Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals
Search results

Search: MSC category 46L ( Selfadjoint operator algebras ($C^$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx] *$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx] * )  Expand all Collapse all Results 51 - 75 of 77 51. CJM 2004 (vol 56 pp. 843) Ruan, Zhong-Jin  Type Decomposition and the Rectangular AFD Property for$W^*$-TRO's We study the type decomposition and the rectangular AFD property for$W^*$-TRO's. Like von Neumann algebras, every$W^*$-TRO can be uniquely decomposed into the direct sum of$W^*$-TRO's of type$I$, type$II$, and type$III$. We may further consider$W^*$-TRO's of type$I_{m, n}$with cardinal numbers$m$and$n$, and consider$W^*$-TRO's of type$II_{\lambda, \mu}$with$\lambda, \mu = 1$or$\infty$. It is shown that every separable stable$W^*$-TRO (which includes type$I_{\infty,\infty}$, type$II_{\infty, \infty}$and type$III$) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for$W^*$-TRO's. One of our major results is to show that a separable$W^*$-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular${\OL}_{1, 1^+}$space (equivalently, a rectangular Categories:46L07, 46L08, 46L89 52. CJM 2004 (vol 56 pp. 225) Blower, Gordon; Ransford, Thomas  Complex Uniform Convexity and Riesz Measure The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue$L^p$spaces and the von~Neumann-Schatten trace ideals. Banach spaces that are$q$-uniformly$\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals$c^p$are$2$-uniformly$\PL$-convex for$1\leq p\leq 2$. Keywords:subharmonic functions, Banach spaces, Schatten trace idealsCategories:46B20, 46L52 53. CJM 2004 (vol 56 pp. 3) Amini, Massoud  Locally Compact Pro-$C^*$-Algebras Let$X$be a locally compact non-compact Hausdorff topological space. Consider the algebras$C(X)$,$C_b(X)$,$C_0(X)$, and$C_{00}(X)$of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on$X$. Of these, the second and third are$C^*$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the$C^\ast$-algebra$C_0(X)$, one can get the other three algebras by$C_{00}(X)=K\bigl(C_0(X)\bigr)$,$C_b(X)=M\bigl(C_0(X)\bigr)$,$C(X)=\Gamma\bigl( K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of$C_0(X)$, respectively. In this article we consider the possibility of these transitions for general$C^\ast$-algebras. The difficult part is to start with a pro-$C^\ast$-algebra$A$and to construct a$C^\ast$-algebra$A_0$such that$A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is possible are called {\it locally compact\/} and we have characterized them using a concept similar to that of an approximate identity. Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's idealCategories:46L05, 46M40 54. CJM 2003 (vol 55 pp. 1302) Katsura, Takeshi  The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A.~Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and$K$-groups of our algebras. Categories:46L05, 46L55, 46L45 55. CJM 2002 (vol 54 pp. 1100) Wood, Peter J.  The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group$G$is operator biprojective if and only if$G$is discrete. Keywords:locally compact group, Fourier algebra, operator space, projectiveCategories:13D03, 18G25, 43A95, 46L07, 22D99 56. CJM 2002 (vol 54 pp. 694) Gabriel, Michael J.  Cuntz Algebra States Defined by Implementers of Endomorphisms of the$\CAR$Algebra We investigate representations of the Cuntz algebra$\mathcal{O}_2$on antisymmetric Fock space$F_a (\mathcal{K}_1)$defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states$\varphi_{P_1}$. We pay corresponding to these representations and the Fock special attention to the vector states on$\mathcal{O}_2$vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra$\mathcal{F}_2$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states$\varphi_{P_1}$. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to$\mathcal{F}_2$. The endomorphisms of$B \bigl( F_a (\mathcal{K}_1) \bigr)$associated with these representations of$\mathcal{O}_2$are also considered. Categories:46L05, 46L30 57. CJM 2002 (vol 54 pp. 138) Razak, Shaloub  On the Classification of Simple Stably Projectionless$\C^*$-Algebras It is shown that simple stably projectionless$\C^S*$-algebras which are inductive limits of certain specified building blocks with trivial$\K$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras. Categories:46L35, 46L05 58. CJM 2001 (vol 53 pp. 1223) Mygind, Jesper  Classification of Certain Simple$C^*$-Algebras with Torsion in$K_1$We show that the Elliott invariant is a classifying invariant for the class of$C^*$-algebras that are simple unital infinite dimensional inductive limits of finite direct sums of building blocks of the form $$\{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\},$$ where$x_1,x_2,\dots,x_N \in \T$,$d_1,d_2,\dots,d_N$are integers dividing$n$, and$M_{d_i}$is embedded unitally into$M_n$. Furthermore we prove existence and uniqueness theorems for$*$-homomorphisms between such algebras and we identify the range of the invariant. Categories:46L80, 19K14, 46L05 59. CJM 2001 (vol 53 pp. 979) Nagisa, Masaru; Osaka, Hiroyuki; Phillips, N. Christopher  Ranks of Algebras of Continuous$C^*$-Algebra Valued Functions We prove a number of results about the stable and particularly the real ranks of tensor products of \ca s under the assumption that one of the factors is commutative. In particular, we prove the following: {\raggedright \begin{enumerate}[(5)] \item[(1)] If$X$is any locally compact$\sm$-compact Hausdorff space and$A$is any \ca, then\break$\RR \bigl( C_0 (X) \otimes A \bigr) \leq \dim (X) + \RR(A)$. \item[(2)] If$X$is any locally compact Hausdorff space and$A$is any \pisca, then$\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$. \item[(3)]$\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$for any nonzero \ca\$A$, and$\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$for any unital \ca\$A$. \item[(4)] If$A$is a unital \ca\ such that$\RR(A) = 0$,$\sr (A) = 1$, and$K_1 (A) = 0$, then\break$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \item[(5)] There is a simple separable unital nuclear \ca\$A$such that$\RR(A) = 1$and\break$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \end{enumerate}} Categories:46L05, 46L52, 46L80, 19A13, 19B10 60. CJM 2001 (vol 53 pp. 809) Robertson, Guyan; Steger, Tim  Asymptotic$K$-Theory for Groups Acting on$\tA_2$Buildings Let$\Gamma$be a torsion free lattice in$G=\PGL(3, \mathbb{F})$where$\mathbb{F}$is a nonarchimedean local field. Then$\Gamma$acts freely on the affine Bruhat-Tits building$\mathcal{B}$of$G$and there is an induced action on the boundary$\Omega$of$\mathcal{B}$. The crossed product$C^*$-algebra$\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$depends only on$\Gamma$and is classified by its$K$-theory. This article shows how to compute the$K$-theory of$\mathcal{A}(\Gamma)$and of the larger class of rank two Cuntz-Krieger algebras. Keywords:$K$-theory,$C^*$-algebra, affine buildingCategories:46L80, 51E24 61. CJM 2001 (vol 53 pp. 592) Perera, Francesc  Ideal Structure of Multiplier Algebras of Simple$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$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoidCategories:46L05, 46L80, 06F05 62. CJM 2001 (vol 53 pp. 631) Walters, Samuel G.  K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism For a dense$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}$.) Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern charactersCategories:46L80, 46L40, 19K14 63. CJM 2001 (vol 53 pp. 546) Erlijman, Juliana  Multi-Sided Braid Type Subfactors We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II$_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 64. 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 65. 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 66. 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 67. 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 68. CJM 2000 (vol 52 pp. 1164) Elliott, George A.; Villadsen, Jesper  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 69. 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 70. 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 71. 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 72. 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 73. 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 74. 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 75. 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
 Page Previous 1 2 3 4 Next
 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2015 : https://cms.math.ca/