176. CJM 1998 (vol 50 pp. 323)
 Dykema, Kenneth J.; RĂ¸rdam, Mikael

Purely infinite, simple $C^\ast$algebras arising from free product constructions
Examples of simple, separable, unital, purely infinite
$C^\ast$algebras are constructed, including:
\item{(1)} some that are not approximately divisible;
\item{(2)} those that arise as crossed products of any of a certain class of
$C^\ast$algebras by any of a certain class of nonunital endomorphisms;
\item{(3)} those that arise as reduced free products of pairs of
$C^\ast$algebras with respect to any from a certain class of states.
Categories:46L05, 46L45 

177. CJM 1997 (vol 49 pp. 1188)
 Leen, Michael J.

Factorization in the invertible group of a $C^*$algebra
In this paper we consider the following problem:
Given a unital \cs\ $A$ and a collection of elements $S$ in the
identity component of the invertible group of $A$, denoted \ino,
characterize the group of finite products of elements of $S$. The
particular $C^*$algebras studied in this paper are either
unital purely infinite simple or of the form \tenp, where $A$ is
any \cs\ and $K$ is the compact operators on an infinite dimensional
separable Hilbert space. The types of elements used in the
factorizations are unipotents ($1+$ nilpotent), positive invertibles
and symmetries ($s^2=1$). First we determine the groups of finite
products for each collection of elements in \tenp. Then we give
upper bounds on the number of factors needed in these cases. The main
result, which uses results for \tenp, is that for $A$ unital purely
infinite and simple, \ino\ is generated by each of these collections
of elements.
Category:46L05 

178. CJM 1997 (vol 49 pp. 1242)
 Randrianantoanina, Beata

$1$complemented subspaces of spaces with $1$unconditional bases
We prove that if $X$ is a complex strictly monotone sequence
space with $1$un\con\di\tion\al basis, $Y \subseteq X$ has no bands
isometric to $\ell_2^2$ and $Y$ is the range of normone projection from
$X$, then $Y$ is a closed linear span a family of mutually
disjoint vectors in $X$.
We completely characterize $1$complemented subspaces and normone
projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q <
\infty$.
Finally we give a full description of the subspaces that are spanned
by a family of disjointly supported vectors and which are
$1$complemented in (real or complex) Orlicz or Lorentz sequence
spaces. In particular if an Orlicz or
Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p <
\infty$ then the only subspaces
of $X$ which are $1$complemented and disjointly supported are the
closed linear spans of block bases with constant
coefficients.
Categories:46B20, 46B45, 41A65 

179. CJM 1997 (vol 49 pp. 963)
 Lin, Huaxin

Homomorphisms from $C(X)$ into $C^*$algebras
Let $A$ be a simple $C^*$algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHFalgebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHFalgebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AFalgebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 

180. CJM 1997 (vol 49 pp. 100)
 Lance, T. L.; Stessin, M. I.

Multiplication Invariant Subspaces of Hardy Spaces
This paper studies closed subspaces $L$
of the Hardy spaces $H^p$ which are $g$invariant ({\it i.e.},
$g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If
$p=2$, the Wold decomposition theorem implies that there is
a countable ``$g$basis'' $f_1, f_2,\ldots$ of
$L$ in the sense that $L$ is a direct sum of spaces
$f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$.
The basis elements $f_j$ satisfy the
additional property that $\int_T f_j^2 g^k=0$,
$k=1,2,\ldots\,.$ We call such functions $g$$2$inner.
It also
follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot
(F_2\circ g)$ where $h_{f,2}$ is $g$$2$inner and $F$ is
outer, generalizing the classical Riesz factorization.
Using $L^p$ estimates for the canonical decomposition of
$H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ
g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite
Blaschke product we obtain, for any $g$invariant
$L\subseteq H^p$, a finite $g$basis of $g$$p$inner
functions.
Categories:30H05, 46E15, 47B38 

181. CJM 1997 (vol 49 pp. 160)