$ are finite.

###### Corollary 3.6

Let $(X,d)$ be a locally compact metric space, $A$ be a Banach algebra and
$\alpha>0$. Then the following statements hold.

_(i)_ $\emph{\hbox{lip}}^{0}_{\alpha}(X,A)$ is contractible if and only if $X$
is finite and $A$ is finite dimensional.

_(ii)_ $\emph{\hbox{lip}}^{0}_{\alpha}X$ is contractible if and only if $X$ is
finite.

Proof. Let $\hbox{lip}^{0}_{\alpha}(X,A)$ be contractible. Hence it is unital
and so $C_{0}(X,A)$ is unital. Thus $X$ is compact. Now, apply Theorem 3.5 to
conclude that (i) holds. The statements (ii) follows at once from
(i).$\hfill\mathchar 1027\relax$

###### Theorem 3.7

Let $A$ be a Banach algebra with $\Delta(A)\neq\emptyset$. Then the following
assertions are equivalent.

_(a)_ $C_{b}(X,A)$ _(_ respectively, $C_{b}(X,A)^{**}$_)_ is contractible.

_(b)_ $C_{0}(X,A)$ _(_ respectively, $C_{0}(X,A)^{**}$_)_ is contractible.

_(c)_ $X$ is finite and $A$ _(_ respectively, $A^{**}$_)_ is contractible.

Proof. Let ${F}(X,A)$ be contractible. Then ${A}(X,A)$ or $C_{0}(X,A)$ is
contractible. Assume that ${A}(X,A)$ is contractible. Then Proposition 2.4.2
in [23] and Lemma 3.2 show that ${A}(X)$ and $A$ are contractible. In view of
Theorem 3.4, $X$ is finite. Now, if $C_{0}(X,A)$ is contractible, then it is
unital. So $X$ is compact and hence $C_{0}(X,A)=C_{b}(X,A)$. This implies that
$X$ is finite and $A$ is contractible.

For the converse, note that if $X$ is finite, then ${F}(X,A)$ is a finite
number of copies ${{C}}\hat{\otimes}A$. So, if $A$ is contractible, then by
Theorem 2.7, ${A}(X,A)$ be contractible.$\hfill\mathchar 1027\relax$

The proof of the following result is similar to Theorem 3.7, and so we omit
it.

###### Theorem 3.8

Let $A$ be a Banach algebra with $\Delta(A)\neq\emptyset$. Then the following
assertions are equivalent.

_(a)_ $\emph{Lip}_{\alpha}(X,A)$ _(_ respectively,
$\emph{Lip}_{\alpha}(X,A)^{**}$_)_ is contractible.

_(b)_ $\emph{lip}_{\alpha}(X,A)$ _(_ respectively,
$\emph{lip}_{\alpha}(X,A)^{**}$_)_ is contractible.

_(c)_ $X$ is finite and $A$ _(_ respectively, $A^{**}$_)_ is contractible.

Theorem 3.7 together with Theorem 2.1 and Lemma 3.2 proves the following
results.

###### Corollary 3.9

Let $A$ be a unital Banach algebra with $\Delta(A)\neq\emptyset$. Then the
following assertions are equivalent.

_(a)_ $C_{b}(X,A)$ _(_ respectively, $C_{b}(X,A)^{**}$_)_ is biprojective.

_(b)_ $C_{b}(X,A)$ _(_ respectively, $C_{b}(X,A)^{**}$_)_ is contractible.

_(c)_ $X$ is finite and $A$ _(_ respectively, $A^{**}$_)_ is contractible.

###### Corollary 3.10

Let $A$ be a unital Banach algebra with $\Delta(A)\neq\emptyset$. Then the
following assertions are equivalent.

_(a)_ $\emph{Lip}_{\alpha}(X,A)$ _(_ respectively,
$\emph{Lip}_{\alpha}(X,A)^{**}$_)_ is biprojective.

_(b)_ $\emph{Lip}_{\alpha}(X,A)$ _(_ respectively,
$\emph{Lip}_{\alpha}(X,A)^{**}$_)_ is contractible.

_(c)_ $\emph{lip}_{\alpha}(X,A)$ _(_ respectively,
$\emph{lip}_{\alpha}(X,A)^{**}$_)_ is biprojective.

_(d)_ $\emph{lip}_{\alpha}(X,A)$ _(_ respectively,
$\emph{lip}_{\alpha}(X,A)^{**}$_)_ is contractible.

_(e)_ $X$ is finite and $A$ _(_ respectively, $A^{**}$_)_ is contractible.

##  4 Homological and cohomological properties of weighted group algebras

Theorem 2.1 shows that there is a closed connection between cohomology and
homology properties. This result enables us to find facts about homological
properties of group algebras and their second dual.

###### Proposition 4.1

Let $\omega$ be a weight function on a locally compact group $G$. Then the
following assertions are equivalent.

_(a)_ $L^{1}(G,\omega)$ is biflat.

_(b)_ $L^{1}(G,\omega)$ is amenable.

_(c)_ $G$ is amenable and $\omega^{*}$ is bounded

Proof. This follows from Theorem 2.1 and the fact that $L^{1}(G,\omega)$ is
amenable if and only if $G$ is amenable and $\omega^{*}$ is bounded; see
[10].$\hfill\mathchar 1027\relax$

Helmeski [14] showed that $L^{1}(G)$ is biprojective if and only if $G$ is
compact. We prove this result for Beurling algebra $L^{1}(G,\omega)$.

###### Proposition 4.2

Let $G$ be a locally compact group. Then the following statements hold.

_(i)_ $L^{1}(G,\omega)$ is biprojective if and only if $G$ is compact.

_(ii)_ $L^{1}(G,\omega)$ is contractible if and only if $G$ is finite.

Proof. (i) Let $L^{1}(G,\omega)$ be biprojective. Then $L^{1}(G,\omega)$ is
biflat and so it is amenable. In this case, it is known from [37] that the
Banach algebras $L^{1}(G,\omega)$ and $L^{1}(G)$ are isomorphic and
homeomorphic as Banach algebras. So Theorem 2.3 shows that $L^{1}(G)$ is
biprojective. Hence $G$ is compact. For the converse, we only remark that if
$G$ is compact, then $L^{1}(G,\omega)=L^{1}(G)$.

(ii) Let $L^{1}(G,\omega)$ be contractible. Then $L^{1}(G,\omega)$ is unital
and so $G$ is discrete. On the other hand, $L^{1}(G,\omega)$ is biprojective.
By (i) $G$ is also compact. Therefore, $G$ is finite. Conversely, if $G$ is
finite, then $L^{1}(G,\omega)$ is unital and by (i), it is biprojective. Thus
$L^{1}(G,\omega)$ is contractible. $\hfill\mathchar 1027\relax$

In the sequel, we investigate homological properties of weighted measure
algebra $M(G,\omega)$.

###### Proposition 4.3

Let $\omega$ be a weight function on a locally compact group $G$. Then the
following assertions are equivalent.

_(a)_ $M(G,\omega)$ is biflat.

_(b)_ $M(G,\omega)$ is amenable.

_(c)_ $G$ is a discrete amenable group and $\omega^{*}$ is bounded.

Proof. Let $M(G,\omega)$ is biflat. In view of Proposition 2.4 ,
$L^{1}(G,\omega)$ is biflat and so $G$ is an amenable group and $\omega^{*}$
is bounded. On the other hand, $M(G,\omega)$ is weakly amenable. Hence $G$ is
discrete; see Theorem 2.4 in [22]. This together with Theorem 2.1 proves the
result.$\hfill\mathchar 1027\relax$

###### Proposition 4.4

Let $\omega$ be a weight function on a locally compact group $G$. Then the
following assertions are equivalent.

_(a)_ $M(G,\omega)$ is biprojective.

_(b)_ $M(G,\omega)$ is contractible.

_(c)_ $G$ is finite.

Proof. Since $M(G,\omega)$ is unital, (a) and (b) are equivalent. Let
$M(G,\omega)$ be contractible. This implies that $M(G,\omega)$ is weakly
amenable. It follows from Theorem 2.4 in [22] that $G$ is a discrete. Thus
$L^{1}(G,\omega)$ is contractible. By Proposition 4.2, $G$ is finite. That is
(b) implies (c). Clearly, (c) implies (a).$\hfill\mathchar 1027\relax$

Let $C_{0}(G,1/\omega)$ be the Banach algebra of all complex-valued functions
$f$ on $G$ such that $f/\omega\in C_{0}(G)$.

###### Theorem 4.5

Let $\omega$ be a weight function on a locally compact group $G$ and ${X}$ be
a Banach algebra containing $C_{0}(G,1/\omega)$. Then the following assertions
are equivalent.

_(a)_ ${X}^{*}$ is biprojective.

_(b)_ ${X}^{*}$ is contractible.

_(c)_ $G$ is finite.

Proof. Let ${X}$ be a Banach algebra containing $C_{0}(G,1/\omega)$. Then

${X}^{*}=M(G,\omega)\oplus C_{0}(G,1/\omega)^{\perp}.$

If ${X}^{*}$ is biprojective, then by Proposition 2.6, $M(G,\omega)$ is
biprojective. Hence $G$ is finite. So (a)$\Rightarrow$ (c). The implications
(c)$\Rightarrow$(b)$\Rightarrow$(a) are clear.$\hfill\mathchar 1027\relax$

###### Theorem 4.6

Let $\omega$ be a weight function on a locally compact group $G$ and ${X}$ be
a Banach algebra containing $C_{0}(G,1/\omega)$. If ${X}^{*}$ is biflat, then
$G$ is a discrete amenable group and $\omega^{*}$ is bounded.

Proof. Let ${X}^{*}$ be biflat. Since $M(G,\omega)$ is a direct summand of
${X}^{*}$, it follows that $M(G,\omega)$ is biflat. Hence $G$ is a discrete
amenable group and $\omega^{*}$ is bounded.$\hfill\mathchar 1027\relax$

Let $C_{b}(G,1/\omega)$ be the Banach algebra of all complex-valued functions
$f$ on $G$ such that $f/\omega\in C_{b}(G)$, and let $LUC(G,1/\omega)$ be the
Banach algebra of all $f\in C_{b}(G,1/\omega)$ such that the map
$x\mapsto\;_{x}(f/\omega)$ from $G$ into $C_{b}(G,1/\omega)$ is norm
continuous. Let also $\hbox{Ap}(G,1/\omega)$ (respectively,
$\hbox{Wap}(G,1/\omega)$) be the set of all functions $f\in C_{b}(G,1/\omega)$
such that

$\\{\frac{{}_{x}f}{\omega(x)\omega}:x\in G\\}$

is relatively norm (respectively, weakly) compact in $C_{b}(G)$, where
${}_{x}f(y)=f(yx)$ for all $x,y\in G$. Let $L_{0}^{\infty}(G,1/\omega)^{*}$
and $M(G,\omega)_{0}^{*}$ be the Banach algebras defined as in [20, 21],
respectively. It is easy to see that $C_{0}(G,1/\omega)$ is a subspace of
these Banach algebras. So, we may apply Theorems 4.5 and 4.6 for them.

###### Theorem 4.7

Let $G$ be a locally compact group. Then the following assertions are
equivalent.

_(a)_ $L^{1}(G,\omega)^{**}$ _(_ respectively, $M(G,\omega)^{**}$_)_ is
amenable.

_(b)_ $L^{1}(G,\omega)^{**}$ _(_ respectively, $M(G,\omega)^{**}$_)_ is
biflat.

_(c)_ $L^{1}(G,\omega)^{**}$ _(_ respectively, $M(G,\omega)^{**}$_)_ is
biprojective.

_(d)_ $L^{1}(G,\omega)^{**}$ _(_ respectively, $M(G,\omega)^{**}$_)_ is
contractible.

_(e)_ $L^{1}(G,\omega)^{**}$ _(_ respectively, $M(G,\omega)^{**}$_)_ is weakly
amenable.

_(f)_ $G$ is finite.

Proof. In view of Theorem 2.1, the statements (a)- (d) imply the statement
(e). Also, by Theorem 3.1 in [22], the statement (e) follows (f). So the proof
will be completed if we note that the statement (f) implies the other
statements.$\hfill\mathchar 1027\relax$

###### Remark 4.8

From Proposition 4.2 and Theorem4.7 we see that if $G$ is an infinite compact
group, then $L^{1}(G)$ is both biprojective and biflat, however,
$L^{1}(G)^{**}$ is not neither biprojective nor bifalt. So biprojectivity
(respectively, biflatness) of $A$ does not imply biprojectivity (respectively,
biflatness) of $A^{**}$.

###### Theorem 4.9

Let $G$ be a locally compact group and $A$ be a Banach algebra with
$\Delta(A)\neq\emptyset$. Then $L^{1}(G,A)$ _(_ respectively,
$L^{1}(G,A)^{**}$_)_ is contractible if and only if $G$ is finite and $A$ _(_
respectively, $A^{**}$ _)_ is contractible.

Proof. Let $\varphi\in\Delta(A)$. Then the mapping $f\mapsto\varphi\circ f$ is
a continuous epimorphism from $L^{1}(G,A)$ onto $L^{1}(G)$. Also, if $x\in G$,
then the mapping $f\mapsto f(x)$ is a continuous epimorphism from $L^{1}(G,A)$
onto $A$. Now, apply Proposition 2.4.2 in [23] and Proposition
4.2.$\hfill\mathchar 1027\relax$

###### Theorem 4.10

Let $G$ be a locally compact group and $A$ be a unital, commutative Banach
algebra. Then the following assertions are equivalent.

_(a)_ $L^{1}(G,A)$ is biflat.

_(b)_ $L^{1}(G,A)$ is amenable.

_(c)_ $G$ and $A$ are amenable.

Proof. Since $A$ is unital and $L^{1}(G)$ has a bounded approximate identity,
(a) and (b) are equivalent. If (c) holds, then $L^{1}(G)$ and $A$ are
amenable. So $L^{1}(G)\hat{\otimes}A$ is amenable. From this and the fact that
$L^{1}(G,A)$ isometrically isomorphic to $L^{1}(G)\hat{\otimes}A$ we infer
that (c) implies (b). By the proof of Theorem 4.9, the Banach algebras
$L^{1}(G)$ and $A$ are continuous homomorphic images of $L^{1}(G,A)$. Hence
amenability of $L^{1}(G,A)$ implies amenability of $G$ and $A$. That is, (b)
implies (c).$\hfill\mathchar 1027\relax$

###### Theorem 4.11

Let $G$ be a locally compact group and $A$ be a Banach algebra with
$\Delta(A)\neq\emptyset$. If $A$ has a non-zero idempotent element, then the
following statements hold.

_(i)_ $M(G)\hat{\otimes}A$ is biprojective if and only if $G$ is finite and
$\ell^{1}(G)\hat{\otimes}A$ is biprojective.

_(ii)_ $M(G)\hat{\otimes}A$ is biflat if and only if $G$ is finite and
$\ell^{1}(G)\hat{\otimes}A$ is biflat.

Proof. (i) Let $M(G)\hat{\otimes}A$ be biprojective. By [28], the Banach
algebras $M(G)$ is biprojective. It follows from Theorem 4.4 that $G$ is
finite and so $M(G)\hat{\otimes}A=\ell^{1}(G)\hat{\otimes}A$ is biprojective.
The converse is clear.

(ii) This proves similar to (i).$\hfill\mathchar 1027\relax$

###### Theorem 4.12

Let $G$ be a locally compact group and $A$ be a Banach algebra with
$\Delta(A)\neq\emptyset$. Then $M(G)\hat{\otimes}A$ is contractible if and
only if $G$ is finite and $A$ is contractible.

Proof. Let $M(G)\hat{\otimes}A$ be contractible. By Theorem 2.7, the Banach
algebras $M(G)$ and $A$ are contractible. It follows from Theorem 4.4 that $G$
is finite.$\hfill\mathchar 1027\relax$

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  * [24] M. S. Moslehian and A. Niknam, Biflatness and biprojectivity of second dual of Banach algebras, Southeast Asian Bull. Math., 27 (1)(2003)129–133.
  * [25] K. Parthasarathy and N. Shravan Kumar, Homological properties of Fourier algebras on homogeneous spaces, Arch. Math. (Basel), 96 (2011), no. 4, 359–367.
  * [26] A. Yu. Yu. Pirkovskii and Ju. V. Selivanov, Homologically trivial Frechet algebras, in: Topological Algebras and Applications, Contemp. Math. 427, Amer. Math. Soc., Providence, RI, 2007, 367–387.
  * [27] P. Ramsden, Homological Properties of Semigroup Algebras, Thesis (Ph.D.)–University of Leeds, 2008.
  * [28] P. Ramsden, Biflatness of semigroup algebras, Semigroup Forum, 79 (2009), no. 3, 515–530.
  * [29] H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, London Math. Society Monographs, 22, Clarendon Press, Oxford, 2000.
  * [30] V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer Verlag, Berlin, 2002.
  * [31] V. Runde, Biflatness and biprojectivity of the Fourier algebra, Arch. Math. (Basel), 92 (2009), no. 5, 525–530.
  * [32] Ju. V. Selivanov, Biprojective Banach algebras, their structure, cohomology and relation with nuclear operators, (Russian) Funkcional. Anal. i Prilozen, 10 (1976), no. 1, 89–90.
  * [33] Ju. V. Selivanov, Cohomological characterizations of biprojective and biflat Banach algebras, Monatsh. Math., 128 (1999), no. 1, 35–60.
  * [34] D. R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., 13 (1963), 1387–1399.
  * [35] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc., 111 (1964), 240–272.
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  * [37] M. C. White, Characters on weighted amenable groups, Bull. London Math. Soc., 23 (1991), no. 4, 375–380.
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Mohammad Javad Mehdipour
Department of Mathematics,
Shiraz University of Technology,
Shiraz 71555-313, Iran
e-mail<EMAIL_ADDRESS>
Ali Rejali
Department of Pure Mathematics,
Faculty of Mathematics and Statistics,
University of Isfahan,
Isfahan 81746-73441, Iran
e-mail<EMAIL_ADDRESS>


#  A Nearly Finitary Matroid that is not $k$-Nearly Finitary

Patrick Tam

###### Abstract

The class of $k$-nearly finitary matroids for some natural number $k$ is a
subclass of the class of nearly finitary matroids. A natural question is
whether this inclusion is proper. We answer this question affirmatively by
constructing a nearly finitary matroid that is not $k$-nearly finitary for any
$k\in\mathbb{N}$.

##  1 Introduction

We settle a question raised in [3]. Let us first review some background
material starting with an axiom system for infinite matroids from [1].

Now, we introduce a set of matroid axioms developed in [1] in 2010 that
captures essential aspects of finite matroid theory while allowing for
infinite matroids to be defined.

Now we let $E$ to be any set and possibly infinite. Note that for shorthand,
we define $A+b:=A\cup\\{b\\}$ and $A-b:=A\setminus\\{b\\}$.

######  Definition 1.1.

A matroid $M$ is a pair $(E,\mathcal{L})$ with $\mathcal{L}\subset 2^{E}$
satisfying the following properties:

  * •

I1: $\emptyset\in\mathcal{L}$.

  * •

I2: If $B\in\mathcal{L}$ and $A\subset B$, then $A\in\mathcal{L}$.

  * •

I3: If $B$ is a maximal element of $\mathcal{L}$ and $A$ is a non-maximal
element of $\mathcal{L}$, then there exists $b\in B\setminus A$ such that
$A+b\in\mathcal{L}$.

  * •

I4: If $A\in\mathcal{L}$ and $A\subset X\subset E$, then the set
$\\{S\in\mathcal{L}:A\subset S\subset X\\}$ has a maximal element.

Here, we use set inclusion as our partial ordering when we talk about
maximality and minimality. We will use this partial order throughout the rest
of the paper. Elements of $\mathcal{L}$ are called independent sets. Maximal
elements of $\mathcal{L}$ are also called bases. These first two axioms are
familiar from finite matroids. The third axiom is different from our third
axiom for finite matroids because of the possibility of infinite independent
sets. It may be possible to extend a countable independent set by another
countable independent set under these axioms. A countable set can be a proper
subset of another countable set. Thus, cardinality does not give us enough
information to determine whether we can extend an independent set by another
one. We thus need axiom I3 which is not reliant on comparing cardinalities.
The fourth axiom ensures that every matroid $M$ has a base and any restriction
of $M$ to any subset of the ground set of $M$ has a base. Elements of
$2^{E}\setminus\mathcal{L}$ are called dependent sets. A minimal dependent set
is called a circuit. A circuit with only one element is called a loop. It is
possible to define a matroid $M=(E,\mathcal{L})$ by specifying a suitable set
of circuits and taking the independent sets to be subsets of $E$ that contain
no circuit. Any pair $M=(E,\mathcal{L})$ that satisfies the first two axioms
I1 and I2 is called an independence system.

The rank of a matroid is given by the cardinality of a base if this
cardinality is finite. If bases of a matroid are infinitely large, we simply
say that the rank of that matroid is infinite. In other words, there is only
one infinite rank for matroids.

Under the above axioms, every matroid has a base and a dual. Let
$M=(E,\mathcal{L})$ be a matroid. We define

$\mathcal{L}^{*}:=\\{S\subset E\colon\exists
B\in\mathcal{L}^{\mathrm{max}}\text{ s.t. }S\subset E\setminus B\\}$

where $\mathcal{L}^{\mathrm{max}}$ is the set of bases of $M$. Then
$M^{*}=(E,\mathcal{L}^{*})$ is the dual of $M$. The authors of [1] who
developed this axiom system showed that this dual is indeed a matroid. In
fact, this duality is an involution and $M^{**}=M$.

A circuit of $M^{*}$ is called a cocircuit of $M$. A loop of $M^{*}$ is also
known as a coloop of $M$. Equivalently, a coloop is an element of $M$ that is
not contained in any circuit of $M$ and thus contained in every base of $M$.

A special class of matroids which are known as the finitary matroids are
better understood than general infinite matroids.

######  Definition 1.2.

A matroid $M$ is finitary if a set $S$ is independent in $M$ if and only if
all finite subsets of $S$ are independent.

For every matroid $M=(E,\mathcal{L})$, there exists an associated finitary
matroid $M^{\mathrm{fin}}=(E,\mathcal{L}^{\mathrm{fin}})$ whose independent
sets are subsets $S$ of $E$ such that every finite subset of $S$ is
independent in $M$. The proof of this relies on Zorn’s lemma and can be found
in [1]. $M^{\mathrm{fin}}$ is also known as the finitarization of $M$.

To understand finitarization, consider the matroid
$M=(\mathbb{N},\mathcal{L})$ where

$\mathcal{L}:=\\{S\subset\mathbb{N}\colon|\mathbb{N}\setminus S|\geq 2\\}.$

Since all finite subsets of $\mathbb{N}$ are independent in $M$,
$M^{\mathrm{fin}}=(\mathbb{N},2^{\mathbb{N}})$. Here, $M^{\mathrm{fin}}$ has
more independent sets than $M$. We can make this precise in the following way.

Every base $B$ of $M$ extends to a base $F$ of $M^{\mathrm{fin}}$. To see why,
note that any base $B$ is independent in $M^{\mathrm{fin}}$. Because of our
fourth independence axiom for matroids, the set
$\\{S\in\mathcal{L}(M^{\mathrm{fin}})\colon B\subset S\subset E\\}$ has a
maximal element $F$. This maximal element is a base of $M^{\mathrm{fin}}$.

Conversely, every base $F$ of $M^{\mathrm{fin}}$ contains a base $B$ of $M$.
To see why, consider $F^{*}:=E\setminus F$. Then $F^{*}$ is a base of
$M^{\mathrm{fin}*}$ . Independent sets of $M^{\mathrm{fin}*}$ are also
independent in $M^{*}$. So $F^{*}$ extends to some base $B^{*}$ in $M^{*}$ by
I4. $B:=E\setminus B^{*}$ is then a base in $M$ and $F$ contains $B$.
Unfortunately, the class of finitary matroids is not closed under duality. The
authors of [2] define the class of nearly finitary matroids which is still not
closed under duality but extends the class of finitary matroids.

######  Definition 1.3.

A matroid $M$ is called nearly finitary if whenever a base $F$ in
$M^{\mathrm{fin}}$ contains a base $B$ in $M$, their set difference is a
finite set.

######  Definition 1.4.

For an integer $k$, a matroid is called $k$-nearly finitary if whenever a base
$F$ in $M^{\mathrm{fin}}$ contains a base $B$ in $M$, their set difference has
cardinality bounded by $k$.

##  2 Counterexample

A natural question raised in [3] is whether every nearly finitary matroid is
$k$-nearly finitary for some $k\in\mathbb{N}$. We now present an example of a
nearly finitary matroid that is not $k$-nearly finitary for any $k$ that is
originally due to Attila Por (personal communication, November 20, 2018).

Before we construct our example, let us first introduce a finitary matroid
that will allow us to construct our counterexample.

Note, our set of natural numbers $\mathbb{N}$ excludes zero and consists of
positive integers. Let $E$ be the Cartesian product
$\mathbb{N}\times\mathbb{N}$. We imagine $E$ to be the integral points of the
first quadrant. We define $A\subset E$ to be in $\mathcal{L}_{1}$ if for every
$n\in\mathbb{N}$, the first $n$ rows of $E$ have at most $n$ elements in A.
Then $M_{1}=(E,\mathcal{L}_{1})$ is a finitary matroid. It is straightforward
to see that $M_{1}$ satisfies I1 and I2. For I3, suppose there is a base
$B_{1}$ of $M_{1}$ and a non-maximal independent set $A_{1}$. Since $A_{1}$ is
non maximal, there is some base $A_{B}$ that properly contains $A_{1}$. Let
$i\in\mathbb{N}$ be a row which contains an element of $A_{B}\setminus A_{1}$.
Since $B_{1}$ is maximal, there is some $j\geq i$ such that $B_{1}$ has
exactly $j$ elements in the first $j$ rows. By construction, $A_{1}$ has
strictly less than $j$ elements in the first $j$ rows. There must be some
$d\leq j$ such that $A_{1}$ has strictly less than $n$ elements in the first
$n$ rows for all $n\geq d$. We pick the smallest such $d$. $B_{1}\setminus
A_{1}$ has at least one element $b$ in rows $d$ to $j$. This element can be
used to augment $A_{1}$ and I3 is satisfied. For I4, suppose $A_{1}\subset X$
with $A_{1}\in\mathcal{L}_{1}$. Let $A_{1}[0]=A_{1}$. We inductively let
$A_{1}[i]$ be $A_{1}[i-1]\cup S[i]$ where $S[i]$ is a maximal subset of row i
of $X$ that keeps $A_{1}[i]$ independent. Taking the union of all these
$A_{1}[i]$ sets will give us our desired maximal independent set that shows
that I4 is satisfied. Finally, to show that $M_{1}$ is finitary, suppose $A$
is any subset of $E$ whose finite subsets are all independent. If $A$ were
dependent, then there would be some $j$ where $A$ has more than $j$ elements
in the first $j$ rows. Picking these elements would give us a finite dependent
set which contradicts our assumption that all of $A$’s finite subsets are
independent. Thus, we conclude that $A$ is independent and that $M_{1}$ is
finitary.

We now define $\mathcal{L}\subset\mathcal{L}_{1}\subset 2^{E}$. Suppose
$A_{1}$ is independent in $\mathcal{L}_{1}$. If $A_{1}$ is finite, then
$A_{1}$ is in $\mathcal{L}$. If $A_{1}$ is infinite, then we define column $l$
to be dominant in $A_{1}$ if all but finitely many elements of $A_{1}$ are in
column $l$. If the $l$-th column of $A_{1}$ is dominant, subsets of the form
$A_{1}\setminus X$ where $X\subset A_{1}$ and $|X|\geq l$ are elements of
$\mathcal{L}$. If there is no such dominant column $l$, then we declare
$A_{1}$ to be an element of $\mathcal{L}$.

######  Theorem 2.1.

$M=(E,\mathcal{L})$ as constructed above is a nearly finitary matroid, but not
$n$-nearly finitary for any $n\in\mathbb{N}$.

###### Proof.

$M$ naturally inherits I1 and I2 from $M_{1}$. It also clearly satisfies I4
because it is nearly finitary. For I3, let $S$ be a non-maximal independent
set and let $B$ be a base. $S$ is properly contained in some base $A$. If $A$
and $B$ both have no dominant column, we can extend $S$ with an element of $B$
because $M_{1}$ is a matroid. If $A$ and $B$ have the same dominant column
$k_{1}$, it is straightforward to see that we can extend $S$ with an element
in $B\setminus S$. We follow [2] and define the truncation matroid $M_{1}[k]$
as follows. $M_{1}[k]$ has the same finite independent sets as $M_{1}$. An
infinite set is independent in $M_{1}[k]$ if it can be obtained by deleting
$k$ elements of an independent set in $M_{1}$. This construction was proven to
be a matroid in [2]. We define $M_{S}(k)$ to be $M_{1}[k]$ restricted to
$A\cup B$. Now, $M_{S}(k_{1})$ is a matroid with independent sets that are
independent in $M$ and thus I3 is satisfied. Let us now assume that at least
one of $A$ and $B$ have a dominant column but do not share the same dominant
column. This is the last remaining case. $B\setminus S$ has elements above the
row $m$ for all $m\in\mathbb{N}$. Since $S$ is properly contained in $A$,
there is some $a\in A\setminus S$. Let $r$ be the index of the row where this
$a$ is. Any element in $B\setminus S$ above row $r$ can be used to augment $S$
to satisfy I3 and finally prove that $M$ is a matroid. It is clear that $M$ is
nearly finitary but not $n$-nearly finitary for any $n\in\mathbb{N}$. ∎

From this example infinite families can be obtained by standard matroid
operations defined in [2]. Some of these are compiled in the following
statements. We omit the (easy) proofs:

######  Theorem 2.2.

Let $M$ be a matroid that is nearly finitary but not $n$-nearly finitary for
any $n\in\mathbb{N}$. Then, the following matroids have the same property:

  1. 1.

The truncation $M[k]$ for any $k\in\mathbb{N}$, as defined in [2].

  2. 2.

The direct sum $M\oplus N$ for any nearly finitary matroid $N$. In other
words, this is the matroid union $M\vee N$ where $M$ and $N$ are defined on
disjoint ground sets.

  3. 3.

The deletion $M-S$ for any finite subset $S$ of the ground set.

  4. 4.

The contraction $M/T$ for any set $T$ of coloops in $M$.

### Acknowledgements

We thank Attila Por for providing the example described in the paper.
Coauthorship was offered to him. We also thank the editor and reviewers for
useful feedback to improve this paper.

## References

  * [1] H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh, and P. Wollan.  Axioms for infinite matroids.  In _Advances in Mathematics_ , 239:18–46.  arxiv:1003.3919, 2013.
  * [2] E. Aigner-Horev, J. Carmesin, J. Fröhlich.  Infinite matroid union.  arxiv:1111.0602, 2012.
  * [3] E. Aigner-Horev, J. Carmesin, J. Fröhlich.  On the intersection of infinite matroids.  _Discrete Mathematics_ , 341(6):1582-1596, 2018.


# Teaching Introductory HRI: UChicago Course
“Human-Robot Interaction: Research and Practice”

Sarah Sebo<EMAIL_ADDRESS>0000-0003-2211-6429 The University of
ChicagoChicagoUSA

(2024)

###### Abstract.

In 2020, I designed the course CMSC 20630/30630 Human-Robot Interaction:
Research and Practice as a hands-on introduction to human-robot interaction
(HRI) research for both undergraduate and graduate students at the University
of Chicago. Since 2020, I have taught and refined this course each academic
year. Human-Robot Interaction: Research and Practice focuses on the core
concepts and cutting-edge research in the field of human-robot interaction
(HRI), covering topics that include: nonverbal robot behavior, verbal robot
behavior, social dynamics, norms & ethics, collaboration & learning, group
interactions, applications, and future challenges of HRI. Course meetings
involve students in the class leading discussions about cutting-edge peer-
reviewed research HRI publications. Students also participate in a quarter-
long collaborative research project, where they pursue an HRI research
question that often involves conducing their own human-subjects research study
where they recruit human subjects to interact with a robot. In this paper, I
detail the structure of the course and its learning goals as well as my
reflections and student feedback on the course.

Human-Robot Interaction, Introductory HRI Course

††copyright: rightsretained††journalyear: 2024††doi: XXXX.XXXX††conference:
Presented at the Designing an Intro to HRI Course Workshop at HRI 2024
(arXiv:2403.05588) Report-no: HRI101/2024/10; March 11, 2024; Boulder,
CO††ccs: Social and professional topics Computer science education††ccs:
Human-centered computing HCI design and evaluation methods††ccs: Computer
systems organization Robotics

##  1\. Introduction

At the University of Chicago, I created the course CMSC 20630/30630 Human-
Robot Interaction: Research and Practice to serve as an introductory course
for both undergraduate and graduate students interested in learning more about
and gaining hands-on experience in human-robot interaction (HRI). Students are
exposed to important topics in HRI through a carefully selected reading list
of HRI publications, where most class periods are devoted to a class
discussion of one of these papers. Students also gain hands-on experience with
HRI by conducting an HRI research project of their own design that they
complete throughout the academic quarter.

From my observations and the feedback I have received from students, this
course has been well-received. I, as the instructor, am able to broaden and
deepen my thoughts about HRI through thoughtful class discussions and gain new
inspiration for research ideas and directions through students’ course
projects. Students reported enjoying the class discussions, feeling challenged
to think about human-robot interaction in new ways. They also gained
tremendously from the course project, which gave them real, tangible
experience conducting HRI research.

This paper focuses on describing the structure of the course and my rationale
for its design. I also include some student feedback about the course and my
own reflections on the strengths of the course and areas for improvement. For
further details, please refer to the 2024 course
website111https://classes.cs.uchicago.edu/archive/2024/winter/20630-1/index.html
and my personal website222https://sarahsebo.com/teaching.html for links to all
of my publicly available course materials.

##  2\. Course Structure

At the University of Chicago, CMSC 20630/30630 Human-Robot Interaction:
Research and Practice enrolls a maximum of 20 students in order to enable an
environment for fruitful in-class discussions where all students are able to
contribute. While this class size may not scale to other institutions, this is
a typical class size for many specialized Computer Science courses at the
University of Chicago. Additionally, the University of Chicago operates on the
quarter system, where each quarter involves 9 weeks of instruction (semester
schedules typically include 13 weeks of instruction) and 1 week for finals. In
the rest of this section I detail the structure of the course, its learning
objectives, the format of class meetings, and details about the quarter-long
class project.

###  2.1. Course Description

Robots are increasingly common in our everyday spaces: tutoring elementary
students, assisting human workers in manufacturing contexts, providing museum
tours, interacting with families within their homes, and helping to care for
the elderly. One critical factor to the success of these robots is their
ability to effectively interact with people: human-robot interactions.

This course focuses on the core concepts and cutting-edge research in the
field of human-robot interaction (HRI), covering topics that include:
nonverbal robot behavior, verbal robot behavior, social dynamics, norms &
ethics, collaboration & learning, group interactions, applications, and future
challenges of HRI. Course meetings involve students in the class leading
discussions about cutting-edge peer-reviewed research HRI publications.
Students also participate in a quarter-long collaborative research project,
where they pursue an HRI research question that often involves conducing their
own human-subjects research study where they recruit human subjects to
interact with a robot.

The course has no prerequisites, however backgrounds in robotics or HCI is
encouraged. Programming knowledge is not necessary, however, is often useful
for the course project.

###  2.2. Learning Objectives

This course has three central learning objectives:

  1. (1)

Students will obtain a broad understanding and exposure to cutting-edge
research in the field of Human-Robot Interaction through course readings and
discussions of those readings.

  2. (2)

Students will cultivate analytical and critical thinking skills when
evaluating research in HRI through leading class discussions and making
analytical comments on the readings.

  3. (3)

Students will gain hands-on HRI research experience through the course
project.

###  2.3. Class Meetings: Discussions about HRI Research Papers

Class meetings occur three times a week (typically Monday, Wednesday, Friday)
for 50 minutes each. As shown in Table 1, each week of the course is focused
on a different core topic in HRI. During each class meeting, the class reads
and discusses one paper. I carefully select these papers to include a
distribution of:

  * •

Foundational papers in the field (e.g., Robinette et al. (2016)’s
investigation of people’s overtrust of robots in emergency evacuation
scenarios, Brščić et al. (2015)’s exploration of robot abuse in shopping
malls, Kahn Jr et al. (2012)’s examination of child responses to a robot being
forced to go in a closet)

  * •

HRI papers published within the last year (e.g., for the Winter 2024 version
of the course I included 8 papers published in 2023, 6 of which were published
at the HRI 2023 conference (Cui et al., 2023; Brawer et al., 2023; Jeong et
al., 2023; Ligthart et al., 2023; Nanavati et al., 2023; Winkle et al., 2023)
and 2 of which were published in other HRI or robotics publication venues
(Mavrogiannis et al., 2022; Natarajan et al., 2023))

  * •

Papers that cover important topics in the field of HRI (e.g., trust (Robinette
et al., 2016), nonverbal robot behaviors such as gaze (Gillet et al., 2021)
and gestures (Rifinski et al., 2021), HRI ethics (Winkle et al., 2023),
multiparty HRI (Erel et al., 2022; Reig et al., 2021; Tennent et al., 2019))

Before each class meeting, each student is expected to read the assigned paper
and submit at least one analytical comment on the paper using the Google
Chrome extension hypothes.is. Analytical comments are intended to demonstrate
critical thinking about the paper and can, for example, highlight and justify
strengths or weaknesses with the methodology or data analysis, suggest a
fruitful avenue for future research, discuss potential implications.

In addition to the required analytical comment, 2-3 students are asked to
write a peer-review of the paper. Guidelines for the peer-review are similar
to those used for the ACM/IEEE International Conference on Human-Robot
Interaction. Each student is asked to write two peer reviews during the
course, which are due before the class discussion of that paper.

During each class meeting one student serves as the discussion leader. The
discussion leader gives a brief presentation, where they provide an overview
of the paper, highlight strengths and weaknesses, and pose discussion
questions for the class. Then, the discussion leader moderates a class
discussion.

For 2-3 class periods during the quarter, I invite the first author of one of
the recently published papers to join our class as a guest presenter. The
guest presenter (via Zoom) presents their paper and then engages in an open
question and answer session with the students. This provides students with the
opportunity to directly learn from top researchers in the field and gain a
deeper insight into the work that goes into the papers we read in class.

Table 1. Class Meeting Schedule for UChicago’s HRI: Research & Practice in the Winter Quarter of 2024 Week | Topic | Paper Sub-Topics & Citations
---|---|---
1 | Course Introduction | Robot Embodiment (Bainbridge et al., 2011)
2 | Verbal Behavior | Emotion (Pelikan et al., 2020)
|  | Self-Disclosure (Martelaro et al., 2016)
3 | Nonverbal Behavior | Gaze (Gillet et al., 2021)
|  | Non-humanoid Gestures (Rifinski et al., 2021)
4 | Social Dynamics | Trust (Robinette et al., 2016)
|  | Conflict Resolution (Babel et al., 2022)
|  | Robot Role (Jeong et al., 2023)
5 | Norms/Ethics | Harsh Robot Treatment (Kahn Jr et al., 2012)
|  | Social Norms (Komatsu et al., 2021)
|  | Ethical Research Practice (Winkle et al., 2023)
6 | Collaboration & | Crowd Navigation (Mavrogiannis et al., 2022)
| Learning | Language Corrections (Cui et al., 2023)
|  | Interactive Policy Shaping (Brawer et al., 2023)
7 | Group Interactions | Shaping Group Dynamics (Tennent et al., 2019)
|  | Carryover Effects (Erel et al., 2022)
|  | Multiple Robots (Reig et al., 2021)
8 | Applications | Public Spaces (Brščić et al., 2015)
|  | Education (Ligthart et al., 2023)
|  | Robot-Assisted Feeding (Nanavati et al., 2023)
9 | Grand Challenges | Human-Robot Teaming (Natarajan et al., 2023)

###  2.4. Hands-On Course Project where Students Conduct their own HRI
Research Project

The hands-on course project is the cornerstone of this course, where students
conduct a quarter-long HRI research project of their own design. Students
complete their projects in groups of 2-4 and are given access to my research
lab’s resources (i.e., robots) to complete their projects. The robots
available to students for their course projects in my lab include: 2 SoftBank
Robotics Nao robots, 1 Hello Robot Stretch robot, 1 Franka Emika Panda robot,
7 Anki Vector Robots, 2 Misty Robotics Misty II robots, 11 ClicBot robots, and
12 Turtlebot3 robots.

Students are told that their project must (1) be focused on the interaction
between physically embodied robots and one or more people and (2) have the
potential to contribute new knowledge to the field of HRI. While most students
choose to conduct an in-person human subjects study, students are allowed to
conduct a project that falls under one of the following categories:

  * •

An in-person human-subjects study: this option gives students the chance to
conduct an in-person human-subjects study of their own design. Students are
given target number of participants to recruit based on their group size
(e.g., 10 participants for groups of 2, 24 participants for groups of 4).

  * •

A human-subjects study evaluated on an online crowd-sourced platform: this
option gives students the chance to conduct an HRI human-subjects study on an
online crowd-sourced platform (e.g., Prolific), that would enable them to
gather a larger sample size.

  * •

Analysis of a human-robot interaction in a pre-recorded dataset: this option
requires students to find an HRI dataset and analyze the data in a novel way.

  * •

Development of a computational model to improve specific human-robot
interactions: this option involves the development of a novel computational
model, algorithm, or framework to improve human-robot interactions. Students
pursuing this option are expected to both build a novel computational tool for
HRI and evaluate it against current state-of-the-art approaches.

  * •

A literature review: this option involves diving into the published work on a
particular topic, conducting a systematic literature review, analyzing the
papers, and writing a review that both presents the work in the chosen area
and highlighting opportunities for future work.

Table 2. Typical course project schedule for teams pursuing an in-person human-subjects study Week |  Course Project Activities
---|---
1 |  Introduction of course project
2 |  Students develop 1-3 course project ideas, pitch them to the class, and form project teams
3 |  Initial project proposal is due, the instructor meets with each team to provide feedback on the projects
4 |  Finalized project proposal is due that incorporates instructor feedback, students finalize their study design
5 |  Students submit an Institutional Review Board (IRB) application for their study
6 |  Progress report #1 is due, students work to program the robot for their study
7 |  Students begin to pilot their studies with the instructor and others
8 |  Progress report #1 is due, students recruit and run human subjects
9 |  Students finish running human subjects, analyze their data, and make a final presentation to the class
10 |  Students finalize their projects and submit a final report

Students are expected to complete an end-to-end HRI project. For those who
choose to pursue a human-subjects study, this entails: developing a study
design, devloping appropriate measures, submitting an Institutional Review
Board (IRB) application, programming a robot, recruiting and running 10-24
human subjects, analyzing the resulting data with appropriate statistical
tests, and writing a final report that resembles an HRI conference paper. An
example project timeline is shown in Table 2.

The teaching staff (instructor, TA) provide students with considerable support
on their course projects. We meet with students (often several times) to
refine study designs, review IRB applications, provide students with help
getting set up and debugging robots, pilot students’ studies before they run
participants, help distribute human subjects payments, and provide guidance
and example R code for how to conduct statistical analyses of HRI data.

###  2.5. Grading

This course is graded as follows:

  * •

60% project

  * •

20% analytical comments on the readings

  * •

10% discussion leadership & peer-reviews of readings

  * •

10% attendance/participation

Students are provided with a final project grading rubric, so they are aware
of how the project grade is allocated. Of the students grade that comes from
the project, 50% comes from project work, 25% from the final report, 15% from
intermediary deliverables, and 10% from their final project presentation.

##  3\. Student Feedback

This course has been well received by students, with 100% of Human-Robot
Interaction: Research and Practice students agreeing with “I would recommend
this course to other students” in UChicago’s student course evaluations.
Below, I report some common themes in the course feedback.

###  3.1. Vibrant Class Discussions

Students thought that the paper readings and class discussions were an
effective way to learn about HRI, with students saying “asking students to
read papers and lead presentations is a good way to lean the key methods and
know about HRI research” and “the course was great and the readings were a
good summary of HRI.” While class meeting consisting entirely of discussions
may have initially sounded dull to students, students reported having
“genuinely interesting” and “thought provoking” discussions. Students found
class meetings “went by quickly and often led to post–class discussions that
would linger out into the hallway” and that “it was quite rare that we would
run out of things to talk about when discussing a paper.”

###  3.2. Appreciation of Hands-On Final Project

Students appreciated the chance to do a hands-on project in the course,
gaining valuable experience both in HRI and in conducing research. Students
described the course project as “the highlight of the course” and that the
“practical research experience [was] very valuable.” One undergraduate
student’s experience in particular highlights the value of the hands-on
research project, who said, “I genuinely appreciated this project a lot. As an
economics major my prior experience in the field of HRI could generously be
described as limited. So really getting the opportunity to get my feet wet
with a research experience through the project was very exciting for me.”

###  3.3. Desire for More Instruction on HRI Methods

Some students noted that it could have been helpful for more class time to
have been spent the methods of conducting HRI research. For example, one
student said that “having one or two class sessions on concrete skills related
to the class project, such as conducting data analysis or designing surveys,
would be helpful.”

##  4\. Reflections on Strengths and Areas of Improvement for the Course

I believe that the core strength of this course is the hands-on HRI research
project that students conduct through the course of the quarter. Students have
the opportunity to truly see how people interact with robots, can test their
own hypotheses about how they believe these human-robot interactions will go,
and gain experience with every aspect of the HRI research process. In fact,
several of these projects have later continued on and have either been
published (Lin et al., 2022; Mazursky et al., 2022; Naggita et al., 2022) or
are in the submission process. The in-class discussions are also a strength of
the course, and provide a format for ideas to be exchanged in an engaging and
respectful format.

With respect to areas for improvement, students do report the desire for
additional support and instruction on HRI methods. In the four years I have
taught this class, I have approached it with the idea that these skills are
better learned “on the job” when you can directly apply them to the course
project. However, there may be opportunities for these methods to be more
formally taught in a way that can help students easily apply these skills to
their course projects without such a high learning barrier.

###### Acknowledgements.

I thank the students who have taken this course, who have engaged meaningfully
in class discussions, changed how I think about about HRI, and helped to
improve of this course each year. I also thank Brian Scassellati, whose CPSC
473 Intelligent Robotics Lab course has greatly influenced the design of this
course.

## References

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