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126. CJM 2001 (vol 53 pp. 592)

Perera, Francesc
 Ideal Structure of Multiplier Algebras of Simple $C^*$-algebras With Real Rank Zero We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mul$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mul$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mul$ is reflected in the fact that $\mul$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion. Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoidCategories:46L05, 46L80, 06F05

127. CJM 2001 (vol 53 pp. 631)

Walters, Samuel G.
 K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$ (containing the rationals) it is shown that the group $K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where $A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) = U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense $G_\delta$ set of parameters $\theta$. The main computational tool in this paper is a group homomorphism $\vtr \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of $H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$ is fully determined for each $\theta$. (We conjecture that this subspace is all of $H^{\ev}$.) Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern charactersCategories:46L80, 46L40, 19K14

128. CJM 2001 (vol 53 pp. 546)

Erlijman, Juliana
 Multi-Sided Braid Type Subfactors We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II$_1$ factor $R$ in [E1]---which arise by considering unitary braid representations with certain properties---to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples---where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A$, $B$, $C$, $D$. We also compute the (first) relative commutants. Category:46L37

129. CJM 2001 (vol 53 pp. 325)

Matui, Hiroki
 Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems Giordano, Putnam and Skau showed that the transformation group $C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C(X)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result. Categories:46L40, 46L80, 54H20

130. CJM 2001 (vol 53 pp. 355)

Nica, Alexandru; Shlyakhtenko, Dimitri; Speicher, Roland
 $R$-Diagonal Elements and Freeness With Amalgamation The concept of $R$-diagonal element was introduced in \cite{NS2}, and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to $R$-diagonality, which relies on freeness with amalgamation. The class of $R$-diagonal elements is enlarged to contain examples living in non-tracial $*$-probability spaces, such as the generalized circular elements of \cite{Sh1}. Category:46L54

131. CJM 2001 (vol 53 pp. 51)

Dean, Andrew
 A Continuous Field of Projectionless $C^*$-Algebras We use some results about stable relations to show that some of the simple, stable, projectionless crossed products of $O_2$ by $\bR$ considered by Kishimoto and Kumjian are inductive limits of type I $C^*$-algebras. The type I $C^*$-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional $C^*$-algebras. Categories:46L35, 46L57

132. CJM 2001 (vol 53 pp. 161)

Lin, Huaxin
 Classification of Simple Tracially AF $C^*$-Algebras We prove that pre-classifiable (see 3.1) simple nuclear tracially AF \CA s (TAF) are classified by their $K$-theory. As a consequence all simple, locally AH and TAF \CA s are in fact AH algebras (it is known that there are locally AH algebras that are not AH). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple TAF \CA s with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and $K_0 (A)_+$ is locally finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is the UHF-algebra with the rational $K_0$. Classification results (with restriction on $K_0$-theory) for the above \CA s are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple TAF \CA s with the unique normalized trace satisfying the UCT and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational (scaled ordered) $K_0$, then $A \cong B$. Similar results are also obtained for some cases in which $K_0$ is non-divisible such as $K_0(A) = \mathbf{Z} [1/2]$. Categories:46L05, 46L35

133. CJM 2000 (vol 52 pp. 1164)

 Perforated Ordered $\K_0$-Groups A simple $\C^*$-algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition---induced by addition of orthogonal projections---form the additive semi-group $$\{0,2,3,\dots\}.$$ (This is a particularly simple instance of the phenomenon of perforation of the ordered $\K_0$-group, which has long been known in the commutative case---for instance, in the case of the four-sphere---and was recently observed by the second author in the case of a simple $\C^*$-algebra.) Categories:46L35, 46L80

134. CJM 2000 (vol 52 pp. 920)

Evans, W. D.; Opic, B.
 Real Interpolation with Logarithmic Functors and Reiteration We present reiteration theorems'' with limiting values $\theta=0$ and $\theta = 1$ for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in [D]. Keywords:real interpolation, broken-logarithmic functors, reiteration, weighted inequalitiesCategories:46B70, 26D10, 46E30

135. CJM 2000 (vol 52 pp. 999)

Mankiewicz, Piotr
 Compact Groups of Operators on Subproportional Quotients of $l^m_1$ It is proved that a typical'' $n$-dimensional quotient $X_n$ of $l^m_1$ with $n = m^{\sigma}$, $0 < \sigma < 1$, has the property $$\Average \int_G \|Tx\|_{X_n} \,dh_G(T) \geq \frac{c}{\sqrt{n\log^3 n}} \biggl( n - \int_G |\tr T| \,dh_G (T) \biggr),$$ for every compact group $G$ of operators acting on $X_n$, where $d_G(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of $X_n$. Several consequences of this estimate are presented. Categories:46B20, 46B09

136. CJM 2000 (vol 52 pp. 849)

Sukochev, F. A.
 Operator Estimates for Fredholm Modules We study estimates of the type $$\Vert \phi(D) - \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D - D_0 \Vert^{\alpha}, \quad \alpha = \frac12, 1$$ where $\phi(t) = t(1 + t^2)^{-1/2}$, $D_0 = D_0^*$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $\calM$, $D - D_0$ is a bounded self-adjoint linear operator from $\calM$ and $(1 + D_0^2)^{-1/2} \in \emt$, where $\emt$ is a symmetric operator space associated with $\calM$. In particular, we prove that $\phi(D) - \phi(D_0)$ belongs to the non-commutative $L_p$-space for some $p \in (1,\infty)$, provided $(1 + D_0^2)^{-1/2}$ belongs to the non-commutative weak $L_r$-space for some $r \in [1,p)$. In the case $\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this result continues to hold under the weaker assumption $(1 + D_0^2)^{-1/2} \in \calC_p$. This may be regarded as an odd counterpart of A.~Connes' result for the case of even Fredholm modules. Categories:46L50, 46E30, 46L87, 47A55, 58B15

137. CJM 2000 (vol 52 pp. 789)

Kamińska, Anna; Mastyło, Mieczysław
 The Dunford-Pettis Property for Symmetric Spaces A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that $\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence spaces with the Dunford-Pettis property, and that in the class of symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only spaces with the Dunford-Pettis property are $L^1$, $L^{\infty}$, $L^1 \cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 + L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1 \cap L^{\infty}$ in $X$. It is also proved that all Banach dual spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces $(L^1 + L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some K\"othe-Bochner spaces. Categories:46E30, 46B42

138. CJM 2000 (vol 52 pp. 633)

Walters, Samuel G.
 Chern Characters of Fourier Modules Let $A_\theta$ denote the rotation algebra---the universal $C^\ast$-algebra generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where $\theta$ is a fixed real number. Let $\sigma$ denote the Fourier automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{-1}$, and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote the associated $C^\ast$-crossed product. It is shown that there is a canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each $\theta$ given by nine canonical modules. The unbounded trace functionals of $B_\theta$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when $\theta$ is irrational. The Chern characters of the nine modules---and more importantly, the Fourier module---are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense $G_\delta$ set of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is generated by the nine classes constructed here. Keywords:$C^\ast$-algebras, unbounded traces, Chern characters, irrational rotation algebras, $K$-groupsCategories:46L80, 46L40

139. CJM 1999 (vol 51 pp. 745)

Echterhoff, Siegfried; Quigg, John
 Induced Coactions of Discrete Groups on $C^*$-Algebras Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a $C^*$-coaction $\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a discrete group $G$ to a $C^*$-coaction $\Ind\delta\colon\Ind D\to \Ind D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions. Category:46L55

140. CJM 1999 (vol 51 pp. 850)

Muhly, Paul S.; Solel, Baruch
 Tensor Algebras, Induced Representations, and the Wold Decomposition Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over $C^{*}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras. Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theoremCategories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35

141. CJM 1999 (vol 51 pp. 566)

Ferenczi, V.
 Quotient Hereditarily Indecomposable Banach Spaces A Banach space $X$ is said to be {\it quotient hereditarily indecomposable\/} if no infinite dimensional quotient of a subspace of $X$ is decomposable. We provide an example of a quotient hereditarily indecomposable space, namely the space $X_{\GM}$ constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we provide an example of a reflexive hereditarily indecomposable space $\hat{X}$ whose dual is not hereditarily indecomposable; so $\hat{X}$ is not quotient hereditarily indecomposable. We also show that every operator on $\hat{X}^*$ is a strictly singular perturbation of an homothetic map. Categories:46B20, 47B99

142. CJM 1999 (vol 51 pp. 309)

Leung, Denny H.; Tang, Wee-Kee
 Symmetric sequence subspaces of $C(\alpha)$, II If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha)$ be the space of all continuous real-valued functions defined on the ordinal interval $[0, \alpha]$. We characterize the symmetric sequence spaces which embed into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy $(E_\alpha)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, $E_\alpha$ embeds into $C(\omega^{\omega^\alpha})$, but does not embed into $C(\omega^{\omega^\beta})$ for any $\beta < \alpha$. Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12

143. CJM 1999 (vol 51 pp. 26)

Fabian, Marián; Mordukhovich, Boris S.
 Separable Reduction and Supporting Properties of FrÃ©chet-Like Normals in Banach Spaces We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals. Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chet-like normals and subdifferentials, supporting properties, Asplund spacesCategories:49J52, 58C20, 46B20

144. CJM 1999 (vol 51 pp. 147)

Suárez, Daniel
 Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$ Let $m$ be a point of the maximal ideal space of $\papa$ with nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra of $\papa$. We characterize the points $m$ for which $L_m$ is a homeomorphism in terms of interpolating sequences, and we show that in this case $\papa \circ L_m$ coincides with $\papa$. Also, if $I_m$ is the ideal of functions in $\papa$ that identically vanish on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$. Categories:30H05, 46J20

145. CJM 1998 (vol 50 pp. 1236)

Kalton, N. J.; Tzafriri, L.
 The behaviour of Legendre and ultraspherical polynomials in $L_p$-spaces We consider the analogue of the $\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the best possible'' result that if $2 Categories:42C10, 33C45, 46B07 146. CJM 1998 (vol 50 pp. 1138) Chalov, P. A.; Terzioğlu, T.; Zahariuta, V. P.  Compound invariants and mixed$F$-,$\DF$-power spaces The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed$F$-,$\DF$-power series spaces, {\it i.e.} the spaces of the following kind $$G(\la,a)=\lim_{p \to \infty} \proj \biggl(\lim_{q \to \infty}\ind \Bigl(\ell_1\bigl(a_i (p,q)\bigr)\Bigr)\biggr),$$ where$a_i (p,q)=\exp\bigl((p-\la_i q)a_i\bigr)$,$p,q \in \N$, and$\la =( \la_i)_{i \in \N}$,$a=(a_i)_{i \in \N}$are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of$F$- and$\DF$-types, respectively. The$m$-rectangle characteristic$\mu_m^{\lambda,a}(\delta,\varepsilon; \tau,t)$,$m \in \N$of the space$G(\la,a)$is defined as the number of members of the sequence$(\la_i, a_i)_{i \in \N}$which are contained in the union of$m$rectangles$P_k = (\delta_k, \varepsilon_k] \times (\tau_k, t_k]$,$k = 1,2, \ldots, m$. It is shown that each$m$-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pe{\l}czynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis). Categories:46A04, 46A45, 46M05 147. CJM 1998 (vol 50 pp. 673) Carey, Alan; Phillips, John  Fredholm modules and spectral flow An {\it odd unbounded\/} (respectively,$p$-{\it summable}) {\it Fredholm module\/} for a unital Banach$\ast$-algebra,$A$, is a pair$(H,D)$where$A$is represented on the Hilbert space,$H$, and$D$is an unbounded self-adjoint operator on$H$satisfying: \item{(1)}$(1+D^2)^{-1}$is compact (respectively,$\Trace\bigl((1+D^2)^{-(p/2)}\bigr) <\infty$), and \item{(2)}$\{a\in A\mid [D,a]$is bounded$\}$is a dense$\ast-$subalgebra of$A$. If$u$is a unitary in the dense$\ast-$subalgebra mentioned in (2) then $$uDu^\ast=D+u[D,u^{\ast}]=D+B$$ where$B$is a bounded self-adjoint operator. The path $$D_t^u:=(1-t) D+tuDu^\ast=D+tB$$ is a continuous'' path of unbounded self-adjoint Fredholm'' operators. More precisely, we show that $$F_t^u:=D_t^u \bigl(1+(D_t^u)^2\bigr)^{-{1\over 2}}$$ is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The {\it spectral flow\/} of this path$\{F_t^u\}$(or$\{ D_t^u\}$) is roughly speaking the net number of eigenvalues that pass through$0$in the positive direction as$t$runs from$0$to$1$. This integer, $$\sf(\{D_t^u\}):=\sf(\{F_t^u\}),$$ recovers the pairing of the$K$-homology class$[D]$with the$K$-theory class [$u$]. We use I.~M.~Singer's idea (as did E.~Getzler in the$\theta$-summable case) to consider the operator$B$as a parameter in the Banach manifold,$B_{\sa}(H)$, so that spectral flow can be exhibited as the integral of a closed$1$-form on this manifold. Now, for$B$in our manifold, any$X\in T_B(B_{\sa}(H))$is given by an$X$in$B_{\sa}(H)$as the derivative at$B$along the curve$t\mapsto B+tX$in the manifold. Then we show that for$m$a sufficiently large half-integer: $$\alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{-m}\Bigr)$$ is a closed$1$-form. For any piecewise smooth path$\{D_t=D+B_t\}$with$D_0$and$D_1$unitarily equivalent we show that $$\sf(\{D_t\})={1\over {\tilde {C}_m}} \int_0^1\Tr \Bigl({d\over {dt}} (D_t)(1+D_t^2)^{-m}\Bigr)\,dt$$ the integral of the$1$-form$\alpha$. If$D_0$and$D_1$are not unitarily equivalent, we must add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form: $$\sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1-F_t^2)^n\Bigr)\,dt$$ for$n\geq{{p-1}\over 2}$an integer. The unbounded case is proved by reducing to the bounded case via the map$D\mapsto F=D(1+D^2 )^{-{1\over 2}}$. We prove simultaneously a type II version of our results. Categories:46L80, 19K33, 47A30, 47A55 148. CJM 1998 (vol 50 pp. 658) Symesak, Frédéric  Hankel operators on pseudoconvex domains of finite type in${\Bbb C}^2$The aim of this paper is to study small Hankel operators$h$on the Hardy space or on weighted Bergman spaces, where$\Omega$is a finite type domain in${\Bbbvii C}^2$or a strictly pseudoconvex domain in${\Bbbvii C}^n$. We give a sufficient condition on the symbol$f$so that$h$belongs to the Schatten class${\cal S}_p$,$1\le p<+\infty$. Categories:32A37, 47B35, 47B10, 46E22 149. 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 150. 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
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