scholarly journals Projective multi-resolution analyses arising from direct limits of Hilbert modules

2007 ◽  
Vol 100 (2) ◽  
pp. 317 ◽  
Author(s):  
Nadia S. Larsen ◽  
Iain Raeburn
Keyword(s):  
Directed Graph ◽  
Hilbert Spaces ◽  
Path Space ◽  
Hilbert Modules ◽  
C Algebras ◽  
Infinite Path ◽  
Direct Limits ◽  
New Applications ◽  
Shed Light

The authors have recently shown how direct limits of Hilbert spaces can be used to construct multi-resolution analyses and wavelets in $L^2(\mathsf R)$. Here they investigate similar constructions in the context of Hilbert modules over $C^*$-algebras. For modules over $C(\mathsf T^n)$, the results shed light on work of Packer and Rieffel on projective multi-resolution analyses for specific Hilbert $C(\mathsf T^n)$-modules of functions on $\mathsf R^n$. There are also new applications to modules over $C(C)$ when $C$ is the infinite path space of a directed graph.

TENSOR PRODUCT SYSTEMS OF HILBERT MODULES AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

2000 ◽  
Vol 03 (04) ◽  
pp. 519-575 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
MICHAEL SKEIDE
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hilbert Modules ◽  
Completely Positive ◽  
Positive Semigroups ◽  
C Algebras ◽  
Product Systems ◽  
Dilation Theory ◽  
Cocycle Conjugacy ◽  
The Right

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E0-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on [Formula: see text], but also extend the results immediately to much more general setups. For instance, Arveson classifies E0-semigroups on [Formula: see text] up to cocycle conjugacy by product systems of Hilbert spaces.5 We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E0-semigroups on [Formula: see text] in dilation theory for CP-semigroups on [Formula: see text] is now played by E0-semigroups on [Formula: see text], the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke27 for CP maps.

C*-Algebras of Infinite Graphs and Cuntz-Krieger Algebras

2002 ◽  
Vol 45 (3) ◽  
pp. 321-336 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Directed Graph ◽  
Infinite Graphs ◽  
C Algebras ◽  
Graph Algebra

AbstractThe Cuntz-Krieger algebra 𝓞B is defined for an arbitrary, possibly infinite and infinite valued, matrix B. A graph C*-algebra G*(E) is introduced for an arbitrary directed graph E, and is shown to coincide with a previously defined graph algebra C*(E) if each source of E emits only finitely many edges. Each graph algebra G*(E) is isomorphic to the Cuntz-Krieger algebra 𝓞B where B is the vertex matrix of E.

scholarly journals Generalized n-circular projections on JB*-triples and Hilbert C0(Ω)-modules

2017 ◽  
Vol 4 (1) ◽  
pp. 109-120
Author(s):  
Dijana Ilišević ◽  
Chih-Neng Liu ◽  
Ngai-Ching Wong
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Orthogonal Projections ◽  
Geometric Properties ◽  
C Algebras ◽  
Structure Theorems ◽  
Special Case

Abstract Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C0(Ω).

scholarly journals AN ALGEBRAIC DUALITY THEORY FOR MULTIPLICATIVE UNITARIES

2001 ◽  
Vol 12 (04) ◽  
pp. 415-459 ◽  
Author(s):  
SERGIO DOPLICHER ◽  
CLAUDIA PINZARI ◽  
JOHN E. ROBERTS
Keyword(s):  
Hilbert Spaces ◽  
Duality Theory ◽  
Duality Theorem ◽  
Separable Hilbert Space ◽  
Regular Representation ◽  
Cuntz Algebra ◽  
Relative Depth ◽  
Minimal Representation ◽  
C Algebras ◽  
Algebraic Duality

Multiplicative unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki–Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra related algebraically to the grading are discussed as is the notion of braided symmetry in a tensor C*-category.

scholarly journals Centers of Cuntz–Krieger C*-algebras

2017 ◽  
Vol 16 (05) ◽  
pp. 1750091
Author(s):  
Adel Alahmadi ◽  
Hamed Alsulami
Keyword(s):  
Directed Graph ◽  
C Algebras

In this paper, for a finite directed graph [Formula: see text], we determine the center of the Cuntz–Krieger [Formula: see text]-algebra CK[Formula: see text].

scholarly journals Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

scholarly journals The irreducibility of C*-algebras acting on Hilbert C*-modules

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2425-2433
Author(s):  
Runliang Jiang
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
C Algebra ◽  
C Algebras ◽  
The Difference

Let B be a C*-algebra, E be a Hilbert B module and L(E) be the set of adjointable operators on E. Let A be a non-zero C*-subalgebra of L(E). In this paper, some new kinds of irreducibilities of A acting on E are introduced, which are all the generalizations of those associated to Hilbert spaces. The difference between these irreducibilities are illustrated by a number of counterexamples. It is concluded that for a full Hilbert B-module, these irreducibilities are all equivalent if and only if the underlying C*-algebra B is isomorphic to the C*-algebra of all compact operators on a Hilbert space.

scholarly journals On multipliers of Hilbert modules over pro-C*-algebras

2008 ◽  
Vol 185 (3) ◽  
pp. 263-277 ◽  
Author(s):  
Maria Joiţa
Keyword(s):  
Hilbert Modules ◽  
C Algebras
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