Quaternions and Functional Calculus

Elham Ghamari; Dan Kučerovský

doi:10.3390/sym11080953

Quaternions and Functional Calculus

Symmetry ◽

10.3390/sym11080953 ◽

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

Mathematica Slovaca ◽

10.1515/ms-2016-0259 ◽

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.

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CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES

Reviews in Mathematical Physics ◽

10.1142/s0129055x13500062 ◽

2013 ◽

Vol 25 (04) ◽

pp. 1350006 ◽

Cited By ~ 73

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.

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Finite Rank Operators and Functional Calculus on Hilbert Modules over Abelian C*-Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-023-3 ◽

1997 ◽

Vol 40 (2) ◽

pp. 193-197 ◽

Cited By ~ 1

Author(s):

Dan Kucerovsky

Keyword(s):

Finite Rank ◽

Functional Calculus ◽

Normal Operator ◽

Hilbert Module ◽

Hilbert Modules ◽

Finitely Generated ◽

C Algebras ◽

Finite Rank Operators ◽

Compact Normal Operator

AbstractWe consider the problem: If K is a compact normal operator on a Hilbert module E, and f ∈ C0(SpK) is a function which is zero in a neighbourhood of the origin, is f(K) of finite rank? We show that this is the case if the underlying C*-algebra is abelian, and that the range of f(K) is contained in a finitely generated projective submodule of E.

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Spectrum and Analytic Functional Calculus for Clifford Operators via Stem Functions

Concrete Operators ◽

10.1515/conop-2020-0115 ◽

2021 ◽

Vol 8 (1) ◽

pp. 90-113

Author(s):

Florian-Horia Vasilescu

Keyword(s):

Clifford Algebra ◽

Complex Plane ◽

Functional Calculus ◽

Clifford Algebras ◽

Cauchy Type ◽

Analytic Functional ◽

Slice Regular Functions ◽

Regular Functions

Abstract The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and also certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.

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