scholarly journals BESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES

2007 ◽  
Vol 50 (1) ◽  
pp. 23-36 ◽  
Author(s):  
Senka Banić ◽  
Dijana Ilišević ◽  
Sanja Varošanec

AbstractIn this paper we give Bessel- and Grüss-type inequalities in an inner product module over a proper $H^*$-algebra or a $C^*$-algebra.

2020 ◽  
Vol 44 (4) ◽  
pp. 571-579
Author(s):  
T. TEIMOURI-AZADBAKHT ◽  
A. G GHAZANFARI

Let X be a Hilbert C∗-module on C∗-algebra A and p ∈ A. We denote by Dp(A,X) the set of all continuous functions f : A → X, which are Fréchet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A,X), and using them some Grüss type inequalities in semi-inner product C∗-module Dp(A,X) and Dp(A,Xn) are established.


2019 ◽  
Vol 52 (1) ◽  
pp. 410-427
Author(s):  
Andrea C. Antunez

AbstractLet 𝒜 be a unital C*-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊ϕ = {x ∈ 𝒜 : ϕ(x*x) = 1} and the projective space ℙϕ = 𝕊ϕ/𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰ϕ(𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙϕ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.


Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 22
Author(s):  
Suzana Bedić ◽  
Otto C. W. Kong ◽  
Hock King Ting

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.


2005 ◽  
Vol 133 (11) ◽  
pp. 3271-3280 ◽  
Author(s):  
Dijana Ilišević ◽  
Sanja Varošanec

1974 ◽  
Vol 26 (5) ◽  
pp. 1272-1280 ◽  
Author(s):  
William L. Paschke

The principal result of this paper states that if X is a pre-Hilbert B-module over an arbitrary C*-algebra B, then the B-valued inner product on X can be lifted to a B-valued inner product on X″ (the B-dual of the B-dual X′ of X). Appropriate identifications allow us to regard X as a submodule of X″ and the latter in turn as a submodule of X′. In this sense, the inner product on X″ is an extension of that on X. As an example (and application) of this result, we consider the special case in which X is a right ideal of B and give a topological description of X″ when in addition B is commutative.


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