A Cohomological Property of π-invariant Elements

2013 ◽  
Vol 56 (3) ◽  
pp. 534-543
Author(s):  
M. Filali ◽  
M. Sangani Monfared
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Orthonormal Basis ◽  
Hochschild Cohomology ◽  
Separable Hilbert Space ◽  
Continuous Representation ◽  
Cohomology Groups ◽  
Cohomological Property ◽  
Coordinate Functions ◽  
Special Case

Abstract.Let A be a Banach algebra and let be a continuous representation of A on a separable Hilbert space H with dim H = m. Let πi j be the coordinate functions of π with respect to an orthonormal basis and suppose that for each and . Under these conditions, we call an element left π-invariant if In this paper we prove a link between the existence of left π-invariant elements and the vanishing of certain Hochschild cohomology groups of A. Our results extend an earlier result by Lau on F-algebras and recent results of Kaniuth, Lau, Pym, and and the second author in the special case where π : A → C is a non-zero character on A.

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}}.

scholarly journals Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

2018 ◽  
Vol 2020 (23) ◽  
pp. 9148-9209
Author(s):  
Domenico Fiorenza ◽  
Niels Kowalzig
Keyword(s):  
Hochschild Cohomology ◽  
Cyclic Homology ◽  
Cyclic Cohomology ◽  
Algebra Structure ◽  
String Topology ◽  
Cohomology Groups ◽  
Poincare Duality ◽  
Homology Groups ◽  
Bv Algebra ◽  
Special Case

Abstract The purpose of this article is to embed the string topology bracket developed by Chas–Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Völcsey–Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincaré duality is given. For negative cyclic cohomology, this in particular leads to a Batalin–Vilkoviskiĭ (BV) algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an $e_3$-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.

Relations between the homologies of C*-algebras and their commutative C*-subalgebras

2002 ◽  
Vol 132 (1) ◽  
pp. 155-168
Author(s):  
ZINAIDA A. LYKOVA
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Banach Algebra ◽  
Left Ideal ◽  
Separable Hilbert Space ◽  
Infinite Dimensional ◽  
C Algebras

The paper concerns the identification of projective closed ideals of C*-algebras. We prove that, if a C*-algebra has the property that every closed left ideal is projective, then the same is true for all its commutative C*-subalgebras. Further, we say a Banach algebra A is hereditarily projective if every closed left ideal of A is projective. As a corollary of the stated result we show that no infinite-dimensional AW*-algebra is hereditarily projective. We also prove that, for a commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space, the following conditions are equivalent: (i) A is separable; and (ii) the C*-tensor product A [otimes ]minA is hereditarily projective. Howerever, there is a non-separable, hereditarily projective, commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space.

scholarly journals A note on local asymptotic behaviour for Brownian motion in Banach spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 669-676
Author(s):  
Mou-Hsiung Chang
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Brownian Motion ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Integral Test ◽  
Real Separable Banach Space ◽  
Real Separable Hilbert Space ◽  
Special Case

In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] whereB=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that whenBis itself a real separable Hilbert space is also illustrated.

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.

Functional Quantum Theory of Free Relativistic Fermi Fields

1970 ◽  
Vol 25 (5) ◽  
pp. 575-586
Author(s):  
H. Stumpf
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Physical Interpretation ◽  
Functional Equations ◽  
Dirac Field ◽  
Functional Hilbert Space ◽  
Free Fermi ◽  
Special Case ◽  
Relativistic Fermi

Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Biorthogonal Systems in lp-Spaces

1969 ◽  
Vol 21 ◽  
pp. 625-638 ◽  
Author(s):  
R. Keown ◽  
C. Conatser
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Natural Generalization ◽  
Standard Basis ◽  
Biorthogonal Systems ◽  
Lp Spaces ◽  
Standard Bases

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).

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