Commuting pairs of self-adjoint elements in C *-algebras

2017 ◽  
Vol 67 (1) ◽  
pp. 209-212
Author(s):  
Osamu Hatori
Keyword(s):  
Continuous Function ◽  
Functional Calculus ◽  
Continuous Functional ◽  
C Algebras ◽  
Commuting Pairs ◽  
The Given

Abstract We give a condition on commutativity of a pair of self-adjoint elements in a C *-algebra with respect to the continuous functional calculus. We also give an answer to the question raised by Jeang and Ko that if a non-constant continuous function totally spans the given C *-algebra.

scholarly journals CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES

2013 ◽  
Vol 25 (04) ◽  
pp. 1350006 ◽  
Author(s):  
RICCARDO GHILONI ◽  
VALTER MORETTI ◽  
ALESSANDRO PEROTTI
Keyword(s):  
Hilbert Spaces ◽  
Functional Calculus ◽  
Normal Operator ◽  
Regular Function ◽  
Continuous Functions ◽  
Continuous Functional ◽  
Unbounded Operators ◽  
C Algebras ◽  
Normal Operators ◽  
The One

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

scholarly journals Dilations of $q$-Commuting Unitaries

2020 ◽  
Author(s):  
Malte Gerhold ◽  
Orr Moshe Shalit
Keyword(s):  
Continuous Function ◽  
Hausdorff Metric ◽  
Natural Structure ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
C Algebras ◽  
Mathieu Operator ◽  
Almost Mathieu Operator ◽  
Continuous Field ◽  
Continuity Of The Spectrum

Abstract Let $q = e^{i \theta } \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, that is, $u$ and $v$ are unitaries such that $vu = quv$. In this paper, we find the optimal constant $c = c_{\theta }$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$where $u_{\theta }, v_{\theta }$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q^{\prime}$-commuting unitaries. The techniques that we develop allow us to give new and simple “dilation theoretic” proofs of well-known results regarding the continuity of the field of rotations algebras. In particular, for the so-called “almost Mathieu operator” $h_{\theta } = u_{\theta }+u_{\theta }^*+v_{\theta }+v_{\theta }^*$, we recover the fact that the norm $\|h_{\theta }\|$ is a Lipschitz continuous function of $\theta $, as well as the result that the spectrum $\sigma (h_{\theta })$ is a $\frac{1}{2}$-Hölder continuous function in $\theta $ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial $p(u_{\theta },v_{\theta })$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

Variation Reducing Properties of Decreasing Rearrangements

1975 ◽  
Vol 27 (2) ◽  
pp. 330-336 ◽  
Author(s):  
Kong-Ming Chong
Keyword(s):  
Continuous Function ◽  
Property A ◽  
Absolutely Continuous ◽  
Absolutely Continuous Function ◽  
Almost Everywhere ◽  
The Difference ◽  
Decreasing Rearrangements ◽  
The Given ◽  
Considerable Detail ◽  
Reducing Properties

One well-established characteristic of the operation of decreasing rearrangement is its variation reducing property. A systematic study of this property has been made in considerable detail by G.F.D. Duff in [5] and [6]. He proved some inequalities related to the operation of rearrangement in decreasing order showing that the total variation of a sequence or an absolutely continuous function is in general diminished by such rearrangement. He also showed that the Lp norm of the difference sequence (or the derivative function) is diminished by this rearrangement operation unless the given sequence (or absolutely continuous function) is already monotonie (or equal to a monotonie function almost everywhere).

COVARIANCE GROUP C*-ALGEBRAS AND GALOIS CORRESPONDENCE

1991 ◽  
Vol 02 (06) ◽  
pp. 673-699 ◽  
Author(s):  
PALLE E. T. JORGENSEN ◽  
XIU-CHI QUAN
Keyword(s):  
Main Body ◽  
Normal Subgroups ◽  
C Algebra ◽  
Group System ◽  
C Algebras ◽  
Galois Correspondence ◽  
Covariance Group ◽  
The Given ◽  
The Way

The main purpose of this paper is to establish a Galois correspondence for a given covariant group system, its associated C*-algebra and Hopf C*-algebra. On the way to this, we first study covariance group C*-algebras and their representations, and prove a result which is simpler but yet very similar to the C*-algebra case in the main body of the paper. We then show that there is a Galois correspondence between the lattice of normal subgroups of the given covariant group system and a corresponding lattice of certain invariant *-subalgebras of the covariant group C*-algebra; in particular, there is a natural Galois correspondence for the group C*-algebra. We further study this Galois correspondence for the Hopf C*-algebras associated with covariant group systems.

scholarly journals The space of ideals in the minimal tensor product of C*-algebras

2010 ◽  
Vol 148 (2) ◽  
pp. 243-252 ◽  
Author(s):  
ALDO J. LAZAR
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Tensor Product ◽  
C Algebras

AbstractFor C*-algebras A1, A2 the map (I1, I2) → ker(qI1 ⊗ qI2) from Id′(A1) × Id′(A2) into Id′(A1 ⊗minA2) is a homeomorphism onto its image which is dense in the range. Here, for a C*-algebra A, the space of all proper closed two sided ideals endowed with an adequate topology is denoted Id′(A) and qI is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A1) × Prim(A2) with values into a T1 topological space can be extended to Prim(A1 ⊗minA2). This enlarges the scope of [7, corollary 3·5] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A1 ⊗minA2 is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A1 ⊗minA2 with certain other properties are presented.

scholarly journals FUNCTIONAL CALCULUS EXTENSIONS ON DUAL SPACES

2009 ◽  
Vol 79 (1) ◽  
pp. 71-77
Author(s):  
VENTA TERAUDS
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Functional Calculus ◽  
Sufficient Conditions ◽  
Continuous Functional ◽  
Absolutely Continuous ◽  
Bounded Linear ◽  
Dual Spaces

AbstractIn this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.

Estimating the home range

1994 ◽  
Vol 31 (3) ◽  
pp. 700-720 ◽  
Author(s):  
L. De Haan ◽  
Sidney Resnick
Keyword(s):  
Continuous Function ◽  
Home Range ◽  
Confidence Region ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Polar Coordinates ◽  
Common Distribution ◽  
The Right ◽  
The Given ◽  
Independent Identically Distributed

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X1,· ··, Xn whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.

Estimating the home range

1994 ◽  
Vol 31 (03) ◽  
pp. 700-720 ◽  
Author(s):  
L. De Haan ◽  
Sidney Resnick
Keyword(s):  
Continuous Function ◽  
Home Range ◽  
Confidence Region ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Polar Coordinates ◽  
Common Distribution ◽  
The Right ◽  
The Given ◽  
Independent Identically Distributed

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X 1,· ··, Xn whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R 1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.

scholarly journals Quaternions and Functional Calculus

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 953
Author(s):  
Elham Ghamari ◽  
Dan Kučerovský
Keyword(s):  
Functional Calculus ◽  
C Algebras ◽  
Regular Functions

In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic C * -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions.

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