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1. CMB Online first
Exact and Approximate Operator Parallelism Extending the notion of parallelism we introduce the concept of
approximate parallelism in normed spaces and then substantially
restrict ourselves to the setting of Hilbert space operators endowed
with the operator norm. We present several characterizations of the
exact and approximate operator parallelism in the algebra
$\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a
Hilbert space $\mathscr{H}$. Among other things, we investigate the
relationship between approximate parallelism and norm of inner
derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the
parallel elements of a $C^*$-algebra by using states. Finally we
utilize the linking algebra to give some equivalence assertions
regarding parallel elements in a Hilbert $C^*$-module.
Keywords:$C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-module Categories:47A30, 46L05, 46L08, 47B47, 15A60 |
2. CMB 2010 (vol 53 pp. 550)
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations |
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations
In this paper we propose a new technical tool for analyzing
representations of Hilbert $C^*$-product systems. Using this tool,
we give a new proof that every doubly commuting representation
over $\mathbb{N}^k$ has a regular isometric dilation, and we also
prove sufficient conditions for the existence of a regular
isometric dilation of representations over more general
subsemigroups of $\mathbb R_{+}^k$.
Categories:47A20, 46L08 |
3. CMB 2008 (vol 51 pp. 545)
$C^{\ast}$-Algebras Associated with Mauldin--Williams Graphs A Mauldin--Williams graph $\mathcal{M}$ is a generalization of an
iterated function system by a directed graph. Its invariant set $K$
plays the role of the self-similar set. We associate a $C^{*}$-algebra
$\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin--Williams graph $\mathcal{M}$
and the invariant set $K$, laying emphasis on the singular points.
We assume that the underlying graph $G$ has no sinks and no sources.
If $\mathcal{M}$ satisfies the open set condition in $K$, and $G$
is irreducible and is not a cyclic permutation, then the associated
$C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is simple and purely
infinite. We calculate the $K$-groups for some examples including the
inflation rule of the Penrose tilings.
Categories:46L35, 46L08, 46L80, 37B10 |
4. CMB 2001 (vol 44 pp. 355)
Hilbert Bimodules with Involution We examine Hilbert bimodules which possess a (generally unbounded)
involution. Topics considered include a linking algebra
representation, duality, locality, and the role of these bimodules
in noncommutative differential geometry
Categories:46L08, 46L57, 46L87 |