THE REPRESENTATIONS AND ENDOMORPHISMS OF A SEPARABLE NUCLEAR C*-ALGEBRA

2003 ◽  
Vol 14 (03) ◽  
pp. 313-326 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Hilbert Space ◽  
Irreducible Representation ◽  
Separable Hilbert Space ◽  
Type I ◽  
C Algebra

It is shown that if A is a separable, non-type I, nuclear simple C*-algebra and π is an irreducible representation of A, then for any representation ρ on a separable Hilbert space of A there is an endomorphism α of A such that ρ is unitarily equivalent to πα.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

Extreme Points of the Convex set of Stochastic Maps on A C*-Algebra

1998 ◽  
Vol 01 (04) ◽  
pp. 599-609 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Bounded Operators ◽  
Completely Positive ◽  
C Algebra ◽  
Permutation Matrices ◽  
Normal Maps

Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.

A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

1991 ◽  
Vol 110 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
C Algebra ◽  
Alternative Characterization ◽  
Image Position ◽  
Algebra Of Operators

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such thatas i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

Factorization in the Invertible Group of a C*-Algebra

1997 ◽  
Vol 49 (6) ◽  
pp. 1188-1205 ◽  
Author(s):  
Michael J. Leen
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Upper Bounds ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Identity Component ◽  
C Algebras ◽  
Number Of Factors ◽  
Finite Products

AbstractIn this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), 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 (A ⊗ K)+, where A is any C*-algebra 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 (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

FRAME-LESS HILBERT C*-MODULES

2018 ◽  
Vol 61 (1) ◽  
pp. 25-31 ◽  
Author(s):  
M. B. ASADI ◽  
M. FRANK ◽  
Z. HASSANPOUR-YAKHDANI
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Faithful Representation ◽  
Central Projection ◽  
Multiplier Algebra ◽  
C Algebra ◽  
Infinite Dimensional

AbstractWe show that if A is a compact C*-algebra without identity that has a faithful *-representation in the C*-algebra of all compact operators on a separable Hilbert space and its multiplier algebra admits a minimal central projection p such that pA is infinite-dimensional, then there exists a Hilbert A1-module admitting no frames, where A1 is the unitization of A. In particular, there exists a frame-less Hilbert C*-module over the C*-algebra $K(\ell^2) \dotplus \mathbb{C}I_{\ell^2}$.

C*-ALGEBRAS ASSOCIATED WITH SETS OF SEMIGROUPS OF ISOMETRIES

1991 ◽  
Vol 02 (03) ◽  
pp. 235-255 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Hilbert Space ◽  
Continuous Time ◽  
Separable Hilbert Space ◽  
Infinite Sequence ◽  
C Algebra ◽  
C Algebras

Let U0, U1, …, Un be a (finite or infinite) sequence of semigroups of isometries which act on the same separable Hilbert space H, n = 1, 2, …, ∞. {Uj} is said to be orthogonal if for all i ≠ j we have [Formula: see text] With every such sequence we associate a separable C*-algebra [Formula: see text]. These C*-algebras [Formula: see text], n = 1, 2, …, ∞, are the "continuous time" analogues of the Cuntz C*-algebras [Formula: see text], n = 2, 3, …, ∞, in the same sense that the Wiener-Hopf C*-algebra is the continuous time analogue of the Toeplitz C*-algebra. For example, we show that they are nuclear unitless C*-algebras which have no closed nontrivial ideals. Indeed, we show that each [Formula: see text] is stably isomorphic to one of the spectral C*-algebras which arise in the theory of E0-semigroups.

WHAT TYPE OF DYNAMICS ARISE IN E0-DILATIONS OF COMMUTING QUANTUM MARKOV SEMIGROUPS?

2008 ◽  
Vol 11 (03) ◽  
pp. 393-403 ◽  
Author(s):  
ORR MOSHE SHALIT
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Type I ◽  
Markov Semigroups ◽  
Finite Dimensional

Let H be a separable Hilbert space, and let ϕ and θ be two strongly commuting CP0-semigroups on B(H). In a previous paper we constructed a Hilbert space K ⊇ H and two (strongly) commuting E0-semigroups α and β such that [Formula: see text] for all s, t ≥ 0 and all A ∈ B(K). In this note we prove that if ϕ is not an automorphism semigroup, then the semigroup α (given by the above mentioned construction) is cocycle conjugate to the minimal *-endomorphic dilation of ϕ, and that if ϕ is an automorphism semigroup, then α is also an automorphism semigroup. In particular, we conclude that if ϕ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional), then α is a type I E0-semigroup.

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

scholarly journals Construction and pure infiniteness of $C^*$-algebras associated with lambda-graph systems

2005 ◽  
Vol 97 (1) ◽  
pp. 73 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Hilbert Space ◽  
Bratteli Diagram ◽  
C Algebra ◽  
C Algebras ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System ◽  
Creation Operators

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced a $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ associated with a $\lambda$-graph system $\mathfrak{L}$ by using groupoid method as a generalization of the Cuntz-Krieger algebras. In this paper, we concretely construct the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ by using both creation operators and projections on a sub Fock Hilbert space associated with $\mathfrak{L}$. We also introduce a new irreducible condition on $\mathfrak{L}$ under which the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ becomes simple and purely infinite.

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

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