Table Of Content305 A 1150
Obstructions
305 (IX.1 1) and by @“(jr) the set of thomotopy classes
Obstructions of mappings in %“(f’) relative to L. The set
&O(f’) consists of a single element ‘because of
the arcwise connectedness of Y, @(f’) is non-
A. History empty, and @‘(f’) (na 2) may be empty. Let ,f”
be an element of @“(f’). If we consider the
The theory of obstructions aims at measuring restriction off” to the boundary Oni1 of an
the extensibility of mappings by means of oriented (n + l)-cell (r”+’ of K, then f”: $‘+l -t
algebraic tools. Such classical results as the Y determines an element c(f”, on+‘) of the
+Brouwer mapping theorem and Hopf’s exten- thomotopy group 7~,,(Y ) (- 202 Homotopy
sion and tclassification theorems in homotopy Theory). This element gives a measure of ob-
theory might be regarded as the origins of this struction for extending f” to the interior of
theory. A systematic study of the theory was C?+I. We obtain an (n+ 1)-tcocycle c”+‘(f”) of
initiated by S. Eilenberg [l] in connection the tsimplicial pair (K, L) with coefficients in
with the notions of thomotopy and tcoho- n,J Y), called the obstruction cocycle off”, by
mology groups, which were introduced at the assigning c(fn,a”+‘) to each (n+ l)-ce11 a”+‘.
same time. A. Komatu and P. Olum [L] ex- This obstruction cocycle c”“(f”) is the mea-
tended the theory to mappings into spaces not sure of obstruction for extending f” to R”“. A
necessarily +n-simple. For mappings of poly- necessary and sufflcient condition for the
hedra into certain special spaces, the +homo- extensibility is given by c”“(f”) == 0. Clearly,
topy classification problem, closely related to c”+l (j”“) is uniquely determined for each ele-
the theory of obstructions, was solved in the ment f of @“(f’). The set of a11 c”+‘(f”) with
following cases (K” denotes an m-dimensional S”E@“(~‘) forms a subset o”+‘(f’) of the group
polyhedron): K”+‘+S” (N. Steenrod [SI), Knt2 of cocycles Z”‘l(K, L; n,(Y)). @“‘l(f) is non-
4s” (J. Adem), Kntk*Y, where ni( Y)=0 for empty if and only if o”+l(,f’) contains the zero
i <n and n < i < n + k (M. Nakaoka). There element 0.
are similar results by L. S. Pontryagin, M. LetKn=KxI,Lu=(KxO)L(LxZ)U
Postnikov, and S. Eilenberg and S. MacLane. (K x 1). Given two mappings SO, fi : K-t Y
Except for the special cases already noted, it is satisfying f. 1L =f, 1L , we cari defïne a natural
extremely difflcult to discuss higher obstruc- mapping F’ : La -) Y such that an element
tions in general since they involve many com- F” of @“(F’) corresponds to a thomotopy
plexities. Nevertheless, it is significant that the h”-’ relative to L connecting f. ( K”-’ with
idea of obstructions has given rise to various fi 1 K”-‘. Given an element F”E@“(F’), we
important notions in modern algebraic topol- have the element P’(F”) of Z”“(Km, L”;
ogy, including cohomology operations (- 64 n,(Y)), which we identify with Z”(K, L; n,(Y))
Cohomology Operations) and characteristic through the natural isomorphism of chain
classes (- 56 Characteristic Classes). groups of the pair (Ko, L”) to those of the
The notion of obstruction is also very useful pair (K, L). Thus we cari regard c”+‘(F”) as an
in the treatment of cross sections of fiber bun- element of Z”(K, L; 7c”( Y)), which is denoted
dles (- 147 Fiber Bundles), tdiffeomorphisms by d”(f,, h”-‘,f,), and cal1 it the separation (or
of differentiable manifolds, etc. difference) cocycle. If ,jo ( R”-l =SI ) i?‘, we
have the canonical mapping F”: L” U (Ko)
* Y, and the separation cocycle is denoted
B. General Theory for an n-Simple Space Y simply by d”(fo,fi). The set of :separation
cocycles corresponding to elements of @“(Fr) is
The question of whether two (continuous) considered to be a subset of Z”(K, L; n,( Y))
mappings of a topological space X into an- and is denoted by o”(.fO,fi). A necessary and
other space Y are +homotopic to each other sufflcient condition for h”-’ to be extensible
cari be reduced to the extensibility of the given to a homotopy on l?” is d”(f,,h”-‘,f,)=O.
mapping: (X x {O))U(X x {l})+ Y to a map- Therefore a necessary and suffccient condition
ping of the product X x 1 of X and the unit for f0 1 R” =fi ( R” (rel L) (i.e., relative to L) is
interval I= [0, 11 into Y. Therefore the prob- O~o”(f,,f,). Givenf;,f;:K”-tYwithfO[L=
lem of classifying mappings cari be treated .fTIL, then d”(&‘,h”-‘,f;) (~o’(f;,f;)) is
in the same way as that of the extension of an element of Z”(K”, L; 7c”( Y)): which is also
mappings. considered to be a cochain of the pair (K, L).
Let K be a tpolyhedron, L a subpolyhedron In this sense, we ca11 @(SO, h” -‘, ,f,“) the sepa-
of K, and R” = LU K” the union of L and the ration (or deformation) cochain over (K, L).
+n-skeleton K” of K. Let Y be an tarcwise The coboundary of the separation cochain
connected n-simple space, and ,f’ be a mapping d”(fl, h”-‘,,f;) coincides (except possibly for
of L into Y. Denote by O”(f’) the set of map- sign) with c”“(&‘)-c”“(f,“).
pings of I?” into Y that are extensions off’, For a tïxed fo E a>“( f’), any n-cochain dn
1151 305 c
Obstructions
of the pair (K, L) with coefficients in n,(Y) general, if On( fo, fi) is nonempty, it is a coset
is expressible as a separation cochain d”= of H”(K, L; x,(Y)) factored by the subgroup
d”(fl,f;) wheref/EQ”(f’) is a suitable map- O”(f,, fo). Combined with the existence theo-
ping such that fil R”-’ =fr” 1R ”-’ (existence rem on separation cochains, this cari be uti-
theorem). lized to show the following theorem.
Therefore if we take an element f”-’ of Assume that O”(f ‘) is nonempty. The set
@-I(f’) whose obstruction cocycle c”(f”-‘) of a11 elements @‘(f ‘) that are extensions of
is zero, the set of a11 obstruction cocycles an element of @-l (f ‘) is put in one-to-one
c”“(J”) of a11 such ~“E@“(S’) that are exten- correspondence with the quotient group of
sions off”-’ forms a subset of O”“(f’) and Hn(I?‘, L; A,( Y)) modulo On( fo, fô) by pairing
coincides with a coset of Z”+‘(K, L; n,(Y)) the obstruction On( fô, f “) with each f” for a
factored by B”+‘(K, L; rcL,Y( )). Thus a coho- fixed fô. Among such elements of @“( f ‘), the
mology class ?+r(f”-~)EH”+~(K, L; rrn( Y)) set off” that are extensible to @+’ is in one-
corresponds to an f”-’ E Qn-l (f’) such that to-one correspondence with the quotient
c”(f”-‘) = 0, and ?“+i(f”-‘) = 0 is a necessary group of H”(R”+1 ,L; ~n(Y))=H”(KL; G(Y))
and sufftcient condition for f”-’ to be exten- modulo the subgroup On( fô”, f$+‘), assuming
sible to I?“” (first extension theorem). that fo is extended to fo”+’ (fkst classification
For the separation cocycle, d”(f,, h”-‘,fJ~ theorem).
H”(K, L;a,(Y)) corresponds to each homotopy
h”-’ on I?n-Z such that d”-‘(f,, h”-‘,f,)=O,
and 6”( f& h”-‘, fi) = 0 is a necessary and C. Primary Obstructions
sufficient condition for h”-’ to be extensible to
a homotopy on R” (tirst homotopy theorem). Assume that H’+‘(K, L; ni( Y))= H’(K, L; ni( Y))
The subset of H”+‘(K, L, K,( Y)) correspond- =O,whereO<i<p(e.g.,rq(Y)=O,O<i<p).In
ing to on+’ (f ‘) is denoted by On+’ (f ‘) and is this case, by consecutive use of the lïrst exten-
called the obstruction to an (n + 1)dimensional sion theorem and the lïrst homotopy theorem,
extension off ‘. Similarly, the subset On( fo, fi) we cari show that each @(f ‘) (i <p) consists of
of H”(K, L, n,( Y)) corresponding to o”( fo, fi) is a single element and OP+‘( f ‘) also consists of a
called the obstruction to an ndimensional single element Pc1 (f ‘) E HP+‘(K, L; xP( Y)). The
homotopy connecting f. with fi. Clearly, a element CP+’ (f ‘), called the primary obstruc-
condition for f' to be extensible to Rn+’ is tion off ‘, vanishes if and only if f’ cari be
given by 0 E O”+l (f ‘), and a necessary and extended to RP+1 (second extension theorem).
sufftcient condition for f. 1K ” = fi 1K ” (rel L) is When Hi+’ (K, L; ai(Y)) = 0 for i > p (for exam-
given by OeO”(fo, fi). ple, when ai(Y) = 0 for p ci < dim(K -L)),
A continuous mapping <p: (K’, L’)+(K, L) f’ is extendable to K if and only if the first ob-
induces homomorphisms of cohomology struction off’ vanishes (third extension
groups ‘p*: H”+‘(K, L; n,( Y))+H”+‘(K’, L’, theorem).
n,(Y)), H”(K, L; w,( Y))+H”(K’, L’; n,(Y)). Then Correspondingly, if H’(K, L; zi( Y))=
for f’:LtY, O”“(f’ocp)~rp*O”+‘(f’), and Hi-’ (K, L; rci( Y)) = 0 (0 < i < p), then for any
for f,, fi : K + Y such that f. 1L = fi ) L, On( f, o two mappings fo, fi : K-+ Y, f. (L = fi 1L ,
cp,f , o cp)~ p*O”( fo, fi). Therefore we also lïnd Op(fo, fi) consists of a single element dP(fo, fi)
that the obstruction to an extension and the E HP(K, L; rcp( Y)), which we cal1 the primary
obstruction to a homotopy are independent difference off0 and fi. This element vanishes
of the choice of subdivisions of K, L, and con- if and only if f0 1l @’ = fi 1& ’ (rel L) (second
sequently are topological invariants. homotopy theorem). Moreover, when H’(K, L;
Let fo, fi, and fi be mappings K-* Y such rci( Y)) = 0 (i > p), the primary difference is zero
that f. 1L = fi 1L = f, 1L . Given homotopies if and only if f. E fi (rel L) (third homotopy
h~;l:f,)~“-l~fi)Rn-l(relL),h;;l:fl)~-l~ theorem).
f2 1R ”-’ (rel L), then for the composite h”,;’ = Assume that the hypotheses of the second
h;;’ o ht;‘, we have extension theorem and second homotopy
theorem are satistïed. If we assign to each
element f P of @‘(f ‘) the primary difference of
f P and the tïxed element f{, then Gi”( f ‘) is in
one-to-one correspondence with HP(RP, L;
and for the inverse homotopy h;;’ : f, 1Z ?‘-l T rcp( Y)) by the lïrst classification theorem (sec-
f. 1R ”-’ of h”,;‘, clearly ond classification theorem). Similarly, assume
that the hypotheses of the third extension
d”(f,,h;o’,fo)= -d”(fo,h&‘,f,).
theorem and third homotopy theorem are
Therefore O”(fO,fO) forms a subgroup of satisfîed. Iff,:K+ Y, f’=& 1L , then homotopy
H”(K, L, nnn(Y )) that is determined by the classes relative to L of extensions f off’ are
homotopy class off0 1e m1 relative to L. In put in one-to-one correspondence with the
305 D 1152
Obstructions
elements of HP(K, L; xp( Y)) by pairing dp(,f;fO) ferential equations by reducing the operations
with ,f (tbird classification theorem). of differentiation and integration into alge-
brait ones in a symbolic manner. The idea
was initiated by P. S. Laplace in his Théorie
D. Secondary Obstructions
analytique des probabilités (18 12), but the
method has acquired popularity since 0.
For simplicity, assume that ni(Y) = 0 (i< p and
Heaviside used it systematically in the late
p < i < q). If the primary obstruction CP+I (f’)~
19th Century to solve electric-circuit problems.
H P+I (K, L; 7~,,(Y )) off’: L-t Y vanishes, we cari
The method is therefore also callec. Heaviside
detïne 04+’ (,f’) c H4+’ (K, L; 7rIq(Y )), which we
calculus, but Heaviside gave only a forma1
cal1 the secondary obstruction off’. When Y =
method of calculus without bothering with
SP, q = p + 1, p > 2, the secondary obstruction
rigorous arguments. The mathematical foun-
Op”(f’) coincides with a coset of HpfZ(K, L;
dations were given in later years, tïrst in terms
Z,) modulo the subgroup Sq2(HP(K, L; Z)),
of +Laplace transforms, then by ap:$ying the
where Sq’ denotes the +Steenrod square
theory of tdistributions. One of the motiva-
operation [S]. In this case, if L = KP, then
tions behind L. Schwartz’s creation of this
Op”(,f’) reduces to a cohomology class,
latter theory in the 1940s was to give a sound
Sq’(i*)-‘f’*(o) with i: L+K, where o is a gen-
foundation for the forma1 method, but the
erator of HP(SP, Z) (in this case (i*)m’f’*(a) #
theory obtained has had a much larger range
0 is equivalent to ?“(f’)=O) [S]. Moreover,
of applications. Schwartz’s theory was based
if Sq2f’*(a) = 0, then there exists a suitable
on the newly developed theory of + topological
extension fp+2:lZp+2 + Y = SP of ,f’. The set of
linear spaces. On the other hand, J. Mikusin-
obstruction cocycles of all such fp” defïnes
ski gave another foundation, based only on
the tertiary obstruction Op+3( f ‘), which coin-
elementary algebrdic notions and on Titch-
cides wit h a coset of Hp+3(K, L; Z,) modulo the
marsh’s theorem, whose proof has recently
subgroup SqZ(HP”(K, L; Z,)). By using the
been much simplifïed.
tsecondary cohomology operation 0 of J.
In this article, we fïrst explain the simple
Adem, it cari be expressed as @((i*)-‘f’*(o))
theory established by Mikusinski [2] and later
(- 64 Cohomology Operations).
discuss its relation to the classical Laplace
All the propositions in this article remain
transform method.
true if we take +CW complexes instead of
polyhedra K.
B. The Operational Calculus of Mikusibski
References
[l] S. Eilenberg, Cohomology and continuous The set % of all continuous complex-valued
mappings, Ann. Math., (2) 41 (1940), 231-251. functions a = {a(t)} defïned on t 2 0 is a tlinear
[2] P. Olum, Obstructions to extensions and space with the usual addition and scalar multi-
homotopies, Ann. Math., (2) 52 (1950) l-50. plication. %?i s a +Commutative algebra with
[3] P. Olum, On mappings into spaces in multiplication a. b detïned by the tconvolution
which certain homotopy groups vanish, Ann. {S;u(t-s)b(s)ds}. Th e ring W has no +zero
Math., (2) 57 (1953) 561-574. divisors (Titchmarsh’s theorem). (There have
[4] N. E. Steenrod, The topology of fibre been several interesting proofs of Titchmarsh’s
bundles, Princeton Univ. Press, 1951. theorem since the tïrst demonstratilon given
[5] N. E. Steenrod, Products of cocycles and by Titchmarsh himself [3]. For example, a
extensions of mappings, Ann. Math., (2) 48 simple proof has been published by C. Ryll-
(1947), 290-320. Nardzewski (1952).) Hence we cari construct
[6] E. H. Spanier, Algebraic topology, the tquotient field -2 of the ring %. An element
McGraw-Hill. 1966. of 2 is called a Mikusitiski operator, or simply
an operator. If we deiïne a(t) = 0 for t < 0 for
the elements {a(t)} in ??, then V? is a subalgebra
of %VIw, hich is the set of all locally integrable
(locally L,) functions in (-a, a) uhose +sup-
306 (X11.20) port is bounded below. Here we identify two
Operational Calculus functions that coincide almost everywhere.
The algebra J& has no zero divisor, and its
quotient tïeld is also 2.
A. General Remarks The unity element for multiplication in %,
denoted by 6 = b/b (b #O), plays the role of the
The term “operational calculus” in the usual +Dirac &function. It is sometimes called the
sense means a method for solving tlinear dif- impulse function. The operator 1= { 1) ~‘6 is the
1153 306 C
Operational Calculus
function that takes the values 0 and 1 accord- sented by
ing as t < 0 or t > 0. This operator is Heavi-
side’s function and is sometimes called the
unit function. Usually it is denoted by l(t)
or simply 1. The value l(0) may be arbitrary, m
M(ni) of
but usually it is detïned as 1/2, the mean of the +C
i=1Zj7Je "
limit values from both sides. The operator I is
an integral operator, because, as an operator where we assume that Âo, 1,). . . , i, exhaust the
carrying a into I. a, it yields roots of the equation o(L) = 0, 1, is a multiple
root of degree 1, and a11 other roots are simple
t
l.a= a(s)ds = the integral of a over [0, t]. (WI = n - 1). The above formula is called the
is 0 1 expansion theorem.
More generally, the operator {t”-‘/I(n)} (ReÂ.
> 0) gives the Âth-order integral. The operator
C. Limits of Operators
s = 611, which is the inverse operator of 1, is a
differential operator. If a E +Z is of tclass C’ ,
then we have A sequence a, of operators is said to converge
to the limit a = b/q if there exists an operator
s~a=a’+a(+0)6=a’+{a(+O)}/{l}. q( # 0) such that q. a, EV and the sequence
of functions q. a, converges tuniformly to b
Similarly, if UEW is of class C”, we have
on compact sets. The limit a is determined
S”.a=a’“)+a’“-“(0)6+a(“-2)(0)s
uniquely without depending on the operator q.
Based on this notion of limits of operators, we
+ . +a(O)s”-1. (2)
cari construct the theory of series of operators
The operator U+S. a cari be applied to func- and differential and integral calculus of func-
tions a that are not differentiable in the ordi- tions of an independent variable i whose
nary sense, and considering the application of values are operators. They are completely
s to be the operation of differentiation, we cari parallel to the usual theories of elementary
treat the differential operator algebraically in calculus (- 106 Differential Calculus; 216
the tïeld 02. In particular, we have s. 1 = 6, and Integral Calculus; 379 Series). A linear partial
this relation is frequently represented by the differential equation in the function V(X, t) of
formula two variables, in particular its initial value
problem, reduces to a linear ordinary differen-
dl(t)/dt=S(t). (3)
tial equation of an operational-valued func-
A rational function of s whose numerator is tion of an independent variable x.
of lower degree than its denominator is an For a given operator w, the solution (if
telementary function of t. For example, we it exists) of the differential equation v’(1) =
have the relations w. <p(Â) with the initial condition C~(O)= 6 is
unique, is called the exponential function of an
l/(s-~)“={t”-‘ea’/(n-l)!},
operator w, and is denoted by ~(1) = e”“. If the
power series
s/(s2 +/32) = {cos~t}. (4)
j. I”w”/n! (6)
The solution of an ordinary linear differ-
ential equation with constant coefficients converges, the limit is identical to the exponen-
C:=,, a,<p(‘)(t) =f(t) under the tinitial condition tial function e”‘“. However, there are several
C~“)(O) = yi (0 < i <n - 1) is thus reduced to an cases in which eAW exists even when the series
equation in s by using formulas (1) and (2), and (6) of operators does not converge.
is computed by decomposing the following For example, for w = - &, we have
operator into partial fractions:
e -d= {(Â/2&P)exp( -L’/4t)}, , (7)
(5) and for w = -s, we have
e-“S=h”=s.H,(t), (8)
whereL i=~,+l~o+~,+2~l + . .. +w~-,-~,
0 < r < n - 1. The general solution is repre- where the function H,(t) takes the values 0 and
sented by (5) if we consider the constants yo, 1 according as t < 1 or t > 1. H,(t) belongs to
. . . ,yn-i or fio, . ,jI-i as arbitrary parameters. the ring @ and is called the jump function at Â.
If the rational function in the right-hand side For f(t)e%, we have
of (5) is M@)/L)(s) and the degree of the numer-
~A~u(t)~=u(t-4~9
ator is less than that of the denominator, then
the right-hand side of (5) is explicitly repre- and hence we cal1 (8) the translation operator
306 D 1154
Operational Calculus
(or shift operator). For w = -s, the series (6) changing the variable from s to p and multi-
does not converge, but if we apply the forma1 plying the former transform by p.
relation e~“s=C~o(-is)“/n! to f(t), we have a
forma1 Taylor expansion
E. Relation to Distributions
For f EV?, an operator of the form h’. sk. f is
identifïed with a distribution of L. Schwartz
The solution of linear tdifference equations
with support bounded from below. We cari
are represented by rational functions of h”.
identify with a Schwartz distribution the limit
The power series C a,h” of operators always
of a sequence f, (or a suitable equivalence class
converges. This fact gives an explicit example
of sequences) of operators of the form h” sk .f
of a representation by forma1 power series.
such that f., fn+l, . are identical in the inter-
Note that the operators em” and e-“JS play
val (-II, n). The notions of Schwartz distri-
an essential role in the solution of the +wave
butions and of MikusiBski operators do not
equation
include each other, but both are generaliza-
a-=z-q a2azp tions of the notions of functions and their
ax2 at2 derivatives. For formulas and examples -
Appendix A, Table 12.
and the +heat equation
azq dacp
~=~~ References
5X2 at
The operator (7) converges to 6 for /2+0, and [l] 0. Heaviside, On operators in physical
this gives a tregularization of the Dirac 6- mathematics 1, II, Proc. Roy. Soc. London,
function. ser. A, 52 (1893), 504-529; 54 (1893), 105-442.
[2] J. Mikusihski, Operational calculus, Per-
gamon, 1959. (Original in Polish, 1953.)
[3] E. C. Titchmarsh, The zeros of certain
D. Laplace Transform
integral functions, Proc. London Math. Soc.,
25 (1926), 283-302.
For every function {f(t)} E %?,t he limit
[4] K. Yosida, Operational calculus: A theory
of hyperfunctions, Springer, 1984.
lim Pe-“‘f(l)dl= m eeAsf(,l)dA (9)
B-+m s o s 0
always exists (in the sense of the limit of oper-
ators), and as an operator coincides with the 307 (XIX.1 3)
original function {f(t)}. Therefore, if the usual Operations Research
Laplace transform (- 240 Laplace Trans-
form) of the function f(n) exists and (9) is a
function g(s), then as a function of the differen- A. General Remarks
tial operator s, g(s) is the operator that is
given by the inverse Laplace transform ,f(t) of Operations research in the most general sense
g(s). Formulas (4) and (7) are indeed typical cari be characterized as the application of
examples of this relation, where the left-hand scient& methods, techniques, and tools to the
side is the usual Laplace transform of the operations of systems SO as to provide those in
right-hand side. In the practical computation control with optimum solutions to problems.
of (5), we cari compute the Laplace inverse This definition is due to Churchman, Ackoff,
transform of the right-hand side. However, if and Arnoff [l]. Operational research began
we took the Laplace transform as the founda- in a military context in the United Kingdom
tion of the theory, it would not only be com- during World War II, and it was quickly taken
plicated but also be subject to the artifïcial up under the name operations research (OR)
restriction caused by the convergence condi- in the United States. After the war it evolved
tion on Laplace transforms. in connection with industrial organization,
In the theory of operational calculus, the and its many techniques have found expand-
transform ing areas of application in the Uniied States,
m the United Kingdom, and in other industrial
dP)=P e-P’f(t)dt (10) countries. Nowadays OR is used widely in
I 0
industry for solving practical problems, such
is sometimesu sed instead of the Laplace trans- as planning, scheduling, inventory, transpor-
form itself. But the difference is not essen- tation, and marketing. It also has various im-
tial; we obtain the latter transform merely by portant applications in the fïelds ojagricul-
1155 307 c
Operations Research
ture, commerce, economics, education, public ming), Markovian decision processes (- 127
service, etc., and some techniques developed in Dynamic Programming, 261 Markov Pro-
OR have influenced other lïelds of science and cesses), and basic tprobability theory are used
technology. to construct models for these problems.
(4) Queuing mode1 (- 260 Markov Chains
H). In a telephone system, calls made when a11
B. Applications
the lines of the system are busy are lost. The
problem of computing the probability of loss
Applications of OR to practical problems are
involved was lïrst solved in the pioneering
often carried out by project teams because
article on tqueuing theory by A. K. Erlang in
knowledge of disparate aspects of the prob-
1917. For systems in which calls cari be put on
lems are required, and interdisciplinary co-
hold when a11 lines are busy, one deals with the
operation is indispensable. The following are
twaiting time distribution instead of the proba-
the major phases of an OR project: (i) formu-
bility of 10s~. In the 1930s F. Pollaczeck and
lating the problem, (ii) constructing a mathe-
A. Ya. Khinchin gave explicit formulas for the
matical mode1 to represent the system under
characteristic function of the waiting time
study, (iii) deriving a solution from the model,
distribution. In many situations, such as in
(iv) testing the mode1 and the solution derived
telephone systems, air and surface traffic,
from it, (v) establishing controls over the solu-
production lines, and computer systems, vari-
tion and putting it to work (implementation).
ous congestion phenomena are often observed,
When the mathematical mode1 that has
and many kinds of queuing models are utilized
been constructed in phase (ii) is complicated
to analyze the congestion. Mathematically,
and/or the amount of data to be handled is
almost a11 such models are formulated by
large, a digital tcomputer is often utilized in
using Markov processes. For practical uses,
phases (iii) and (iv).
approximation and computational methods
are important as well as theoretical results.
C. Mathematical Models [2] (5) Scheduling mode1 (- 376 Scheduling and
Production Planning). Network scheduling is
Typical mathematical models and tools that used to schedule complicated projects (for
appear frequently in OR are: example, construction of buildings) that
(1) Optimization mode1 (- 264 Mathematical consist of a large number of jobs related to
Programming). This mode1 is characterized by each other in some natural order. PERT (pro-
one or more real-valued functions, which are gram evaluation and review technique) and
called objective functions, to be minimized (or CPM (critical path method) are popular com-
maximized) under some constraints. According putational methods for this mode1 (- 281
to the number of objective functions, the types Network Flow Problems). Job shop scheduling
of objective functions, and the types of con- is used when we have m jobs and n machines
straints, this mode1 is classilïed roughly as and each job requires some of the machines in
follows: (i) Single-objective model, which in- a given order. The mode1 allows us to lïnd an
cludes linear, quadratic, convex, nonlinear and optimal order (in some certain sense) of jobs to
integer programming models (- 215 Integer be implemented on each machine.
Programming, 255 Linear Programming, 292 (6) Replacement model. There are two typical
Nonlinear Programming, 349 Quadratic Pro- cases. One is the preventive maintenance model,
gramming); (ii) multi-objective model; (iii) sto- which is suitable when replacements are done
chastic programming mode1 (- 408 Stochastic under a routine policy because a replacement
Programming); (iv) dynamic programming or a repair before a failure is more effective
mode1 (- 127 Dynamic Programming); (v) than after a failure. Probabilistic treatments
network flow mode1 (- 281 Network Flow are mainly used, and this mode1 resembles
Problems). those for queues and tMarkov processes. The
(2) Game-theoretic mode1 (- 173 Game other is a mode1 for deciding whether to re-
Theory). Game theory is a powerful tool for place a piece of equipment in use. In this case,
deriving a solution to practical problems in we need to compare costs of both used and
which more than one person is involved, with new equipment, and evaluations of various
each player having different objectives. types of present cost are important.
(3) Inventory mode1 (- 227 Inventory Con- (7) Simulation. This is a numerical experi-
trol). It is necessary for most lïrms to control ment in a simulated mode1 of a phenomenon
stocks of resources, products, etc., in order to which we want to analyze. Simulation is one of
carry out their activities smoothly; various the most popular techniques in OR.
inventory models have been developed for (8) Other models. Besides the models listed
such problems. Mathematically, optimization above, many problems are formulated by way
techniques (- 264 Mathematical Program- of various other models in OR. In modeling,
307 Ref. 1156
Operations Research
tprobability theory, and mathematical tstatis- weak operator topology, but not with respect
tics, especially, tMarkov chain, tmultivariate to the strong operator topology. The opera-
analysis, tdesign of experiments, tregression tion from 2 x J to &? of taking the product,
analysis, ttime series analysis, etc. often play (A, B)-tAB, is continuous with respect to the
important roles. uniform operator topology, is continuous with
respect to the strong operator topology when
the first factor is restricted to a set bounded in
References
the operator norm, but is not contmuous on
.-A x %. It is continuous with respect to the
[l] C. W. Churchman, R. L. Ackoff, and E. L.
weak operator topology when one of the fac-
Arnoff, Introduction to operations research,
tors is iïxed (i.e., it is separately continuous).
Wiley, 1957.
The set D is a TBanach space with respect to
[2] H. M. Wagner, Princip]es of operations
the operator norm, or, more precisely, a +C*-
research, Prentice-Hall, 1975.
algebra. It is a +locally convex topological
linear space with respect to the strong or weak
operator topology.
The Banach space g is the +dual of the
308 (X11.19) Banach space ~4, of all tnuclear operators in
S, (- 68 Compact and Nuclear Operators 1).
Operator Algebras
The weak* topology in 33 as the dual of %, is
called the <T-weak topology.
A. Preliminaries
Let 5 be a +Hilbert space. The set of +bounded C. Von Neumann Algebras
linear operators on @ is denoted by .S(sj) = J.
It contains the identity operator 1. The notions
A subset ./H of .S is called a *-subalgebra if it
of opera.tor sum A + B, operator product AB,
is a subalgebra (i.e., A, BE.~ implies i.A+pB,
and +adJoint A* are defïned on it. A subalge-
ABE.~) and contains the adjoint ,4* of any
bra of U(sj) is called an operator algebra. In
AEM. The commutant .d’ of a subset .d of Z8
this article we consider mainly von Neumann
is the set of operators that commu1.e with both
algebras. For C*-algebras - 36 Banach Alge-
A and A* for AE.~. The commutant is a *-
bras G-K.
subalgebra, and .d’ = SZZ”‘.
Any tHermitian operator A (i.e., an operator
A von Neumann algebra .,&! is a =-subalgebra
such that A = A*) has the property that (Ax, x)
of %Yt hat is delïned by one of the follow-
is always real for any XE$~. If (Ax,x)>O for
ing four equivalent conditions: (i) ,@ is a *-
any x, A is called positive, and we Write A 3 0.
subalgebra of 8 containing 1, closed under
When Hermitian operators A and B satisfy A
the weak operator topology; (ii) & is a *-
~ B 3 0, we Write A > B. Thus we introduce an
subalgebra of Ua, containing 1, closcd under
ordering A > B between Hermitian operators.
the strong operator topology; (iii) .&Y is the
A set {A,) of positive Hermitian operators is
commutant of a subset of 9 (or, equivalently,
said to be an increasing directed set if any two
.4? = &“‘); (iv) .4? is a *-subalgebra of .%Yc on-
of them A,, A, always have a common major-
taining Z, closed under the uniform operator
ant A,, that is, A,< A, and A, < A,. If a Her-
topology, and, as a Banach space, coinciding
mitian operator A satisfies (Ax, x) = sup(A,x, x)
with the conjugate space of some Banach
for such a set, it is called the supremum and is
space. Note that a *-subalgebra of g, closed
denoted by sup A,. The supremum sup A,
under the uniform operator topology, is a
exists if and only if the sup(A,x, x) is finite for
C*-algebra. Von Neumann a1gebra.s are also
any x, and then A, converges to A with respect
called rings of operators or W*-algebras. The
to the weak and strong operator topologies.
latter term is usually used for a C*-.algebra
*-isomorphic to a von Neumann algebra in
B. Topologies in .#I contrast to a concrete von Neumann algebra.
The study of these algebras was :started by J.
Various topologies are introduced in % = von Neumann in 1929. He showed the equiva-
.a(%): the +uniform operator topology, the 1 lente of conditions (i)-(iii) (von Neumann%
+strong operator topology, and the tweak 1 density theorem), and established a foundation
operator topology (- 251 Linear Operators). for the theory named after him [l] The notion
These topologies are listed above in order of of von Neumann algebras cari be regarded as
decreasing îïneness. The operation in % of a natural extension of the notion oFmatrix
taking the adjoint, A-t A*, is continuous with algebras in a lïnite-dimensional space, and
respect to the uniform operator topology and therein lies the importance of the theory. The
1157 308 D
Operator Algebras
fourth condition of the defmition was given by plete description of a11 possible normal repre-
S. Sakai (Pacifie J. Math., 1956). sentations (i.e., a normal *-homomorphism
The following theorem is of use in the the- into some B) of a von Neumann algebra.
ory of von Neumann algebras: Given a *-
subalgebra d of LB containing 1, its closure &
with respect to the weak or strong operator
D. States, Weigbts, and Traces
topology is von Neumann algebra; and when
we denote the set of elements of operator norm
< 1 in d (resp. J&‘) by ~2, (resp. &ZJ, A1 is A state <p of a C*-algebra & is a complex-
likewise the closure of ~2, with respect to the valued function on ~-4 that is (1) complex
weak or strong operator topology (Kaplan- linear: q(A+B)=cp(A)+cp(B), ~(CA)=~(A)
sky’s density tbeorem (Ann. Math., 1951). for A, BE.&, C~C, (2) positive: q(A*A)>O for
If E is a projection operator in a von Neu- AE &, and (3) normalized: 11cp \( = 1 (equiva-
mann algebra 4, then EJE = {EAE 1A Idi’} lent to ~(1) = 1 if I E &). For any positive
is a *-subalgebra of %? closed with respect to linear functional <p on d, there exists a triplet
the weak operator topology. It is not a von (sj., xc, 5,) (unique up to the unitary equiva-
Neumann algebra because it does not contain lente) of a Hilbert space &,, a representa-
I, but since its elements operate exclusively in tion ne (i.e., a *-homomorphism into B(&,,))
the closed subspace E!& we cari regard it as of LX~, and a vector 5, in a,+, such that <p(A) =
an algebra of operators’on E!-j. In this sense, (n,(A)&, 5,) and sj, is the closure of K,(&‘)&.
EdE cari be regarded as a von Neumann The space $J,,, is constructed by detïning the
algebra on E4j, which we cal1 the reduced von inner product (q(A), q(B)) = @?*A) in the
Neumann algebra of &’ on ES, and Write J?~. quotient of J& by its left ideal {Alq(A*A)=
If E is a projection operator in A’, EkfE = 0}, where q is the quotient mapping, and by
Ed restricted to the subspace E$j is called completion. Then Z, is defined by n,(A)@) =
the induced von Neumann algebra of & on q(AB). This is called the GNS construction
Etij and is denoted also by dE. In the latter after its originators 1. M. Gel’fand, M. A.
case, the mapping A E J%’ + EA E &fE is a *- Naïmark, and 1. E. Siegel.
homomorphism and is called the induction of A weigbt <p on a von Neumann algebra Jz’ is
dl ont0 AE. a function deiïned on the positive elements of
The tensor product 5jj, @ & of two Hilbert A, with positive real or infinite values, which
spaces Gi (i = 1,2) is the tcompletion of their is additive and homogeneous (q(A + B) =
ttensor product as a complex linear space q(A)+rp(B) and &A)=~~I(A) for a11 A, BE
equipped with the unique tinner product satis- .,&’ and I > 0 with the convention 0. CO = 0 and
fying Ch Q.Lgl Og,)=(f,,g,),(f,,g,)* for cc + a = CO). It is said to be faitbful if it does
a11 fi, gis!&. For any AicB(Sji), there exists not vanish except for rp(O)=O, normal if q(A)
a unique operator in a(!& 0 !&) denoted = sup <p(A,) whenever A, is an increasing net
by A, 0 A, satisfying (A, 0 A,)(f, 0 fi) = of positive elements of J%’ and A = sup A,, and
A,f, @ A2f2 for a11 fic!&. For von Neumann semihite if the left ideal ‘!Rq, consisting of a11
algebras Ai c %(!&), the von Neumann alge- elements A~dt’ for which ‘p(A*A) is finite, has
bra generated by A, Q A, with Ai~Mi is de- the property that the linear span !IX, of %m%,
noted by A1 0 .,+Y2a nd is called the tensor is dense in A. The restriction of cp to positive
product of JZ!, and .L&. The *-isomorphism AE elements of W, has a unique extension to a
&+A @ 1 E & @ 1, where 1 is the trivial von linear functional on !JJl,, which we denote by
Neumann algebra consisting solely of complex the same letter <p. Canonically associated with
multiples of the identity operator, is called an a normal semilïnite weight <p, there exists a
amplification. Hilbert space Ç&, a normal *-representation z,+,
For two von Neumann algebras .Jz’~c @ji) of &!, and a complex linear mapping q from
(i = 1,2), a *-isomorphism n from A1 into &Y2 ‘9&, into a dense subset of $j, such that (q(B’),
is called spatial if there exists a unitary (i.e., a @))=cp(B*B), and q&A)t@)=q(AB), where
bijective isometric linear) mapping U from !& B, B’E’%, and AE 4. If cp is finite (i.e., ‘!R,,, =
to $j2 such that UAU*=nA for a11 Ae.M1, A), then its extension to &’ is a positive linear
and a *-homomorphism R is called normal if functional for which the triplet (!$,, z,,,, q( 1))
supanA, = ~(sup, AJ whenever A, is a bounded is given by the GNS construction.
increasing net in A. Any normal *-homomor- The linear span of a11 normal states of a von
phism is continuous in the strong and weak Neumann algebra A is a norm-closed sub-
operator topologies and is a product of an space of its dual A*, called its predual and
amplification, a spatial *-isomorphism, and an denoted by J&, because 4’ turns out to be the
induction. Its kernel is of the form E&i!, where dual of A.+.
E is a projection operator belonging to the A trace t on a von Neumann algebra .4? is a
tenter ZJ‘= J?! n 4’ of 4. This gives a com- weight satisfying t(UAU*) = t(A) for U unitary
308 E 1158
Operator Algebras
in A and for a11 positive A in A (equivalently, finite), II, (i.e., of type 11 and not fïnite), and
t(A*A)= t(AA*) for a11 A EA). III. A factor of type 1, is isomorphic to the
algebra B(5) of a11 bounded operators on an
n-dimensional Hilbert space sj. Since the dis-
E. The von Neumann Classification
covery of two nonisomorphic examples of
factors of type II, by F. J. Murray and von
A von Neumann algebra for which a semi-
Neumann (1943), classification of factors has
fmite normal trace does not exist is called a
been a central problem in the theory of von
purely infinite von Neumann algebra or von
Neumann algebras. After 1967, great progress
Neumann algebra of type III. In contrast to
was made in the investigation of isomorphism
this, a von Neumann algebra & is called
classes of factors, and we have uncountably
semifinite (resp. finite) if for each positive Her-
many nonisomorphic examples of factors of
mitian operator A (#O) in .I there is a semi-
types II,, II,, and III. After the discovery of
fïnite (resp. tïnite) normal trace t such that
the third to ninth nonisomorphic examples of
t(A)#O. Every Abelian von Neumann algebra
factors of type II, by J. Schwartz (1963) (the
is lïnite. If there are no central projection oper-
third example, 1963), W. Ching (the fourth),
ators E #O such that &E is finite, A is called
Sakai (the fïfth), J. Dixmier and E. (3. Lance
properly infinite. Purely infinite &? and .%(4j)
(the sixth and seventh), and G. Zeller-Meier
for inlïnite 5j, for example, are properly in-
(the eighth and ninth), D. McDuff showed that
fmite. A nonzero projection operator E in 4 is
there exist countably many nonisomorphic
called Abelian when &E is Abelian. We cal1 &
examples of factors of type II 1, and lïnally
a von Neumann algebra of type 1 (or discrete)
McDuff (Ann. Math., 1969) and Sakai (J. Func-
when it contains an Abelian projection E for
tional Anal., 1970) showed the existN:nce of
which 1 is the only central projection P cover-
uncountably many nonisomorphic examples
ing E (i.e., E < P). A von Neumann algebra is
of factors of type II,. For type III factors -
of type II if it is semitïnite and contains no
Section 1.
Abelian projection. A von Neumann algebra
of type 1 t is called of type II 1 if it is fïnite and
of type II, if it is properly infinite. A fïnite von
G. Tbe Integral Direct Sum and
Neumann algebra is also characterized as a
Decomposition into Factors
von Neumann algebra in which the operation
of taking the adjoint is continuous with re-
The Hilbert spaces considered in this section
spect to the strong operator topology on
are a11 tseparable. Let (‘%R,d , p) be a tmeasure
bounded spheres (Sakai, Proc. Japan Acad.,
space; with each [E~JI we associate a Hilbert
1957). A properly infinite von Neumann alge-
space $j([). We consider functions x(5) on 9JI
bra d is characterized by the property that
whose values are in 5j([) for each <. Let K be
.M and J&? @ g(sj) for any separable 5j is *-
a set of these functions having the following
isomorphic.
properties: (i) Ilx([)ll is measurable for x([)EK;
Given a von Neumann algebra A, there
(ii) if for a function y([), the numerical func-
exist mutually orthogonal projections E,, E,,,
tion (x(c), y([)) is measurable any x(LJEK,
E,,, in the tenter d of .M such that E, + E,, +
then yak; (iii) there is a countable family
E,,, = 1, and AE,, AE,,, ME,,, are von Neumann
{x1 (<), x,(l), . . } of functions in K such that
algebras of type 1, type II, type III, respec-
for each fixed [c!IR, the set {xl([),xz([), . ..} is
tively. This decomposition is unique. There
dense in b(c). Then K is a linear space. We cal1
also exist unique central projections E, and
each function in K a measurable vector func-
Ei such that -AE, is f-mite, AE, is properly in-
tion. We introduce in the set of measurable
finite, and Es + Ei = 1. The two decompositions
vector functions x(i) with
cari be combined. (If some of the projections E
are 0, the condition on the corresponding .AE
is to be waived.)
an equivalence relation by delïning x(c) and
F. Factors y([) as equivalent when
A von Neumann algebra whose tenter consists
exclusively of scalar multiples of the identity
operator is called a factor. Von Neumann’s Thus we obtain a space of equivalence classes
reduction theory (- Section G) reduces the which we denote by $. $5 is a Hilbert space
study of arbitrary von Neumann algebras on w(ithy xt=he ),i(nn e rx yprod(u4c(t ii)4)>)0
s
a separable Hilbert space more or less to the
study of factors. Factors are classified into
types 1, (n = CO, 1,2,. ), II, (i.e., of type II and
