%% @texfile{
%%     filename="amst-mon.tex",
%%     version="2.1",
%%     date="30-JUL-1991",
%%     filetype="AMS-TeX: documentation",
%%     copyright="Copyright (C) American Mathematical Society,
%%            all rights reserved.  Copying of this file is
%%            authorized only if either:
%%            (1) you make absolutely no changes to your copy
%%                including name; OR
%%            (2) if you do make changes, you first rename it to some
%%                other name.",
%%     author="American Mathematical Society",
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%%     telephone="401-455-4080 or (in the USA) 800-321-4AMS",
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%%     codetable="ISO/ASCII",
%%     checksumtype="line count",
%%     checksum="286",
%%     keywords="amstex, ams-tex, tex",
%%     abstract="This file contains input for a sample monograph
%%            chapter illustrating the proper way to prepare such
%%            input for electronic submission to the AMS."
%%     }
%*********************************************************
%
% AMS-TeX 2.0+ file for a sample monograph chapter for
% electronic submission.
%
\input amstex
\documentstyle{amsppt}
\Monograph
\NoBlackBoxes
%
%  Macros required to support the ability to process this file with
%  amsppt.sty 2.0 (these macros were not defined until amsppt.sty 2.1);
%  see Guidelines for Preparing Electronic Manuscripts (AMS-TeX).
%  Macro for roman text in non-roman environments.
\ifx\undefined\rom \define\rom#1{{\rm #1}}\fi
%  Macro for current address.
\ifx\undefined\curraddr
  \def\curraddr#1\endcurraddr{\address {\it Current address\/}: #1\endaddress}
\fi

\topmatter
\title\chapter{4} Sample \AmSTeX{} Electronic Manuscript for a Chapter
 of a Monograph,\\
 Matrix Algebras\endtitle
\leftheadtext{SAMPLE \AmSTeX{} ELECTRONIC MONOGRAPH CHAPTER}
\endtopmatter

\document

\head 1. Introduction                % bold, centered;
\endhead                             % don't type final punctuation
This sample paper illustrates the use of the \AmSTeX{} Version~2.0 (or later)
preprint style for a chapter of a monograph.  The file {\bf amst-mon.tex}
was used to prepare this sample; an author should use the coding in
that file as a model.

In this sample chapter, brief instructions to authors are interspersed
with mathematical text extracted from (purposely unidentified)
published papers.  For instructions on preparing mathematical text,
the author is referred to {\it The Joy of \TeX}, by Michael Spivak
\cite{Spi}.  Detailed instructions for preparation of a monograph are
given in the {\bf Guidelines for Preparing Electronic Manuscripts}, to
which this sample chapter is one of several appendixes, and in
particular, in the section {\bf Monograph formatting}.

Two additional ``chapters,'' illustrating the table of contents and
bibliography of a monograph, were produced with the separate files
{\bf amst-mct.tex} and {\bf amst-mbi.tex} respectively.  These
sections are discussed in more detail below.


\subhead Top matter\endsubhead
The input format and content of the top matter can be best understood
by examining the first part of the sample file {\bf amst-mon.tex}, up
through the {\tt\char`\\document} instruction.

The top matter includes both elements that must be input by the author
and a few that are provided automatically.  For details, see the
above-mentioned {\bf Guidelines}.

Note that the monograph style does not include any author information,
subject information, or an abstract.  These elements are usually part
of the front matter of a book, which is prepared separately, and which
is not provided for by the preprint style.  This general information
should be placed in a separate file, called {\bf bookinfo.fil}, using
the tags described in the {\bf Top matter} section of the {\bf Guidelines}.

Note also that the sample table of contents is not part of the top matter,
but a separate chapter and a separate file, as identified above.  It
should include little, if any, text.  Detailed instructions for
entering the table of contents appear in the {\bf Guidelines}; however,
note in the input for this sample the differences between titles and
headings used for the monograph body and for the table of contents.


\subhead Fonts\endsubhead
The fonts used in this sample chapter are from the Computer Modern family;
they should be available to all authors preparing papers with these macros.
However, the final copy may be set by the AMS using other fonts.

\subhead A mathematical extract\endsubhead
The mathematical content of this sample chapter has been extracted from
published papers, with no effort made to retain any mathematical sense.
It is intended only to illustrate the recommended manner of input.

Mathematical symbols in text should always be input in math mode as
illustrated in the following paragraph.

A function is invertible in $C(X)$ if it is never zero and in $C^*(X)$ if
it is bounded away from zero. In an arbitrary $A(X)$, of course, there
is no such description of invertibility which is independent of the 
structure of the algebra. Thus in \S 2 we associate to each noninvertible
$f\in A(X)$ a $z$-filter $\Cal Z (f)$ that is a measure of where
$f$ is ``locally'' invertible in $A(X)$. This correspondence extends to
one between maximal ideals of $A(X)$ and $z$-ultrafilters on $X$.
In \S 3 we use the filters $\Cal Z (f)$ to describe the intersection of 
the free maximal ideals in any algebra $A(X)$. Finally, our main result
allows us to introduce the notion of $A(X)$-compactness of which 
compactness and realcompactness are special cases. In \S 4 we show how
the Banach-Stone theorem extends to $A(X)$-compact spaces.

\head 2. Theorems, lemmas, and other proclamations\endhead
%
Theorems and lemmas are varieties of proclamations.  Either may have a
proof or a ``demonstration.''  The lemma and proof below illustrate
the use of a ``roster'' or itemized list; the first item in the proof
roster is run in.  Note that both proclamations and demonstrations
have their beginnings and ends marked in the file.

\proclaim{Lemma 1} Let $f, g\in  A(X)$ and let $E$, $F$ be cozero
sets in $X$.
\roster
\item"(a)" If $f$ is $E$-regular and $F\subseteq E$, then $f$ is $F$-regular.

\item"(b)" If $f$ is $E$-regular and $F$-regular, then $f$ is $E\cup F$-%
regular.

\item"(c)" If $f(x)\ge c>0$ for all $x\in E$, then $f$ is $E$-regular.
\endroster
\endproclaim

\demo{Proof}
\roster\runinitem "(a)" Obvious.

\item"(b)" Let $h, k\in A(X)$ satisfy $hf|_E=1$ and $kf|_F=1$. Let
$w=h+k-fhk$. Then $fw|_{E\cup F}=1$.

\item"(c)" Let $h=\max\{c,f\}$. Then $h|_E=f|_E$ and $h\ge c$. So $0<h^{-1}
\le c^{-1}$. Hence $h^{-1} \in C^*(X)\subseteq A(X)$, and 
$h^{-1} f|_E=1$. 
\endroster
\enddemo

\definition{Definition}
For $f\in A(X)$, we define
$$
\Cal Z (f)=\{E\in Z[X]\: \text{$f$ is $E^c$-regular}\}.
\tag 2.1
$$
\enddefinition

\head 3. Roman type\endhead
%
Numbers, punctuation, (parentheses), [brackets], \{braces\}, and
symbols used as tags should always be set in roman type.  The following
sample theorem illustrates how to code for roman type within the
statement of a theorem.

\proclaim{3.1. Theorem}
Let $\Cal G$ be a free nilpotent-of-class-\rom{2} group of rank
$\ge 2$ with carrier $G$ and let
$$m : G\times G \to Z$$
satisfy \rom{(2.21)}, \rom{(2.22)}, and \rom{(2.24)}, and define
$\kappa$ by \rom{(2.23)}.  Then this kappa-group is kappa-nilpotent
of class \rom{2} and kappa-metabelian, that is to say, it satisfies
\rom{S2} and \rom{S3}, but it is kappa-abelian if, and only if,
$$m(x,y) = -1\quad\text{for all $x, y \notin G'$}.
\tag 3.1$$
\rom{(}Thus \rom{(3.1)} implies the trivial consequence
\rom{(2.1)}.\rom{)}  Assume now that \rom{(3.1)} does not hold,
so that the kappa-group is kappa-nonabelian.  Assume further that $m$
is not constant outside $G'$ \rom{(}inside $G'$ the values of $m$
clearly do not matter\rom{)}.  Then $\kappa$ is neither left nor right
linear, that is to say, neither \rom{S4} nor \rom{S5} holds:
\rom{I1} again holds, but none of \rom{I2--I5}.  As before,
\rom{I6} is equivalent to \rom{(2.25)}.  Now \rom{I7$'$}, however,
is equivalent to a condition similar to \rom{(2.25)}, namely
$$m(xz\sigma, yz\sigma) = m(x,y)\,.
\tag 3.2$$
\endproclaim

Other elements should always be set in roman type.  Control sequences
should be used for common mathematical functions and operators like
$\log$ and $\lim$ \cite{Spi\rm, Chapter 14}, and {\tt\char`\\cite}
should always be used when citing a reference.  Detailed instructions
are given in the {\bf Document body, roman type} section in the
above-mentioned {\bf Guidelines}.


\head 4. References\endhead
%
Detailed instructions for the input of references are given in the
{\bf Guidelines} mentioned above.  The bibliography of a monograph
is a separate chapter, and thus a separate file.  The sample
bibliography which follows this sample chapter was produced from the
file {\bf amst-mbi.tex},

These references have been chosen to illustrate the coding of the most
common types of references.  The sample references have been labeled
with ``key''-type labels, as {\tt\char`\\key AVG1}, etc.  It is also
possible to use numbers for labels; see the instructions in the
{\bf Guidelines} and also the sample article that accompanies them,
where that style has been used.


\head 5. Figures\endhead
%
Figures are handled as inserts, with an amount of space left that
should equal the exact height of the figure.  Extra space around the
figure will be provided automatically.  The positioning of figures may
need to be changed to obtain the best possible page layout.  Thus, it
is important to label your figures and use the labels in the text when
referring to them.  The figure caption should be positioned below
the figure.

Additional instructions for figures and other inserts, including
instructions for preparing art work, are given in the {\bf Guidelines}.

\example{Example 5}
For the link in Figure 5a, the Massey product $\langle u_1, u_2, u_3,
u_4, u_5\rangle$ in $S^3-L$ is defined and consists of all
integer multiples of $\gamma_{1,5}$.  For the link in Figure 5b,
the Massey product $\langle u_1, u_2, u_3, u_4, u_5\rangle$ in
$S^3-L$ contains the single element $\gamma_{1,5}$.  Since the links
in Figures 5a and 5b are homotopic, the example indicates that Massey
products in $S^3-L$ with distinct $u_j$'s do not, in general,
determine homotopy invariants of the link.  For the link in Figure 5a
and the link in Figure 5b, the Massey product $\langle u_1, u_2, \dots,
u_5\rangle$ in $\{S^3-L_i\}_{i=1}^5$ contains the single element
$\gamma_{1,5}$.
\endexample

%  art work measures 11.5pc for figure 5a, 7pc for figure 5b

\topinsert
\vskip 11.5pc
\botcaption{Figure 5{\rm a}}\endcaption
\endinsert
\topinsert
\vskip 7pc
\botcaption{Figure 5{\rm b}}\endcaption
\endinsert

\head 6. Other headings\endhead
%
\subhead A subheading\endsubhead % bold, run-in; do not type ending punctuation
We conclude by noting that another characterization of $A$-compactness
follows from Mandelker \cite 5. We call a family $\Cal S$ of closed sets in
$X\ A$-stable if every $f\in A(X)$ is bounded on some member of $\Cal S$.
Then one can show (as in \cite 5) that a space is $A$-compact if and only if 
every $A$-stable family of closed sets with the finite intersection property
has nonempty intersection.

\subsubhead A second-level subheading\endsubsubhead
This paragraph is included only to illustrate the appearance of a
sub-subheading.

\subsubhead Comment on the reference keys\endsubsubhead
The references were set and printed once without paying attention to
the width of the keys.  Then the widest key was identified and an
instruction added just before the references heading to establish the
``widestnumber''.  See the {\bf Bibliographic references} section in
the {\bf Guidelines}.

\enddocument

% [end of file amst-mon.tex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%