Internat. J. Math. & Math. Sci.
VOL. II NO. 2
(1988)
401-402RESEARCH NOTES A CONVEX OPERATOR FUNCTION
401
DERMING
WANGDepartment of Mathematics and Computer Science California State University, Long Beach
Long Beach, California 90840
(Received September 23,
1987)
ABSTRACT. It is shown that inversion is a convex function on the set of strictly positive elements of a
C*-algebra.
KEY
WORDSAND
PHRASES. Con:,e.x function,C*-alqebra.
1980 S[BJECT CLASSIFICATION CODE. 47C15
1. INTRODUCTION.
A real-valued function
f
defined on a real interval issaid to be convex if f(ks+
(1 k)t)<
kf(s)+
(1 k)f(t)for s,t61 and 0<kK1. Convex functions play a fundamental role in the study of the Lebesgue
L
p spaces [1], [2]. Geometrically, a functionf
is convex if the chord joining the points (s,f(s)) and (t,f(t)) lies above the graph off. An
interesting example of a convex function is the function f(t)=t-I,
t6I=(0,). Thus inversion is a convex function on the set of positive reals. The notion of convexity has been generalized to functions with domain and range more general than reals.For
instance, through a diagonalization process it is shown in [3] that inversion is a convex function on the set of positive-definite real symmetric matrices.In
this note we will show that this result holds in a C’-ilgebra.More
precisely, we use Banach algebra techniques to show that inversion is a convex function on the set of strictly positive elements of a C’-algebra.2.
PRELIMINARIES.
Throughout this article t will denote a complex C’-algebra with identity e. An element x6.A is said to be self-adjoint if
x’=x,
where x* is the adjoint of z. A self-adjoint element z is said to be non-negative, in notation x0, if its spectrum a(x) lies in the interval [0,o).For
self-adjoint elements x and y, we write x Ky if y--x >0.An
element x will be termed strictly positive if it is non-negative and invertible. Thus x is strictly positive if x is self- adjoint and r(z) lies in the interval (0,oo). If c is an invertible element then we use x-t to denote its inverse.A subalgebra
B
of t is said to be self-adjoint if x 6 impliesx’6B.
The main tools we need to establish our result are:(A) If
B
is a closed self-adjoint subalgebra of t and z 6B, thencry(x)=cry(x).
Herer,BCx)
and;t(c}
denote thespectra o[ x relative to and t, respectively.402 D. WANG
(B) If
B
is a commutative Banach algebra andx
thencr(x)--{9(x)l
a complexhomomorphism on ).
Proofs of (A) and (B) may be found, for example, in [4].
3.
MAIN
RESUI,T.I.EMMA:
If u: is a strictly positive element of t, then[ke+(l --k)w]-t <ke+(l --k)u"-I for 0<k<l.
PROOF: Let
B
be the closed subalgebra generated by w and e. Since w is self-adjoint,B
is self-adjoint and commutative. Clearly w and ke4-(I--k)w are elements of B. Since these elementsby (A). Ourare invertiblegoal is to show thatin
., u--[Xe4-(1--k)w]-: aB(V-
u) lies inand v--- ke[O,c).In
view4-(1-- k) wof (B)-
itaresuffices toelements showof B that p(u.)K(v) for complex homomorphismso
on.
Since p(u) [k+
(1-- k)(w)]-
andp(v)---k+(1 --k)((w))
-t,
the result follows from the fact that f(t)--t-
is a convex functionon (0,_-,:.).
THEOREM:
If x and y are strictly positive elements oft, then [kx+(1 --k)y]- <:hx-:+(1 --k)y-
for 0<<_:1.
PROOF: First we recall that if p and q are self-adjoint elements of t with PKq, then
r*pr
r*qr for any r6t. This fact from C*-algebra theory will be used twice in the proof.Now,
since x is strictly positive, it possesses a unique strictly positive square root, say z, in.
Thenw--z-yz - is strictly positive. By the lemma, we have
[ke
-v
(1--k) w]- <:
ke+
(1-- k)w-
Thus
z
-
[he4-(1h)w]-:
z- z-:[he+
(1 h)w-llz -
This in turn gives
The proof is thus complete.
[kx+(1--h)y]-:
Khx-
+(l-h)y-:
REFERENCES
[1] Royden,
H.L.,
Real Analysis, 2nd ed., Macmillan,New
York, 1968.[2] Rudin,
W.,
Real and Complex Analysis, 2rid ed., McGraw-Hill, 1974.[3]
Moore, M.H.,
A convex matrix function, Amer. Math. Monthly, 80 (1973) 408-409.[4] Douglas,
R.G.,
Banach Algebra Techniques in Operator Theory, AcademicPress,
New
York, 1972.Boundary Value Problems
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