CONTRACTIVE PERTURBATIONS IN C*-ALGEBRAS

1. INTRODUCTION
The geometric structure of the unit ball of a C-algebra has been an object
of interest from the beginning of the theory of operator algebras. In his study of
isometries between C-algebras R. Kadison characterized the extreme points of
the unit ball of a unital C-algebra [17]. From his result it follows that the extreme
points of the unit ball of B(H) are the isometries and the co-isometries.
In [4], the n-th contractive perturbations of a subset S of the unit ball of a
C-algebra A, denoted cpn(S), were introduced. An element a in the unit ball
of A is an extreme point precisely when cp2(a) def = cp2(fag) is the whole unit
ball. On the other hand, if H is a Hilbert space, a bounded contraction A on H is a compact (respectively finite rank) operator if and only if cp2(A) is a compact
(respectively finite dimensional) subset of B(H). The main result of [4] is that if
a is an element of the unit ball of a C-algebra A then the image of a under some
faithful representation of A is compact if and only if cp2(a) is compact.
54 M. ANOUSSIS, V. FELOUZIS AND I.G. TODOROV
In [5] and [6] the contractive perturbations are used in the study of compact
and finite rank operators in a nest algebra and in the algebra of adjointable operators
on a Hilbert module, and in [18] in the study of the facial structure of the unit
ball of an operator algebra. An analogous approach is used in [3] and [13] to give
a geometric characterization of partial isometries in a C-algebra and of tripotents
in a JB*-triple. The set cp1(S) is also related to the notion of M-orthogonality introduced
and studied in [12].
In this work we extend the study of contractive perturbations in several
directions. We are interested in characterizing various objects in C-algebras in
terms of the size and the location of the contractive perturbations. In Section 3 we
examine the joint second contractive perturbations of a precompact subset S of
the unit ball of a C-algebra A and show that there exists a faithful representation
p of A such that p(a) is compact for each a 2 S if and only if cp2(lS) is compact,
for each 0 < l < 1. This result extends Theorem 2.2 of [4]. As a corollary, we
obtain a characterization of compact elementary operators.
In Section 4 we provide a geometric characterization of the hereditary C-
subalgebras and the essential ideals of a C-algebra A. We also show, using a
result of L. Brown, that if A is a separable C-algebra and M(A) its multiplier
algebra, then an element a of the unit ball of M(A) belongs to A if and only if
cp2(a) is separable.
In Section 5 we show that a compact face of the unit ball of a C-algebra is
necessarily finite dimensional. We also determine the affine hull of the smallest
face containing a fixed element of the unit ball of a C-algebra, in the case it is
finite dimensional.
It is natural to ask if the notion of contractive perturbations may be used to
describe compact operators on a general Banach space X. In Section 6 we present
some examples which show that this cannot be achieved. We show that if X is
any of the spaces lp, 1 6 p < +¥, p 6= 2, c0 or C(K), then there exists a rank one
contraction A acting on X such that cp2(A) is not compact. We also show that
if A is a compact contraction on c0, then cp2(A) is compact in the weak operator
topology, but not vice versa.
We introduce some notation. If X is a Banach space we will denote by X1
the closed unit ball of X. If S  X, we denote by [S] the linear span of S and by
S the closure of S.
If H is a Hilbert space, B(H) denotes the C-algebra of all bounded linear
operators on H and K(H) the closed ideal of all compact operators. For A 2
B(H) we denote by jAj the square root of AA. If P is an (orthogonal) projection
on H, we set P? = I