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\topmatter
\title
FOUNDATIONS OF SUPERMANIFOLD THEORY: \\ THE AXIOMATIC APPROACH \dag
\endtitle
\author
C\. Bartocci,\ddag\ U\. Bruzzo,$\star$\\
D\. Hern\'andez Ruip\'erez\P \  {\rm and}\  V.G\. Pestov\S
\endauthor
\affil
\ddag\thinspace Dipartimento di Matematica, Universit\`a
di Genova, Italia \\ \P\thinspace Departamento de
Matem\'atica Pura y Aplicada, \\ Universidad de Salamanca, Espa\~na \\
\S Department of Mathematics and Statistics, \\ University of Victoria, B.C\.,
             Canada
\endaffil
\address{(\ddag) Dipartimento di Matematica, Universit\`a
di Genova, Via L. B. Alberti 4, 16132 Genova, Italy. E-Mail:
{\smc bartocci\@ matgen.ge.cnr.it}}
\address{($\star$) Dipartimento di Matematica, Universit\`a
di Genova, Via L. B. Alberti 4, 16132 Genova, Italy. E-Mail:
{\smc bruzzo\@ matgen.ge.cnr.it}}
\address{(\P) Departamento de Matem\'atica Pura y Aplicada,  Universidad
de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca,
Spain. E-Mail: {\smc sanz\@ relay.rediris.es}, subject ``for Hernandez
Ruiperez''}
\address{(\S) Department of Mathematics and
Statistics, University of Victoria, P.O\. Box 1700, Victoria, B.C\.,
             Canada V8W 2Y2. E-Mail: {\smc Vladimir.Pestov\@ vuw.ac.nz}}
\subjclass{58A50, 58C50}
\keywords{supermanifolds, axiomatics, infinite-dimensional algebras}
\abstract{We discuss an axiomatic approach to supermanifolds
valid for arbitrary ground graded-com\-mu\-ta\-tive Banach algebras $B$.
Rothstein's axiomatics is revisited and completed by a
further requirement which calls for the completeness of the rings of sections
of the  structure sheaves, and allows one to dispose of some undesirable
features of Rothstein supermanifolds. The ensuing system of axioms
determines a category of supermanifolds which coincides with graded
manifolds when $B=\R$, and with G-supermanifolds when $B$ is a
finite-dimensional exterior algebra. This category is studied in detail. The
case of holomorphic supermanifolds is also outlined.}
\thanks{\dag\ Research partly supported by the joint  CNR-CSIC
research project `Methods and applications of differential geometry in
mathematical phys\-ics,' by `Gruppo Nazionale per la Fisica Matematica' of CNR,
by the Italian Ministry for University and Research through
the research project
`Metodi geometrici in relativit\`a e teorie di campo,'  and by the
Spanish CICYT
through the research project `Geometr\'{\i}a de las teor\'{\i}as gauge.'}
\endtopmatter

\document   \doskip \heading Introduction \endheading
Supergeometry has been developed along two different
guidelines:
Berezin,  Le\u\i tes and Kostant introduced the
so-called {\sl graded manifolds} via algebro-geometric
techniques (cf\. \cite{10,17,22,23,8}), while
DeWitt and Rogers's treatment
(\cite{16,31,32}; cf\. also \cite{20,37})
relies on more intuitive local
models expressed in the language of differential geometry.
As a matter of fact, this pretended dichotomy has no {\sl
raison  d'\^etre}, for at least two motivations. First of
all, it is  our opinion that the relative formulation of
graded manifold theory \cite{25}  in some sense
includes su\-per\-mani\-folds {\sl \`a la} DeWitt-Rogers;
secondly, and more concretely, in order to provide  a
sound mathematical basis to the DeWitt-Rogers theory, one
need use sheaf theory as well \cite{33,8}, at
least  when the ground algebra is finite-dimensional.
Anyway, the precise relationship between the two models is still unclear.


In his paper \cite{33}, Rothstein   devised
a set of four axioms which any sensible category of
su\-per\-mani\-folds should verify; however,  it turns out that the
category of su\-per\-mani\-folds singled out by his
axiomatics (that we call $R$-su\-per\-mani\-folds) is
too large, in the sense
that, contrary to what is asserted in \cite{33},  it is
neither true that if the ground algebra is com\-mu\-ta\-tive
the axiomatics reduces to Berezin-Le\u\i tes-Kostant's
graded manifold theory  (see Example 3.2 of this paper), nor
that when the ground algebra is a finite-dimensional
exterior algebra, the axiomatics singles out the
category of su\-per\-mani\-folds that are extensions of Rogers's
$G^\infty$ su\-per\-mani\-folds.

The purpose  of the present work is to analyze Rothstein's
axiomatics,  discussing the interdependence among the
axioms and  singling  out the additional axiom  necessary
to characterize those Rothstein su\-per\-mani\-folds  which are
free from the aforementioned drawbacks. The new axiom
calls for the completeness of the rings of sections of the `structure sheaf'
with respect to a certain natural topology.

The ensuing  system of five axioms
can be reorganized into  four statements, defining a
category of su\-per\-mani\-folds, called $R^\infty$-su\-per\-mani\-folds,
that coincide with graded
manifolds when the ground algebra is either $\R$ or $\C$,
and provide the most natural generalization of
differentiable or  complex manifolds. When the
ground algebra is a finite-dimensional exterior algebra,
the resulting category of su\-per\-mani\-folds 
is equivalent to the
category of G-su\-per\-mani\-folds that some of the authors
have independently
introduced and discussed  elsewhere \cite{2-8,14,15}.
This means that G-su\-per\-mani\-folds (in the case of a finite-dimensional
ground algebra)
are the unique concrete model for su\-per\-mani\-folds
fulfilling the extended axiomatics, or
alternatively, that they can   be  defined through
that axiomatics, thus stressing their relevance in supergeometry.
This also means that G-su\-per\-mani\-folds are exactly those Rothstein
su\-per\-mani\-folds that extend Rogers's
$G^\infty$ su\-per\-mani\-folds in the sense of \cite{33}.

Other results that we present in this paper are the following:
any $R$-su\-per\-mani\-fold morphism is continuous as
a morphism between the rings of sections of the relevant
structure sheaves; any $R^\infty$-su\-per\-mani\-fold morphism is
also $G^\infty$;
any $R$-su\-per\-mani\-fold can be in one sense
completed to yield an $R^\infty$-su\-per\-mani\-fold.

Finally, in the last section the case of complex analytic
su\-per\-mani\-folds is discussed.

Many of the results contained in this paper have already been presented
in \cite{8} in the case of a finite-dimensional ground algebra $B$.


We briefly recall the basic definitions and
facts  we shall need. We consider  $\Bbb
Z_2$-graded  (for brevity, simply `graded') algebraic
objects; any morphism of graded objects is assumed to
be homogeneous. (For details, the reader may consult
\cite{22-24,8}).
Let $B$ denote a graded-com\-mu\-ta\-tive Banach algebra with
unit;  so $B_0$ and $B_1$ are, respectively, the even and
odd part of $B$.
With the exception of Section 6,
we consider the case of a real $B$.
The analysis of the properties of supermanifolds
is greatly simplified when $B$ is local and, moreover, satisfies
a very natural additional property that we discuss in Section 3:
that of being a Banach algebra of Grassmann origin.
Some of our results are true only under this additional assumption,
which however does not seem to be truly restrictive, in that
all examples of graded-commutative Banach algebras
that have been used as ground algebras for supermanifolds
are actually Banach algebras of Grassmann origin.

We define the $(m,n)$ dimensional
`su\-per\-space' $B^{m,n}$ as $B_0^m\times B_1^n$ with the
product topology.

By {\sl graded  ringed $B$-space} we mean a pair $(X,\A)$,
where $X$ is a topological space and $\A$ is a sheaf of
graded-com\-mu\-ta\-tive $B$-algebras on $X$. A graded ringed
space is said to be {\sl local}, as it occurs in the most
interesting examples,  if the stalks $\A_z$ are local
graded rings for any $z\in M$ (a graded
ring is said to be local if it has a unique maximal graded
ideal).
The {\sl sheaf $\sh {D}er\A$ of derivations} of $\A$ is by
definition  the completion of the presheaf of $\A$-modules
$U\mapsto \bigl\lbrace\text{graded derivations of}\
                    \A_{\vert U}\bigr\rbrace$, 
where a graded derivation of $\A_{\vert U}$ is an
endomorphism  of sheaves of graded $B$-algebras $D\colon
\A_{\vert U}\to \A_{\vert U}$  which fulfills the graded
Leibniz rule, sc\.  $D(a\cdot b)= D(a)\cdot b +
(-1)^{\vert a\vert \vert D\vert}a\cdot D(b)$.

Furthermore, $\sh D{er}^\ast\A$ denotes the dual
sheaf to $\sh D{er}\,\A$, i.e. $\sh D{er}^\ast\A
=\sh H{om}_{\A}\allowmathbreak (\sh D{er}\,\A,\,\allowmathbreak
\A)$. A  morphism of
sheaves of graded  $B$-modules $d\colon \A\to\sh
D{er}^\ast\A$ --- called the  {\sl exterior differential}
--- is defined by letting $df(D)=(-1)^{\vert
f\vert\,\vert D\vert}\,D(f)$ for all homogeneous
$f\in\A(U)$, $D\in\sh D{er}\,\A(U)$ and all open $U\subset
M$.


\doskip
\heading Rothstein's axiomatics revisited \endheading

In order to state Rothstein's axioms for su\-per\-mani\-folds,
we consider triples $(M,\A,\allowmathbreak\ev)$, where  $(M,\A)$ is
a graded ringed space over
a graded-com\-mu\-ta\-tive Banach algebra $B$, the
space $M$ is assumed to be (Hausdorff) paracompact, and
$\ev\colon \A\to\Cc_M$ is a morphism of sheaves of
graded $B$-algebras, called the `evaluation morphism;'
here $\Cc_M$ is the sheaf of continuous $B$-valued
functions on $M$. Such a triple will be called
an {\sl $R$-su\-per\-space\/}. We shall denote by a tilde the action of
 $\ev$, i.e\. $\tilde
f=\ev(f)$.  A morphism of $R$-su\-per\-spaces is a pair
$(f,f^\sharp)\colon
(M,\allowmathbreak\A,\allowmathbreak \text{\it ev}^M) \to(N,\B,\text{\it ev}^N
)$, where $f\colon M\to N$ is a continuous map
and $f^\sharp\colon\B\to f_\ast\A$ is  a morphism of sheaves
of graded $B$-algebras,
such that $\text{\it ev}^M\circ f^\sharp=f^\ast\circ\text{\it ev}^N$.

After fixing a pair  $(m,n)$ of nonnegative integers, one
says that an $R$-su\-per\-space $(M,\A,\ev)$ is an $(m,n)$
dimensional $R$-su\-per\-mani\-fold if and only if
the following four axioms are
satisfied.

\noprocnumber\proclaim{Axiom 1} $\sh D{er}^\ast\A$ is a
locally free $\A$-module of rank $(m,n)$. Any $z\in M$ has
an open  neighbourhood $U$ with sections $x^1,\dots,
x^m\in\A(U)_0$, $y^1,\dots, y^n\in\A(U)_1$ such that
$\{dx^1,\dots, dx^m,dy^1,\dots, dy^n\}$ is a graded basis
of $\sh D{er}^\ast\A(U)$. \endproclaim
\noindent The
collection $(U,\gcoor xm,yn)$ is called a \sl coordinate
chart \rm for the su\-per\-mani\-fold. This axiom implies
evidently that $\sh D{er}\,\A$ is locally free of rank
$(m,n)$, and is locally generated by the derivations $\pd
,{x^i}$, $\pd ,{y^\alpha}$ defined by duality with the
$dx^i$'s and $dy^\alpha$'s.


\noprocnumber\proclaim{Axiom 2} If $(U,\gcoor xm,yn)$ is a
coordinate chart, the mapping $$\eqalign { \psi\colon
U&\to B^{m,n}\cr z&\mapsto  (\tilde x^1(z) ,\dots, \tilde
x^m(z),\tilde y^1(z),\dots, \tilde y^n(z))\cr}$$ is a
homeomorphism onto an open subset in  $B^{m,n}$. \endproclaim

\noprocnumber\proclaim{Axiom 3} {\rm (Existence of Taylor
expansion)} Let $(U,\gcoor xm,yn)$ be a coordinate chart.
For any $z\in U$ and any germ $f\in\A_z$ there are germs
$g_1,\dots, g_m,h_1,\dots, \allowmathbreak  h_n\in\A_z$
such that $$f=\tilde f(z)+\sum_{i=1}^mg_i\,(x^i-\tilde
x^i(z)) +\sum_{\alpha=1}^nh_\alpha\,(y^\alpha-\tilde
y^\alpha(z))\,. $$ \endproclaim \noprocnumber\proclaim{Axiom 4}
Let $\D(\A)$ denote the
sheaf of differential operators over $\A$,
i.e\., the graded $\A$-module generated
multiplicatively by $\sh D{er}\,\A$ over $\A$,  and let
$f\in\A_z$, with  $z\in M$. If $\widetilde{L(f)}=0$ for
all $L\in\D(\A)_z$, then $f=0$.\endproclaim
The sections of $\A$ will be called {\sl superfunctions.}
Morphisms of $R$-su\-per\-mani\-folds are just $R$-su\-per\-space morphisms.

It is convenient to restate this axiomatics in a
slight different manner, more suitable for dealing with
the topological completeness  of the rings of
sections of $\A$. Let us consider, as before, an $R$-su\-per\-space
$(M,\A,\ev)$.
For any $z\in M$ define a  graded
ideal $\frak{L}_z$ of $\A_z$  by letting
$$\frak{L}_z=\{f\in\A_z\bigm\vert\tilde f(z)=0\}.$$  Axiom
3 can  be obviously reformulated as follows:

{\sl Let $(U,\gcoor xm,yn)$ be a coordinate chart. For any
$z\in U$ the ideal $\frak{L}_z$  is generated by  $\lbrace
x^1-\tilde x^1(z),\dots,x^m-\tilde x^m(z),y^1-\tilde
y^1(z),\dots,y^n-\tilde y^n(z) \rbrace$.}

Axiom 1 allows one to replace this axiom by a weaker
requirement; to this  aim we need some preliminary
discussion.
\proclaim{Lemma} There is an isomorphism of
$\A_z/\frak{L}_z$-modules
$$\eqalign{
\frak{L}_z/\frak{L}_z^2&\to\sh
D{er}^\ast\A_z\otimes_{\A_z}\A_z/\frak{L}_z\cr \bar
f&\mapsto df\otimes 1 \cr} $$
where a bar
denotes the class in the quotient. \endproclaim
\proof It can be easily shown that $df\otimes\bar g\mapsto
\overline{(f-\tilde f(z)) g}$ defines a morphism $\sh
D{er}^\ast\A_z\otimes_{\A_z}\A_z/\frak{L}_z\to\frak{L}_z/\frak{L}_z^2$
which inverts  the previous one. \qed \enddemo


If we denote by $d_zf$ the class  of the element $f-\tilde
f(z)\in\frak{L}_z$ in
$\frak{L}_z/\frak{L}_z^2$, then
Axiom 1 for $(M,\A,\ev)$ implies
that --- given a coordinate chart $(U,\gcoor xm,yn)$ ---
the elements $\lbrace d_zx^i$, $d_zy^\alpha\rbrace$ are a
basis for the  $\A_z/\frak{L}_z$-module
$\frak{L}_z/\frak{L}_z^2$.

Let us suppose until the end of this Section that
$(M,\A)$ is a {\sl graded locally ringed space\/}. Since
in that case any graded  ideal of $\A_z$ is contained in
its radical, one can apply a graded version of Nakayama's
lemma (cf\.  \cite{8}). Thus we obtain
\proclaim{Lemma} Assume that  $\frak{L}_z$ is  finitely
generated. Then the elements  $\lbrace x^i-\tilde x^i(z)$,
$y^\alpha-\tilde y^\alpha(z)\rbrace$ are generators for
$\frak{L}_z$ if and only if their classes  $\lbrace
d_zx^i$, $d_zy^\alpha\rbrace$ generate the
$\A_z/\frak{L}_z$-module $\frak{L}_z/\frak{L}_z^2$.
\qed\endproclaim Thus, we have proved the following result.
\proclaim{Proposition}  If the graded rings $\A_z$ are local, and
$\frak{L}_z$ is  finitely generated, then Axiom 1 implies
Axiom 3.\qed\endproclaim

We are therefore led to consider the apparently weaker axiom
\noprocnumber\proclaim{Axiom $\hbox{{\sl 3\/}}'$} For every  $z\in M$ the
ideal $\frak{L}_z$ is finitely generated. \endproclaim
It is an important fact that Axiom $\hbox{{\sl 3\/}}'$ does not depend on
the choice of a coordinate chart. So, while in order to
check Axiom 3 one has to prove the existence of a Taylor
expansion for any  coordinate chart,  if $(M,\A)$ is a
graded locally ringed space it is sufficient to show that
there is  one coordinate chart for which a Taylor expansion does
exist.

We can summarize this discussion  as follows.
\proclaim{Proposition} If an $R$-su\-per\-mani\-fold is
also a graded locally ringed space, we can replace Axiom 3
by Axiom 3\/$'$.  \qed\endproclaim

\rem{Example} Here we show that Rothstein's Axiom 3 is independent of
Axioms 1, 2, and 4.
Let  $B = B_0 = \Bbb R$, $M = B^{1,0} = \Bbb R$.
Let us fix a continuous
function $\phi\: \Bbb R \to \Bbb R$ such that
for every open and non-empty $U\subset\Bbb R$  the restriction
$\rest{\phi},U$ is neither constant nor one-to-one; an
example of such a function is Weierstrass' nowhere differentiable
continuous function \cite{34}.
We denote $\Cal F=\phi^{-1}\Cal C_{\Bbb R}$ and by
$i_{\Cal F}:\Cal F \hookrightarrow \Cal C_{\Bbb R}$ the canonical injection.
Let $i_{\Cal P}$ be the embedding of the sheaf $\Cal P$ of
germs of real polynomial functions on $\Bbb R$ into
$\Cal C_{\Bbb R}$, and
let $ \Cal A = \Cal F \otimes_{\Bbb R} \Cal P$.
Implicit function arguments enable one to show that
the morphism
$\ev:= i_{\Cal F}\otimes i_{\Cal P}\:\Cal A \to \Cal C_{\Bbb R}$
is injective;
thus, the $R$-su\-per\-space
$(\Bbb R, \Cal A,\ev)$ satisfies Axiom 4.
Let ${\frak K}_x=\frak L_x\cap(\F_x\otimes 1)$;
since for each $x\in\Bbb R$ one has $\frak K_x^2 = \frak K_x$, then
for any  open and non-empty
$U\subset\Bbb R$,
each derivation of the algebra $\Cal A(U)$
is trivial on $\Cal F_U$.
Thus, the sheaves of derivations $\sh Der\,\Cal A$ and $\sh Der\,\Cal P$
are canonically isomorphic,
there is a global coordinate
system $\{x\}$ on $M$, and
Axioms 1 and 2 are satisfied.
Now, let us suppose that $\phi$ admits a decomposition
as in Axiom 3; then
$\phi$ is $C^1$,
and since it is not constant, there are points of
local injectivity for $\phi$, contrary to the assumed properties of $\phi$.
Thus, Axiom 3 is violated.
\endrem

We conclude this Section by noticing that
morphisms of $R$-supermanifolds can behave in a rather unsatisfactory way, as
the following Example shows.
\rem{Example} Consider the $R$-supermanifolds $(M,\Pc,\Id)$ and $(M,\Cc,\Id)$,
where $M=\R$, $\Pc$ is the sheaf of polynomials on $\R$, and $\Cc$ is the sheaf
of smooth functions on $\R$. The only $R$-supermanifold morphisms
$(f,f^\sharp)\colon(M,\Pc)\to(M,\Cc)$ are given by {\sl constant\/} maps
$f\colon\R\to\R$ with $f^\sharp=f^\ast$, as one can  check directly.\endrem
\par
\doskip
\heading $G^\infty$ su\-per\-mani\-folds and $Z$-expansion \endheading
We wish now to introduce the notion of $G^\infty$ function
\cite{27,13,36,16,20}. Let $U\subset B^{m,0}$ be an open set;
a $C^\infty$ map $f\colon U\to B$ is said to be $G^\infty$
if its Fr\'echet differential is $B_0$-linear; the resulting sheaf
of functions on $B^{m,0}$ will be denoted by $\hat\G^\infty$.
A $G^\infty$ function $f(x,y)$ on $B^{m,n}$ is a smooth map
that can be written in the form $f(x,y)=\sum_{\mu\in \Xi_n}
f_\mu(x)\,y^\mu$ for some (in general not uniquely defined) $G^\infty$
functions $f_\mu(x)$. Here
$\Xi_n$ is  the set of  sequences
$\mu=\{\mu(1),\dots,  \mu(r)\}$ of integers  such that
$1\leq \mu(1)<\dots<\mu(r)\leq n$,  including the empty
sequence $\mu_0$, and we let  $y^\mu=y^{\mu(1)}\cdot
\dots\cdot y^{\mu(r)}$. The sheaf of $G^\infty$ functions
on $B^{m,n}$ will be denoted by $\G^\infty$.

\proclaim{Definition} An $(m,n)$ dimensional $G^\infty$
su\-per\-mani\-fold is a graded ringed space
$(M,\A^\infty)$ locally isomorphic with $(B^{m,n},\G^\infty)$, with
$M$ (Hausdorff) paracompact.\endproclaim

One should notice that, generally speaking, a $G^\infty$ su\-per\-mani\-fold
is not an $R$-su\-per\-mani\-fold \cite{13,33,8}, in that
Axiom 1 may be violated.

It is natural to ask whether, given an $R$-su\-per\-mani\-fold
$(M,\A,\ev)$, the pair $(M,\A^\infty)$, where $\A^\infty=\Im\ev$,
is a $G^\infty$ su\-per\-mani\-fold;
contrary to what asserted
in \cite{33}, this question in general has a negative answer.
Indeed,  the sheaf $\A$ may
not be topologically complete with  respect to the even
coordinates; the following Example should clarify what we
mean.

\rem{Example} Let us take $B=\R$, $n=0$ and $M=\R^m$. If
we consider the sheaf $\A=\R[x^1,\dots,x^m]$ of polynomial
functions on $\R^m$ and the trivial evaluation morphism
$\ev\colon\A\hookrightarrow\Cc_\R$, $\ev(f)=f$, then
$(M,\A,\ev)$ is an $R$-su\-per\-mani\-fold of dimension
$(m,0)$. But  $(M,\ev(\A))=(M,\R[x^1,\dots,x^m])$ is
certainly not an $(m,0)$-dimensional
$G^\infty$ su\-per\-mani\-fold, which in this case would
be an $m$-dimensional smooth manifold. \endrem


Thus, there are $R$-su\-per\-mani\-folds which do not
satisfy Rothstein's {\sl structural definition\/} of
su\-per\-mani\-folds \cite{33}. In order to
characterize those  $R$-su\-per\-mani\-folds which
fulfill that definition,
a further axiom must be imposed. This will be discussed in next Section.

In the rest of this Section we discuss a method that,
to a large extent, enables one
to reduce the study of $G^\infty$ functions to that of
$B$-valued functions on Euclidean space, namely, the
so-called {\sl $Z$-expansion}
\cite{7,8,18,20,26,31,32}. We show that the $Z$-expansion is applicable
to a larger class of graded-commutative
Banach algebras than it was known earlier.

\proclaim{Theorem}
Let $B$ be a graded-commutative Banach algebra. The following
conditions are equivalent:
\roster\item $B$ is local and
the linear span of products of odd elements
is  dense in the radical $\Rad B$ of $B$;
\item Any closed unital subalgebra of $B$ containing $B_1$
coincides with $B$;
\item The reflection of $B$ in the category of purely even Banach algebras
is $\Bbb R$. (In other terms,
for any graded Banach algebra morphism $h$ from $B$ to
a purely even Banach algebra, the image $h(B)$ is isomorphic to
$\Bbb R$.)
\item For an appropriate cardinal number $\eta$, there
exists a submultiplicative
seminorm $p$ on a Grassmann algebra $B_{\eta}$ with $\eta$
anticommuting generators such that $B$ is isomorphic to the
Banach algebra associated with $(B_\eta , p)$.
(That is, $B$ is isomorphic to the completion of the quotient normed algebra
of $B_\eta$ by the ideal $\{x\in B_\eta \vert p(x)=0\}$.)
\endroster\endproclaim
\proof
(1) $\Leftrightarrow$ (2): obvious.
(2) $\Leftrightarrow$ (3): it follows from the fact that any graded Banach
algebra morphism $h$ from a graded Banach algebra
$B$ to any purely even Banach algebra can be
factored through the quotient algebra of $B$
by the closed ideal generated by the odd part $B_1$; now,
the quotient algebra is $\R$ if and only if (2) is true.
(3) $\Rightarrow$ (4): let $\eta$ be the cardinality of $B_1$. Denote
by $\pi$ the graded algebra morphism from $B_\eta$ to $B$ such that
the image under $\pi$ of the set of generators coincides with $B_1$,
and for all $x\in B_\eta$ set $p(x)=\Vert \pi (x) \Vert_B$.
(4) $\Rightarrow$ (3): Let $\pi\: B_\eta \to B$ be the projection,
and let $h$ be any morphism from $B$ to a purely even Banach
algebra. Then the composite morphism $h\circ \pi$ is a graded algebra
morphism from a Grassmann algebra $B_\eta$ to an even algebra;
clearly, the image of $h\circ \pi$ is $\Bbb R$, and at the same time
it is dense in the image of $h$.
\qed\enddemo

 Jadczyk and Pilch were the first to consider the above property
(in their paper \cite{20}
this feature, in the form (1), was one of the two conditions determining the
class of Banach-Grassmann algebras). One of the authors of the present paper
has studied the algebras satisfying this property
 under the name of `supernumber
algebras'\cite{26,27,29,30}. 
Here we propose to call the graded-commutative Banach
algebras $B$ satisfying one of the equivalent conditions (1)-(4)
{\sl Banach algebras of Grassmann origin} because of (4);
we shall shorten this into `BGO-algebras.'
Seemingly, these algebras form the most important class of local
graded-commutative Banach algebras; as a matter of fact,
all ground algebras for supermanifolds that have been so far introduced
are BGO-algebras.
So are indeed the finite-dimensional Grassmann algebras
(in this paper we denote them by $B_L$, $L$ being the number of generators)
and Rogers's infinite-dimensional $B_\infty$ algebra
\cite{31} (that in particular is a Banach-Grassmann algebra).
A large number of new examples of Banach-Grassmann algebras
is described in \cite{29,30}.
The so-called Grassmann-Banach algebras \cite{18} are also
BGO-algebras. Moreover, any algebra
of superholomorphic functions on a purely
even graded Banach space \cite{35} can be made into a
BGO-algebra.

Let $B$ be a local Banach algebra.
We will denote by $\sigma_B$ or simply
$\sigma$ the augmentation morphism (that is, the unique
character) $\sigma\: B\to \Bbb R$, and by $s\:B\to \Rad  B$ the
complementary mapping, $s+\sigma=Id_B$.
The mappings $\sigma^{m,n}$ (body map)
and $s^{m,n}$ (soul map) from $B^{m,n}$ to
$\Bbb R^m$ and $(\Rad  B)^{m,n}$, respectively, are defined as direct
sums of copies of the former two mappings.
For a subset $X\subset \R^m$, we denote
$X\,\widetilde{\,} = (\sigma^{m,0})^{-1}(X)$, and call {\sl DeWitt open} sets 
the open subsets
of $B^{m,n}$ of the form $U\,\widetilde{\,},~U\subset \Bbb R^m$
\cite{16,8}.

For any $U\subset \R^m$, the $Z$-expansion is
the morphism of graded
algebras
$$Z \colon \Cal F\to
\Cc^\infty ((\sigma^{m,0})^{-1}(U)),$$
(where $\Cal F$ is a  dense subalgebra of
the graded algebra $\Cc^\infty (U)$ of
$B$-valued $C^\infty$
functions on $U$)
defined by the formula
$$ Z(h)(x)=
\sum_{j=0}^\infty {1\over j !}
D^{(j)}h_{\sigma^{m,0}(x)}(s^{m,0}(x)) $$
 for $h\in \Cc^\infty_{L'}(U)$ and all
$x\in U\,\widetilde{\,};$
here the $j$-th Fr\'echet differential $D^{(j)}h_{\sigma^{m,0}(x)}$ of $h$
at the point  $\sigma^{m,0}(x)$ acts on $B^{m,0}\times \dots\times
B^{m,0}$ ($j$ times) simply by extending by $B_0$-linearity
its action on $\R^m\times\dots \times\R^m$.
When $B$ is finite-dimensional one can take
$\Cal F = \Cc^\infty (U)$.
The $Z$-expansion can be written in another form by using partial
derivatives:
$$ Z(h)(x)=
\sum_{\vert J\vert=0}^\infty {1\over  J !}\left(
\frac{\partial^{\vert J\vert} h}
{\partial x^{ J}}\right)_{\sigma^{m,0}(x)}(s^{m,0}(x))^ J\,, $$
where $ J$ is a multiindex.
 
The proof of the following result is the same as in
\cite{20} where $B$ is a Banach-Grassmann algebra; actually, in that
proof only the property of being a BGO-algebra is used.

\proclaim{Theorem}
Let $B$ be a BGO-algebra, let $m$ be a positive
integer and $V$ be an open subset of $B^{m,0}$. An arbitrary
$G^\infty$ function $f$ on $V$ admits a unique extension to a
$G^\infty$ function over the DeWitt open set
$(\sigma^{m,0}(V))\,\widetilde{\,}$.
\qed\endproclaim
We  study now the convergence of the $Z$-expansion.
\proclaim{Theorem}
Let $B$ be a Banach algebra of Grassmann origin, let $m$ be a positive
integer and $U$ be an open subset of $\R^m$. For an arbitrary
$G^\infty$ function $f$ on $U\,\widetilde{\,}$, the $Z$-expansion of the
restriction $f_\vert$ of $f$ to $U$ converges to $f$.
The convergence is uniform on compacta lying in any
`soul fibre' $\{x\}\,\widetilde{\,}$, with $x\in U$.
For any $i=1, \dots , m$ the following holds:
$\partial f/\partial x^i = Z(\partial f_\vert/\partial x^i)$.
\endproclaim
\proof
Denote by $\hat B$ a unital subalgebra of $B$ generated by
the odd part $B_1$; $\hat B$ is local and dense in $B$.
Taylor formula for a $G^\infty$ function $f$
\cite{36} shows that the $Z$-expansion of $f_\vert$ 
converges to $f(z)$ at any $z \in \hat B^{m,0}$ 
since the remainder of the series
vanishes for $\vert J\vert$ large enough.
Now, fix $x\in U$; since the $Z$-expansion converges pointwise on the
`nilpotent fibre' $\{x\}+(\Rad\hat B)^{m,0}$ over $x$
to a continuous function,
and the terms of the $Z$-expansion restricted to this space are
polynomials on a normed linear space,
the convergence
is {\sl normal at zero}  \cite{12}.
This means that for some neighbourhood $U$ of zero in
$\{x\}+(\Rad\hat B)^{m,0}$ the convergence of the $Z$-expansion to $f$
is uniform on $U$.
As a consequence, the $Z$-expansion converges uniformly
to $f$ on the closure of $U$ in the `quasinilpotent fibre'
$\{x\}\,\widetilde{\,}$, that is, it converges to $f$ normally at
zero in the Banach space $\{x\}\,\widetilde{\,}$ as well.
(Remark that $\{x\}+(\Rad\hat B)^{m,0}$ is dense in
$\{x\}\,\widetilde{\,}=\{x\}+(\Rad B)^{m,0}$.)
Due to quasinilpotency of $\Rad B$,
this implies the pointwise convergence of the $Z$-expansion
to $f$ on any fibre $\{x\}\,\widetilde{\,}$ and thus on the
whole of $U\,\widetilde{\,}$. The first statement is thus proved.
On the other hand, this implies that for any $x\in U$ the restriction
$f\vert\{x\}\,\widetilde{\,}$ is an entire function \cite{12},
hence is analytic ({\sl ibid.}, Prop. 8.2.3.)
Since the Taylor series of an analytic function converges to it
uniformly on compacta lying in the interior of the domain
of convergence, the second statement follows as well.
Finally, the claim regarding partial
derivatives follows from the fact that restriction 
of $\partial f/\partial x^i$
to $U$ coincides with $\partial f_\vert/\partial x^i$.
\qed\enddemo

A Banach space-valued function $f$ on an open subset $U$ of $\Bbb R^m$ 
is said to be {\sl Pringsheim
regular} if its Taylor series converges in a
neighbourhood of every point $x\in U$ (not necessarily to $f$
itself). One can show \cite{28} that
for $B=B_\infty$ all $G^\infty$ functions are
obtained by $Z$-expansion of Pringsheim regular
functions, and that whenever a $C^\infty$ function has a convergent
$Z$-expansion then its sum  is a $G^\infty$
function. Thus, for $B=B_\infty$
the algebra $\Cal F$ is formed by all Pringsheim regular
$B$-valued mappings.




\doskip
\heading $R^\infty$-su\-per\-mani\-folds \endheading

Let  $(M,\A,\ev)$ be an $R$-su\-per\-space
over a graded-commu\-ta\-ti\-ve Banach
algebra $B$, that in this and the next Sections
is assumed to be {\sl real}, and let $\norm{\ }$ denote the norm in $B$;
the rings of sections $\A (U)$ of $\A$ on every open
subset $U\subset M$ can be  topologized by means of the
seminorms $p_{L,K}\colon\A (U)\to\R$ defined by
$$
p_{L,K}(f)=\max_{z\in K}\norm{\widetilde{L(f)}(z)}\,,\tag
$$
where $L$ runs over the differential operators of
$\A$ on $U$, and  $K\subset U$  is  compact (cf\. \cite{17,22}).
The resulting
topology in $\A(U)$, that we call the $R^\infty$ topology,
endows it with a structure
of locally convex graded $B$-algebra (possibly non-Hausdorff).
In the case where $(M,\A,\ev)$ is an $R$-su\-per\-mani\-fold, one obtains
as a consequence of the axioms that the $R^\infty$-topology
is alternatively defined by the family of seminorms
$$p_K^I(f)= \max\Sb z\in K\\ \gr  J \leq I ,\,\mu\in
\Xi_n\endSb\;
\norm{\shave{\ev\left(\shave{\left(\pd{},{x}\right
)^ J\left (\pd{},{y}\right )_\mu f}\right) (z)\;}}\quad,
$$
where $K$ runs over the compact subsets of a coordinate
neighbourhood $W$  with coordinates $\gcoor xm,yn$ (as a matter
of fact, in this case Axiom 4 means that $\A(U)$ is Hausdorff).

The $R^\infty$-topology on an algebra $\A(U)$ of superfunctions
can also be described as the coarsest topology with the properties:

(i) the evaluation map $\ev_U$ from $\A(U)$ to the space $\Cc_M(U)$ of all
continuous $B$-valued functions on $U$ endowed with the topology of
compact convergence is continuous;

(ii) all the differential operators $L\in\sh Der \A(U)$ are continuous.


\proclaim{Theorem} Let  $(f,f^\sharp)$ be
an $R$-su\-per\-space morphism between
two $R$-su\-per\-ma\-ni\-folds
$(M,\allowmathbreak\A,\allowmathbreak \ev^M)$ and $(N,\B,\ev^N)$.
Then $f^\sharp_V\colon\B(V)\to\allowmathbreak\A(f^{-1}(V))$
is continuous for
every open subset $V\subset N$. \endproclaim
\proof
It suffices to verify the property for the case where $V$
is a coordinate neighbourhood. Fix a coordinate system $\varphi =
\gcoor xm,yn$ on $V$. Let $L$ be an arbitrary
differential operator over $f^{-1}(V)$ of order
$k\geq 0$ and let $K$ be a compact subset of $f^{-1}(V)$.
For  multi-indices $ J\in\N^m$ and $\mu\in\Xi_n$
such that the total length does not exceed $k$, i.e\.,
$\vert  J\vert+\vert\mu\vert\leq k$, we let
$C_{ J,\mu}:= \max_{z\in K}\Vert \ev^M\bigl({L(f^\#(x^ Jy^\mu))}\bigr)(z)
\Vert_B$;
because of the continuity of the map
$x\mapsto \Vert \ev^M\bigl({L(f^\#(x^ Jy^\mu))}\bigr)(p)\Vert_B$, the
nonnegative real numbers $C_{ J,\mu}$ are well defined.
We will now prove that if a superfunction $g\in\B(V)$ is such that for
every $ J$ and $\mu$ with $\vert  J\vert+\vert\mu\vert\leq k$ one has
$$\max_{z\in K}\Vert C_{ J,\mu}\ev^N\bigl({\partial^{ J,\mu} g}\bigr)(f(z))
\Vert \leq 1,$$
where
$$\partial^{ J,\mu}=\frac{\partial^{ J_1}}{\partial(x^1)^{ J_1}} \dots
\frac{\partial^{ J_m}}{\partial(x^1)^{ J_1m}}\dots
\frac{\partial}{\partial y^{\mu_1}} \dots \frac{\partial}{\partial y^{\mu_n}}$$
with $ J=( J_1,\dots, J_m)$ and $\mu=\mu_1,\dots,\mu_n$;
then for all $z\in K$ one also has
$$\max_{z\in K}\Vert \ev^M\bigl({L(f^\#(g))}\bigr)(z)\Vert \leq \exp(m+n),$$
which observation will obviously complete the proof.
To prove this,
let $z\in K$ be fixed.
By repeated application of Axiom 3 we represent $g$ in a small neighbourhood
of $f(z)$ as follows:
$$\split g=&\sum_{\vert  J\vert+\vert\mu\vert < k}
\frac{1}{ J!}\ev^N\bigl({\partial^{ J,\mu}g}\bigr)(f(z))\,
(x-f(z))^ J(y-f(z))^\mu \\ &
+ \sum_{\vert  J\vert+\vert\mu\vert = k}\nu_{ J,\mu}\,
(x-f(z))^ J(y-f(z))^\mu ,\endsplit$$
where the $\nu_{ J,\mu}$'s
are some superfunctions whose evaluations vanish at
the point $f(z)$; one can verify that
$\ev^M\bigl({f^\#\nu_{ J,\mu}}\bigr)(z) = \ev^N\bigl({\nu_{ J,\mu}}\bigr)(f(z)) =
(1/ J !)\ev^N\bigl({\partial^{ J,\mu} g}\bigr)(f(z))$.
Thus, for any $g\in\B(V)$ with the above properties
the following holds:
$$\split \Vert& \ev^M\bigl({L(f^\#(g))}\bigr)(z)\Vert_B \\ & =
\biggl\Vert \sum_{\vert  J\vert+\vert\mu\vert < k}\frac{1}{ J!}
\ev^N\bigl({\partial^{ J,\mu}g}\bigr)(f(z))
 \ev^M\bigl({L(f^\#((x-f(z))^ J(y-f(z))^\mu))}\bigr)(z) \\
&\hbox to1.5cm{\hfill} + \sum_{\vert  J\vert+\vert\mu\vert = k}
\ev^M\bigl({L(f^\#(\nu_{ J,\mu} f^\#((x-f(z))^ J(y-f(z))^\mu)))}\bigr)(z)\biggr
\Vert_B \\
&\leq
\sum_{\vert  J\vert+\vert\mu\vert < k}\frac{1}{ J!}C_{ J,\mu}  \\
&\hbox to1.5cm{\hfill} +  \sum_{\vert  J\vert+\vert\mu\vert = k}
        \Vert \ev^M\bigl({f^\#(\nu_{ J,\mu}}\bigr)(z)
\ev^M\bigl({L(f^\#((x-f(z))^ J(y-f(z))^\mu))}\bigr)(x)\Vert_B
 \\ & \leq
\sum_{\vert  J\vert+\vert\mu\vert\leq k}\frac{1}{ J!} < \exp(m+n)\,.
\endsplit $$
\qed\enddemo
Let $(M,\A,\ev)$ be an $(m,n)$ dimensional
$R$-su\-per\-mani\-fold, and let $(U,\varphi)$ be a
coordinate chart on it with  $\varphi=\gcoor xm,yn$.
Define $\hat\A_\varphi$ as the subsheaf of $\rest\A,U$
whose sections `do not depend on the odd variables,' in
the sense that $$ \hat\A_\varphi(V)=\bigl\{f\in A(V)
\bigm\vert {\partial f\over\partial y^\alpha}=0,
\quad\alpha=1,\dots,n\bigr\}\,, $$ for every open subset
$V\subset U$. We have the following canonical isomorphism
(cf\. \cite{33}): $$
\hat\A_\varphi\otimes_\R\medwedge_\R\R^n\to\rest\A,U\tag
$$ having identified $\hbox{$\bigwedge_\R$}\R^n$ with the
Grassmann algebra generated by the $y$'s. Moreover, the
restriction of $\ev$ to $\hat\A_\varphi$ is injective.

\proclaim{Lemma} The isomorphism (4.2),
$\A(V)\iso\hat\A_\varphi(V)\otimes_\R\medwedge_\R\R^n$,
is a topological isomorphism for every open subset $V\subset U$.
\endproclaim \proof
Since the tensor product on the right hand side can be 
identified with the topological linear space
$\hat\A_\varphi(V)^{2^n}$ with the usual product topology,  in order
to check that the algebraic isomorphism is also a homeomorphism,
it remains to verify that all the projection maps
(under the above identification)
$\A(V)\to \hat\A_\varphi(V)$ which are labelled by multiindices
$\mu$ and given by
$y^\mu f(x)\mapsto f(x)$ are continuous. But this follows from the
very definition of the $R^\infty$-topology because the projection maps
are represented as compositions of evaluation maps with
differential operators.
\qed\enddemo

We wish now to investigate the question of the topological completeness
of the rings of sections of the structure sheaf of an $R$-su\-per\-mani\-fold.
The discussion of the previous Section leads us to introduce  the
following supplementary axiom.
\noprocnumber\proclaim{Axiom
5} {\rm (Completeness)} For every open subset $U\subset M$,
the topological algebra $\A(U)$ is complete.
\endproclaim Axioms 4 and 5, taken together, are
equivalent to still another axiom:
\noprocnumber\proclaim{Axiom 6} For every open subset
$U\subset M$, the topological algebra $\A(U)$ is complete Hausdorff.
\endproclaim

Thus, it turns out that in order to determine a class of
su\-per\-mani\-folds whose rings of sections are topologically
complete, it is enough to replace Axiom 4 by Axiom 6. We
therefore consider the following axiomatic
characterization of su\-per\-mani\-folds.

\proclaim{Definition} An $R^\infty$-su\-per\-mani\-fold
over $B$ is an $R$-su\-per\-mani\-fold $(M,\A,\ev)$
over $B$ satisfying additionally Axiom 5;  or,
equivalently, it is an $R$-su\-per\-space fulfilling
Axioms 1, 2, 3, and 6. \endproclaim

We have shown in the previous Section that
Axiom 3 can be replaced by the simpler Axiom $3'$
provided that $(M,\A)$ is a graded locally ringed space.
As a matter of fact, in the case of $R^\infty$-supermanifolds
a simpler assumption, that of locality of the ground algebra
$B$, can be made.

\proclaim{Theorem}
Let $B$ be a local graded-com\-mu\-ta\-tive Banach algebra.
An $R$-su\-per\-space $(M,\A,\ev)$ over $B$ satisfying  axioms
1, 2, $\hbox{{\sl 3\/}}'$ and 6 is an $R^\infty$-supermanifold.
\endproclaim
\proof One needs to show that $(M,\A)$
is a graded {\sl locally\/} ringed space.
Let $p\in M$; we shall prove that the
ideal $$\Cal  J_{p} := \{g\in\Cal \A_p : \tilde g(p)\in \Rad  B\}$$
is the only maximal ideal
in $\A_p$; it suffices to show that
any $g\notin\Cal  J_p$ has a multiplicative inverse.
Pick a representative  $g'\in\A(U)$ of $g$, where $U$
is a suitable coordinate neighbourhood of $p$. Since the map
$q\mapsto \tilde{g}'(q)$ from $U$ to $B$ is continuous,
and since
the invertible elements of a Banach algebra $B$ are exactly those not
belonging to the radical $\Rad  B$, then one can assume
that $\tilde{g}'(p) = 1$ and that for all
$q\in U$ one has
$\Vert \tilde{g}'(q) - 1 \Vert_B < 1$
(in particular, $\tilde{g}'(q)$ is invertible in $B$).
We shall show  that the series $\sum_{j=0}^\infty h^j$,
where $h=1-g'$,
converges in the $R^\infty$-topology on the
algebra $\A(U)$; the germ of the sum of this  series will be
a multiplicative inverse to $g$.

Let $K\subset U$ be compact and $L$ be a differential operator
of order $k$ on $U$. It suffices to prove that
the  series with nonnegative real terms
$\sum_{j=0}^\infty \max_{q\in K} \Vert L(h^j) \Vert_B$
converges.

By using local coordinates one can write
$$L=
\sum_{i_1+\dots+i_{m+n}=k}
L^{i_1}_1\dots L^{i_{m+n}}_{m+n}\,;$$
here $(m,n)=\dim (M,\A,\ev)$,
and each $L_{i}$ is a first-order differential operator.
Then one has
$$L(h^j)=\sum_{r=1}^k\
\sum_{\vert  J_1\vert +\dots+\vert  J_r\vert =k}P_{ J_1,\dots, J_r}(j)\,h^{j-r}
L^{ J_1}(h)\dots L^{ J_r}(h)\,;$$
here the $ J$'s are multiindices,
the number of summands depends on $k$ (and hence on $L$) only,
and $P_{ J_1,\dots, J_r} (j)$  are integer polynomials in $j$
(and in $k$, but $k$ is fixed) of combinatorial origin.
Notation is such that $L^ J=L_1^{j_1}\dots L_N^{j_N}$
if $ J=(j_1,\dots,j_N)$.
Let
$$C_{ J_1,\dots, J_r}=\max_{q\in K}\Vert
\widetilde{L^{ J_1}(h)}(q)\dots \widetilde{L^{ J_r}(h)}(q)
\Vert_B\,.$$
Since
$\max_{q\in K}\Vert \tilde h(q) \Vert_B = t < 1$, one has:
$$\eqalign{
\sum_{j=0}^\infty \max_{q\in K} \Vert L(h^j) \Vert_B
& \leq \sum_{j=0}^\infty\ \sum_{r=1}^k\
\sum_{\vert  J_1\vert +\dots+\vert  J_r\vert =k}
P_{{ J_1,\dots, J_r}} (j)\, C_{{ J_1,\dots, J_r}} \,t^{j-r}\cr
& = \sum_{\vert  J_1\vert +\dots+\vert  J_r\vert =k}
\ \sum_{r=1}^k
C_{{ J_1,\dots, J_r}} \sum_{j=0}^\infty P_{{ J_1,\dots, J_r}} (j)\, t^{j-r}\,,
\cr}$$
the last series being convergent.  \qed\enddemo

We wish now to check that $R^\infty$-supermanifolds
can be defined by means of a local condition. This implies that
Rothstein's structural definition \cite{33} singles
out the category of $R^\infty$ supermanifolds, rather
than the wider category of $R$-supermanifolds.
In other terms, $R^\infty$ supermanifolds coincide with
Rothstein's $C^\infty(B)$-manifolds.
Another
consequence is that only for $R^\infty$ supermanifolds
it is true that the pair $(M,\ev(\A))$ is a $G^\infty$
supermanifold in the sense of Rogers.

During the proof we shall need to assume that $B$ is a BGO-algebra.
We start by stating  the completeness axiom in an alternative
way. The following result is proved straightforwardly.

\proclaim{Proposition} An $R$-su\-per\-mani\-fold
is an $R^\infty$-su\-per\-mani\-fold if and only if every point is
contained in a coordinate chart $(U,\varphi)$ such that
the rings $\hat\A_\varphi(V)$  are complete in the $R^\infty$-topology.
\qed\endproclaim

We define the {\sl standard  $R^\infty$-su\-per\-mani\-fold\/} over
$B^{m,n}$ as the graded ringed space $(B^{m,n},\G)$, where
$\G=p^{-1}\hat\G^{\infty}\otimes_\R\bigwedge\R^n$; here $p$ is the
projection $B^{m,n}\to B^{m,0}$.
The evaluation morphism is given by $\ev(f\otimes a)=fa$.
One proves that $(B^{m,n},\G,\ev)$ is an $R^\infty$ supermanifold;
the only nontrivial thing to be checked (when
$B$ is infinite-dimensional) is the following.

\proclaim{Lemma} The algebra $\G(U)$ is complete in the $R^\infty$ topology
for every open $U\subset B^{m,n}$.\endproclaim
\proof In view of the isomorphism (4.2) we may consider only
the case $n=0$, so that we may identify $\G(U)$ with an algebra
of $G^\infty$ functions of even variables.
Let $\overline{\G(U)}$ be the completion of $\G(U)$ in the $R^\infty$
topology; all differential operators on $\G(U)$
extend to $\overline{\G(U)}$. Being a metric space, $U$
is a $k$-space \cite{21} and therefore
$\overline{\G(U)}$ may be regarded as a subalgebra of
$\Cc_M(U)$, the algebra of $B$-valued continuous functions
on $U$. One needs to check that any function $f\in\overline{\G(U)}$
at any point $p\in U$
is Fr\'echet differentiable and that its differential is given
by multiplicative action of the
partial derivatives of $f$
with respect to the $x$'s, formally extended by continuity from $\G(U)$.
 Since the locally convex space $B^{m,n}$
is Banach, `Fr\'echet' can be replaced by `G\^ateaux,'
that is, one can restrict to an arbitrary 1-dimensional
subspace $K$ of $U$ passing through $p$.
The space of $C^\infty$ $B$-valued functions
on $K$ is complete with respect to its standard topology and therefore
$f\vert_K$ is in this space. This means that the G\^ateaux
differential $d_pf$ of $f$ at $p$ exists ({\sl a priori} not
necessarily bounded). Pick a net $(f_\alpha)$ of functions
$f_\alpha\in\G(U)$ converging to $f$ in the $R^\infty$
topology. Clearly $f_\alpha\vert_K\to f\vert_K$ in the
$C^\infty$ topology over $K$.
Let $K=\{ p+at: t\in \Bbb R\}, ~a=(a_1, \dots, a_m)\in B^{m,0}$.
For all $\alpha$, due to the usual
chain rule for $G^\infty$ functions,
one has $$d_p(f_\alpha\vert_K)(a) = \sum a_i \left(
\frac{\partial f_\alpha}{\partial xi}\right)_p\,.$$
As $f_\alpha\to f$, the above equality turns by continuity into the
following:
$$d_p(f\vert_K)(a) = \sum a_i 
\left(\frac{\partial f}{\partial
x^i}\right)_p\,,$$
which implies that for an arbitrary $h\in B^{m,0}$ the
desired property holds:
$$d_pf(h) = \sum h_i \left(\frac{\partial f}{\partial
x^i}\right)_p\,$$
\qed\enddemo

Quite evidently, 
any $R$-su\-per\-space  $(M,\A,\ev)$ which is locally isomorphic
to the standard $R^\infty$-su\-per\-mani\-fold over $B^{m,n}$
is an $(m,n)$ dimensional $R^\infty$-su\-per\-mani\-fold.
By means of Proposition 4.6 we may prove the converse:
\proclaim{Proposition} Any $(m,n)$
dimensional $R^\infty$-supermanifold  $(M,\A,\ev)$ over
a\break BGO-algebra $B$ is locally isomorphic
to the standard $R^\infty$-su\-per\-mani\-fold over $B^{m,n}$.\endproclaim

To prove this result we need a preliminary Lemma, which can be proved
essentially as in \cite{29} (cf\. also \cite{8}), and a result on the
density of polynomials in the rings of superfunctions.

\proclaim{Lemma}
Let $(M,\A,\ev)$ be an $(m,n)$ dimensional R-su\-per\-mani\-fold, and let
$(U,\varphi)$ be a local chart for it. For all
$f\in\A(U)$, the composition $\tilde f\circ\tilde\varphi^{-1}$ is a $G^\infty$
function on $\tilde\varphi(U)\subset\BLmn$. \qed\endproclaim



Let $(M,\A,\ev)$ be an $R$-su\-per\-mani\-fold,
and let, for a fixed coordinate system $\varphi=\gcoor xm,yn$ in $U$,
$\Pc_\varphi(U)$ be the graded $B$-subalgebra of $\A(U)$
generated by the coordinates. The following result may be considered
as a graded analogue of the Weierstrass approximation theorem.
We do not know whether it remains true when
$B$ is an arbitrary graded-commutative Banach algebra.

\proclaim{Theorem}
Let $B$ be a BGO-algebra.
Then $\Pc_\varphi(U)$ is dense in $\A(U)$.\endproclaim
\proof The demonstration of this result is very lengthy
and has been postponed to an Appendix. \qed\enddemo

\noindent {\it Proof of Proposition 4.7.} \ Let $(U,\varphi)$ be a coordinate
chart for $(M,\A,\ev)$, with $\varphi=(x^1,\dots,x^m,
y^1,\dots,y^n)$. In view of the isomorphism (4.2) one can define an
injection  $$\hat T_\varphi\colon\hat\A_\varphi\hookrightarrow
       \tilde\varphi^{-1}\hat\G_{\vert\tilde\varphi(U)}$$
by letting $\hat T_\varphi(f)=\tilde f\circ\tilde\varphi^{-1}$;
by Lemma 4.8 $\hat T_\varphi(f)$ is a $G^\infty$ function
and therefore is a section of
$\tilde\varphi^{-1}\hat\G_{\vert\tilde\varphi(U)}$. Furthermore, $\hat
T_\varphi$ is a topological isomorphism with its image, so that  $\hat
T_\varphi(\hat\A_\varphi)$ is complete.
Since this space contains the $G^\infty$
functions that are polynomials in the even coordinates, it contains all the
$G^\infty$ functions by virtue of Theorem 4.9;  that is, $\hat T_\varphi$ is
an isomorphism. The morphism $\hat T_\varphi$ determines a topological
isomorphism $$
T_\varphi\colon \rest
\A,U\to\tilde\varphi^{-1}\G_{\vert\tilde\varphi(U)}$$ simply by letting
$T_\varphi(\sum f_\mu\otimes y^\mu)=\sum \hat T_\varphi(f_\mu)\otimes y^\mu$.
Now, the com\-mu\-ta\-tive diagram $$ \CD \rest\A,U @.\longiso
\tilde\varphi^{-1}\G_{\vert\tilde\varphi(U)} \\ @V\ev^U VV @VV \ev V \\
\rest\A^\infty,U @.\longiso
        \tilde\varphi^{-1}\G^\infty_{\vert\tilde\varphi(U)} \\
@VVV @VVV \\
0 @. 0 \\
\endCD
$$
proves the thesis.
\qed\enddemo

\proclaim{Corollary} If $(M,\A,\ev)$ is an $R^\infty$-su\-per\-mani\-fold
over a BGO-algebra,
then $(M,\ev(\A))$ is a $G^\infty$ su\-per\-mani\-fold.\endproclaim
\proof This result holds evidently for the standard
$R^\infty$-su\-per\-mani\-fold over $B^{m,n}$, and therefore, by
local isomorphism,
also for an arbitrary $R^{\infty}$-su\-per\-mani\-fold.\qed\enddemo

Finally, we consider the coordinate description of morphisms.
What follows generalizes results already known for graded
manifolds \cite{23} and for finite-dimensional ground algebras
\cite{33,8}.
Let $(M,\A,\ev^M)$ be an $R^\infty$-su\-per\-mani\-fold
over a BGO-algebra $B$,
let $U$ be an open set in $B^{m,n}$, and denote by
$(U,\G,\ev)$ the restriction to $U$ of the standard $R^\infty$-supermanifold
over $B^{m,n}$.
\proclaim{Lemma} Let $B$ be a BGO-algebra.
If $(f,\phi)\colon(M,\A,\ev^M)\to(U,\G,\ev)$
and $(f,\psi)\colon(M,\A,\ev)\to(U,\G,\ev)$  are 
R$^\infty$-su\-per\-mani\-folds morphisms, and
$\phi(x^i)=\psi(x^i)$ for $i=1,\dots,m$, $\phi(y^\alpha)=\psi(y^\alpha)$
for $\alpha=1,\dots,n$, then $\phi=\psi$.\endproclaim
\proof
$\phi$ and $\psi$ coincide over the sheaf of polynomials in the coordinates,
and therefore by continuity they also coincide over its completion
$\G$.
\qed\enddemo
\proclaim{Proposition} Let $B$ be a BGO-algebra, and let
$U\subset B^{m,n}$ be an open subset.
\roster
\item A family of sections $\gcoor um, vn$ of $\G$ on $U$
is a coordinate system for $(U,\G_{\vert U},\ev)$ as an R-su\-per\-mani\-fold if and
only the evaluations $\gcoor{\tilde u}m,{\tilde v}n$ yield
a $G^\infty$ coordinate system.
\item Let $\gcoor um,{v}n$ be  a
coordinate system for $(U,\G,\ev)$, let $f\colon
U\to W\subset\BLmn$ be the homeomorphism $z\mapsto(\tilde{u}^1(z),
\allowmathbreak\dots,\tilde u^m(z),\allowmathbreak\tilde{v}^1(z),
\allowmathbreak \dots,\tilde v^n(z))$,
and let $\gcoor xm,yn$  be a coordinate system on $W$.
There exists a unique isomorphism of R$^\infty$-su\-per\-mani\-folds
$(f,\phi)\colon(U,\rest{\G},U,\delta)\to(W,\rest{\G},W,\delta)$
such that
$\phi(x^i)=u^i$ for $i=1,\dots,m$, and $\phi(y^\alpha)=v^\alpha$
for $\alpha=1,\dots,n$.
\item  Every isomorphism $g:U\to V\subset B^{m,n}$
can be extended  (in many ways) to an
isomorphism of R$^\infty$-su\-per\-mani\-folds $(g,\phi)\colon
(U,\rest{\G},U)\iso (V,\rest{\G},V)$. Here `extension' means that the
diagram $$
\CD
\rest{\G},V@>\phi>>g_\ast\rest{\G},U\\
@V\ev VV@VV\ev V\\
\rest{\G^\infty},V@>>g^\ast>g_\ast\rest{\G^\infty},U
\endCD
$$
commutes.
\endroster
\endproclaim
\proof \therosteritem1 Since $\Ker\ev$ is nilpotent, a matrix of
sections of $\G$ is invertible if and only if its evaluation is invertible
as well, thus proving the statement.
 
\therosteritem2 One
can define a ring morphism $\phi\colon\Pc\to g_\ast\G$,
where $\Pc$ is the sheaf of polynomials in $x$ and $y$,
by imposing that $\phi(x^i)=u^i$,
$\phi(y^\alpha)=v^\alpha$ for $i=1,\dots,m$, $\alpha=1,\dots,n$.
Since the topology of $\G$ can be described by the seminorms
associated with any coordinate chart, 
$\phi$ is continuous and therefore induces
a morphism between the completions, $\phi\colon\G\to g_\ast\G$.
To see that $(g,\phi)$ is an isomorphism, we can construct, by the same
procedure, an `inverse' morphism $(g',\psi)$;
then, we have two morphisms of R$^\infty$-su\-per\-mani\-folds
$(\Id,\Id),(\Id,\psi\circ\phi)\colon(U,\rest{\G},U,\ev)\to(U,\rest{G},U,\ev)$ 
that coincide on a coordinate system, thus finishing
the proof by the previous Lemma.

\therosteritem3 follows from \therosteritem1 and \therosteritem2 since a
$G^\infty$ isomorphism transforms $G^\infty$ coordinate systems into
$G^\infty$ coordinate systems. \qed\enddemo
 



If $B=B_L$, then $R^\infty$ su\-per\-mani\-folds reduce to the
G-su\-per\-mani\-folds
introduced by some of the authors \cite{2}; they have been extensively
studied in \cite{8}. This on the one hand shows the relevance of
G-supermanifolds, in that they are the unique examples of supermanifolds over
$B_L$ satisfying the extended axiomatics, and, on the other hand, demonstrates
that that axiomatics admits concrete models.

\doskip
\heading From $R$-su\-per\-mani\-folds to $R^\infty$ su\-per\-mani\-folds
\endheading

In this section we show that with any $R$-su\-per\-mani\-fold one can associate
an $R^\infty$-su\-per\-mani\-fold in a functorial way.
We assume that the ground algebra $B$ is a BGO-algebra.
Let $(M,\A,\ev)$ be an $R$-su\-per\-mani\-fold; for any open set $U\subset M$,
let $\Q(U)$ be the completion of $\A(U)$ in the $R^\infty$-topology.
This defines a presheaf $\Q$; let us denote by $\bar \A$ the associated sheaf.
Let $W$ be a coordinate neighbourhood, with coordinates
$\varphi=\gcoor xm,yn$; since the polynomials are dense in $\A$
(Theorem 4.9),
there is a presheaf isomorphism
$\tilde\varphi^{-1}\rest{\G},{\tilde\varphi(W)}\simeq\rest{\Q},W$. This means
that $\rest{\Q},W$ is isomorphic with its associated sheaf $\rest{\bar\A},W$
for each coordinate neighbourhood $W$,
so that $\bar \A$ can be endowed with a structure of a sheaf of {\sl complete\/}
Hausdorff locally convex graded $B$-algebras. The evaluation morphism $\ev$,
being continuous, induces a morphism $\ev\colon\bar\A\to\Cc_M$, so that
$(M,\bar\A,\ev)$ is an $R$-su\-per\-space over $B$, which is locally isomorphic
with the standard $R^\infty$-supermanifold over $B^{m,n}$. Hence, by
Proposition 4.7, we obtain the following result.

\proclaim{Theorem} The triple $(M,\bar\A,\ev)$
is an $R^\infty$ su\-per\-mani\-fold.\qed\endproclaim

Quite obviously, there is a canonical $R$-su\-per\-space
morphism $(f,f^\sharp)\colon (M,\bar\A,\ev)\to (M,\A,\ev)$,
with $f=\Id$. Moreover, in view of Theorem 4.1, this correspondence
between the two categories of su\-per\-mani\-folds is functorial.

In accordance with Corollary 4.10 and with the previous Theorem,
any $R$-su\-per\-mani\-fold determines an `underlying' $G^\infty$
su\-per\-mani\-fold; thus, one can prove the following result.


\proclaim{Proposition} Let
$(f,f^\sharp)\colon (M,\A,\ev^M) \to(N,\B,\ev^N)$
be an $R$-su\-per\-mani\-fold morphism.
Then $f\colon M\to N$ is a $G^\infty$ map.
\endproclaim
\proof One can assume that $M$ and $N$ are coordinate neighbourhoods, in which
case the result is proved by Lemma 4.8.
 \qed\enddemo



\doskip
\heading Holomorphic su\-per\-mani\-folds \endheading

Let $(M,\A, \ev)$ be a complex $R$-su\-per\-space,
that is, an $R$-su\-per\-space over a {\sl complex} graded
com\-mu\-ta\-tive Banach algebra $B$.
We introduce a topology on the algebra $\A(U)$ for every open $U\subset M$,
which we call the $R^\omega$-{\sl topology}, as
the coarsest topology with the properties:

(i) the evaluation map $\ev_U$ from $\A(U)$ to the space $\Cc_M(U)$ of all
continuous $B$-valued functions on $U$ endowed with the topology of
compact convergence is continuous;

(ii) all  odd differential operators $L\in\sh Der A(U)$ are continuous.

One can describe this topology by means of seminorms as
it was done for the $R^\infty$-topology.
The $R^\omega$-topology makes $\A(U)$ into a
locally convex complex topological $B$-algebra.
It can be easily seen that in the non-graded case ($B_1=0$),
and when $(M,\A,\ev)$ is an $R$-supermanifold,
this topology coincides with the customary compact-open topology.

We say that a complex $R$-su\-per\-ma\-ni\-fold $(M,\A, \ev)$ is
an $R^\omega$-{\sl su\-per\-ma\-ni\-fold} if it fulfills Axioms 1 to 4
and the following Axiom.

\noprocnumber\proclaim{Axiom
5$_{\Bbb C}$} For every open subset $U\subset M$,
the topological algebra $\A(U)$ is complete Hausdorff 
in the $R^\omega$-topology.
\endproclaim

Arguing as in the case of $R^\infty$-supermanifolds, and appealing to
results on holomorphic maps between complex Banach spaces,
(see, e.g., \cite{12})
one can reformulate in this context all the
results of Sections 3 and 4.



\subheading{Acknowledgements} It is a pleasure to thank
Mitchell Rothstein for valuable
discussions and suggestions. V.G.P\. wishes to thank
the National Group for Mathematical Physics of C.N.R\. for
providing him support through its Visiting Professorship Scheme,
and the Department of Mathematics and Statistics of the
University of Victoria --- especially Professor Albert Hurd ---
for their hospitality.

\doskip\heading Appendix \endheading
\noindent{\it Proof of Theorem 4.9.\ }
By virtue of the isomorphism (4.2) it is sufficient to consider the case
$n=0$ only. Let $f$ be a $G^\infty$ function defined over an open subset
of $B^{m,0}$; by force of Theorem 3.4  this set may be taken 
of the form $U\,\widetilde{\,},~U\subset \Bbb R^m$ with no loss
of generality.
Let $K\subset U\,\widetilde{\,}$ be a compact set; one may assume that it is
of the form $I\times C,~I$ being an $m$-cube in $\Bbb R^m$ and
$C$ a compact set in $\Rad B$.

Let $\epsilon >0$.
By virtue of Theorem 3.5, we can pick for any $x\in U$ a number $N_x$
such that for all $y\in K$ with $\sigma (y)=x$ one has
$\Vert f(y) -  \sum_{ \vert J\vert =i}^{N(x)} {1\over  J!} D^{( J)}(f)(x)
(s^{m,0}(y))^ J \Vert_B < \epsilon$.
Denote by $p_x(y)$ the polynomial in $y$ of the form
$\sum_{ \vert J\vert=i}^{N(x)} {1\over  J!}
D^{( J)}(f)(x)
 (s^{m,0}(y))^ J$.
The set $U_x = \{y\in U\,\widetilde{\,} :
\Vert f(y) -p_x(y)  \Vert_B < \epsilon$
is a neighbourhood of a compact set $\{x\}\times C$, and hence it
contains a `rectangular' neighbourhood of the form
$V_x\times W_x,~x\in V_x\subset U,~C\subset W_x\subset \Rad B$
(see \cite{21}).
Pick a finite subcover $V_{x_1}, \dots , V_{x_k}$ of the open cover
$\{V_x: x\in I\}$ of $I$.
There is a partition of unity $\{h_i\}_{i=1}^k$ subordinated to
the cover $V_{x_1}, \dots , V_{x_k}$.
Since all the functions $h_i$ may be chosen to be Pringsheim regular
(for example, so are the usual `bell' functions),
the $Z$-expansions  $Z(h_i)$ converge
to $G^\infty$ functions (see \cite{18} where this
result was proved for Grassmann-Banach algebras; however, the proof
is true {\sl verbatim\/} for BGO-algebras).
The collection $\{Z(f_i)\}_{i=1}^k$ of $G^\infty$ functions
forms a partition of unity for the family of DeWitt open sets
$V_{x_1}\,\widetilde{\,} , \dots , V_{x_k}\,\widetilde{\,} $.
The function $g=\sum_{i=1}^k Z(f_i)p_{x^i}$ is $G^\infty$
and $\epsilon$-approximates $f$ on $K$.
The totality of $C^\infty$ functions on $U$
such that for some $\alpha>0$
$$\sum_{n=1}^\infty {1\over n!}\alpha^n
\sum_{\vert J\vert=n}\max_{x\in I}\Vert D^{( J)}(f)(x)\Vert
<+\infty$$
forms an algebra which we denote by $\Cal{UP}^\infty(U)$; it
 contains polynomials and  `bell' functions.
Thus, we can assume that $g\in\Cal{UP}^\infty(U)$.

Turning back to the hypothesis of the first paragraph of our
proof, we may assume now that $f\vert U\in \Cal{UP}^\infty(U)$.
In this case
the $Z$-expansion converges to $f$ {\sl uniformly} on $K$.
Indeed, taking into account the quasinilpotency of elements of $\Rad B$ and
compactness of $C$, one can prove that
for each $\alpha$ with $0<\alpha <1$, there exists a constant
$M_\alpha >0$ such that
for every $\theta\in C$,
where $\theta = (\theta_1, \dots, \theta_m)$, every $i=1,\dots, m$,
and every $n\in \Bbb N$ the inequality
$\Vert \theta_i^n \Vert < M_\alpha\cdot \alpha^n$ holds.


Given an $\epsilon>0$ and a natural number $k$, we can
find a natural number $N$
and a polynomial $p(x)$
on $\R^m$ with coefficients in $B_0$
such that for all $x\in K$ and all
$ J'$ with $\vert J'\vert\leq k$ one has:
$$\sum_{ J=0}^{N+1} {1\over  J!} D^{( J+ J')}(f-p)(\sigma^{m,0}(x))
 (s^{m,0}(x))^ J < e\cdot\epsilon $$
$$ \sum_{ J=N+1}^\infty {1\over  J!} D^{( J+ J')}p(\sigma^{m,0}(x))
 (s^{m,0}(x))^ J < \epsilon $$
$$\sum_{ J=N+1}^\infty {1\over  J!} D^{( J+ J')}f(\sigma^{m,0}(x))
 (s^{m,0}(x))^ J < \epsilon $$
Because of the uniform convergence of the $Z$-expansion on $K=I\times C$,
the last inequality is true for all
$ J'$ with $\vert J'\vert\leq k$  as soon as $N>N_0$
for some $N_0$ large enough.

In order to choose a polynomial $p$, we resort to
the classical proof of the Weierstrass approximation
theorem \cite {19}, going back to Weierstrass himself.
Usually that proof is applied to real-valued functions,
but the case of Banach-valued functions defined on subsets of
$\R^m$ makes no difference at all.

A careful analysis of the proof \cite{19} 
shows that for any finitely supported continuous
function $f$ in $\Bbb R^m$ taking values in
a Banach space and any
compact set $I\subset \Bbb R^m$ there exist real positive
constants $C_1, C_2, C_3 $
(which do not depend on $f$ but rather on $I$)
and a sequence of polynomials $p_n(f),~n\in\Bbb N$
on $\Bbb R^m$ with the properties:

1. For each $\epsilon>0$, if $n$ is such that
$$ {C_1 n^{m\over 2}
((\sum_i\Vert \partial f/\partial x^i \Vert_I)^2 - \epsilon^2)^n
(\Vert f \Vert_I+C_2)  \over
(\sum_i \Vert \partial f/\partial x^i \Vert_I)^{2n}}
< \epsilon\,, $$
where
$$\Vert \partial f/\partial x^i \Vert_I = \max_{x\in I}\Vert f(x) \Vert\,, $$
then
$$ \Vert f - p_n(f) \Vert_I < \epsilon\,. $$

2. The degree of $p_n(f)$ is $n$, and for any multiindex $ J$ with
$\vert J\vert\leq n$ one has
$$\frac{\partial^{\vert J\vert}p_n(f)}{\partial x^{ J}} =
p_n\bigl(\frac{\partial^{\vert J\vert}f}{\partial x^{ J}}\bigr)$$
and
$$ \Vert  p_n(f)  \Vert_I \leq C_3\Vert f\Vert_I\,. $$
As a corollary of 2), for all $N>N'_0$ the third inequality is
fulfilled for all
$ J'$ with $\vert J'\vert\leq k$ as soon as $N>N'_0$
 for some $N'_0$ large enough, if one substitutes
$p_n(f)$ for $p$ (this number $N'_0$ does not depend on $n$).
Put $N=\max\{N_0, N'_0\}$.
Set $$n= \epsilon^{-3}\bigl[(C_0+C_1+C_2)^4
\sum_{\vert J\vert\leq N+k+1}
\bigl\Vert \frac{\partial^{\vert J\vert}f}{\partial x^J} \bigr\Vert_I
\bigr]\,,$$
where the square brackets stand for the
integer part of a number.
Applying 1), one can show that  for all $ J$ with
$\vert J\vert \leq N+k$ one has
$$ \Vert f^{( J)} - (p_n(f))^{( J)} \Vert_I < \epsilon $$
This implies the first inequality with $p=p_n(f)$.

Since $p$ is a polynomial function on $\Bbb R^m,~ m\in \Bbb N$
taking values in $B$,
then $Z(p)$ is a polynomial function on $B^{m,0}$ (with the same
coefficients) and thus belongs to $\Pc_\varphi(U)$.
The three inequalities above imply that
for all $x\in K$ and every
 $ J'$ with $\vert J'\vert\leq k$
one has $\Vert (f-Z(p))^{( J')}(x) \Vert \leq (2+e)\epsilon$.
This proves that $\Pc_\varphi(U)$ is dense in $\A(U)$ in the
$R^\infty$ topology.
\qed\enddemo



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\enddocument
\bye