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176. CJM 1998 (vol 50 pp. 323)

Dykema, Kenneth J.; Rørdam, 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 non-unital 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. 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 norm-one 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 norm-one 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

178. 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

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 UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra 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 AF-algebra. 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. 160)

Rieffel, Marc A.
The Classical Limit of Dynamics for Spaces Quantized by an Action of ${\Bbb R}^{\lowercase{d}}$
We have previously shown how to construct a deformation quantization of any locally compact space on which a vector group acts. Within this framework we show here that, for a natural class of Hamiltonians, the quantum evolutions will have the classical evolution as their classical limit.

Categories:46L60, 46l55, 81S30

181. 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
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