Download (*) f + fog-1 - American Mathematical Society

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 52, October 1975
ON FUNCTIONAL EQUATIONSRELATED TO MIELNIK'S
PROBABILITY SPACES
C. F. BLAKEMORE AND C. V. STANOJEVIC
ABSTRACT. It is shown that the method used by C. V. Stanojevic
to obtain a characterization
Mielnik probability
space
tion to dimension
n > 2.
Let
of inner product
spaces
in terms of a
of dimension
2 does not admit a generaliza-
/: [0, 2] —>[0, l] be continuous
and strictly
= 0 and f(2) = 1. The class of all such functions
Likewise,
let
g: [0, 2] —» [0, 2] be continuous
increasing
with
/ will be denoted by E.
but strictly
decreasing
g(0) = 2 and g(2) = 0. Similarly, the class of all such functions
denoted
by G.
In [l] it is proved
(*)
/(0)
that the functional
with
g will be
equation
f + fog-1
where
(f°g)(t)
volution,
i.e.,
this result
product
ability
= f[g(t)]
g°g
= e where
it is also
space
shown
2.
to the case
appropriate
function
g
n is some positive
of inner product
this
g
integer.
We shall
prob-
as a
In this
note
of inner prod-
generated
g: I —>/ where
e is the identity
show
(*) served
by an
> 2.
of a function
= e where
is a Mielnik
spaces.
function
Using
N is an inner
characterization
p is a probability
m iterations
suppose
on [0, 2].
space
equation
/ and (S, p) is of dimension
denote
Also,
linear
f £ F, (S, f(\x + y\))
of extending
where
function
real
The functional
a new characterization
the possibility
uct spaces
Let
e is the identity
that a normed
[2] of dimension
we consider
/ £ F if and only if g £ G is an in-
if and only if for some
space
tool to obtain
interval.
has a solution
/ is some
function
that the generalized
on / and
functional
equation
(**)
/ + /og
(where
classes
/ and
g are functions
E and
from [1] cannot
G defined
be extended
(S, p) is of dimension
homeomorphisms
+ /og(2)+-..+
>2.
/og("-1)=l
belonging
earlier)
to a suitable
collapses.
In other
in a straightforward
The following
generalization
manner
theorem
words,
of the
the method
to the case
(for a similar
when
result
for
see [3]) is the key to our result:
Received by the editors July 16, 1974.
AMS (A/OS) subject classifications
(1970).
Primary 39A15, 26A18; Secondary
46C10.
Key words
and phrases.
Functional
equations,
Mielnik
probability
spaces.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Copyright © 1975, American Mathematical
315
Society
316
C. F. BLAKEMORE AND C. V. STANOJEVIC
Theorem.
continuous
Let
h: I —►/ be a function
and if for some
m > 2, h
where
= e, then
I is an interval.
If h is
h is an involution,
i.e.,
h°h = e.
Proof.
h
hix) = hiy).
(y) implies
tinuous,
Then
Next
< y, then
First
Hence,
we consider
the case
Hence
where
and zr Kx) < h( \y).
that
Hence
hr
e.
h
A is an involution.
Therefore
In particular,
our generalized
becomes
n odd.
zr
= e and
our theorem
functional
nif + f°g)/2
=1
Now if we want
case
we have
[2].
This
to have
shows
the procedure
that
shows
equation
(**)
there
and
in + l)//2
since
x in / and
is strictly
If x
increasing.
to h(2) we get
g: I —» / appearing
Thus
+ Un - l)/2)f°g
in
(**)
= 1 for
from [l] to the re-dimensional
it is equivalent
is not a trivial
also
decreasing.
the first case
the function
the result
h is
'ix) >
(**) must be an involution.
for zz even
to extend
that
h is con-
where
x = h
hix) = x for all
But (A(2))(m) = ih{m)){2) = e(2)=
= e.
since
=
The contradiction
h is strictly
Applying
h(m)ix)
the case
is a contradiction.
hix) < x.
we consider
hix) > hiy)
h{m) = e, we have
from hix) > x it follows
hix) > x which
from the assumption
= e.
since
h is one-to-one.
monotone.
increasing.
ix) >...>
follows
Then,
x = y and thus
h is strictly
strictly
h
h°h
Let
to Axiom (C) of Mielnik
extension
to dimension
n using
from [l].
REFERENCES
1.
C. V. Stanojevic,
Mielnik's
probability
spaces
and characterization
product spaces, Trans. Amer. Math. Soc. 183 (1973), 441—448.
2. B. Mielnik, Geometry of quantum states, Comm. Math. Phys.
80.
of inner
9 (1968), 55 —
MR 37 #7156.
3.
N. McShane,
On the periodicity
Math. Monthly 68 (1961), 562-563.
of homeomorphisms
of the rent line,
Amer.
MR 24 #A199.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEW ORLEANS, NEW ORLEANS,
LOUISIANA 70122
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI AT ROLLA, ROLLA,
MISSOURI65401
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use