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Register a new userThe Go{l}k{a}b-Schinzel and Goldie functional equations in Banach algebras
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We are concerned below with the characterization in a unital commutative real Banach algebra $mathbb{A}$ of continuous solutions of the Go{l}k{a}b-Schinzel functional equation (below), the general Popa groups they generate and the associated Goldie functional equation. This yields general structure theorems involving both linear and exponential homogeneity in $mathbb{A}$ for both these functional equations and also explict forms, in terms of the recently developed theory of multi-Popa groups [BinO3,4], both for the ring $C[0,1]$ and for the case of $mathbb{R}^{d}$ with componentwise product, clarifying the context of recent developments in [RooSW]. The case $mathbb{A}=mathbb{C}$ provides a new viewpoint on continuous complex-valued solutions of the primary equation by distinguishing analytic from real-analytic ones.
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For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${mathcal B}$, its {em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]in pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${mathcal B}$. The pre-image of $p(A)$ in ${cc}^{n+1}$ is denoted by $P(A)$. When ${mathcal B}$ is the $ktimes k$ matrix algebra $M_k(cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${mathcal B}$ is reflexive or is a $C^*$-algebra, the {em projective resolvent set} $P^c(A):=cc^{n+1}setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${mathcal B}$ is a $C^*$-algebra with a trace $phi$, then $phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), cc)$.
The use of the properties of actions on an algebra to enrich the study of the algebra is well-trodden and still fashionable. Here, the notion and study of endomorphic elements of (Banach) algebras are introduced. This study is initiated, in the hope that it will open up, further, the structure of (Banach) algebras in general, enrich the study of endomorphisms and provide examples. In particular, here, we use it to classify algebras for the convenience of our study. We also present results on the structure of some classes of endomorphic elements and bring out the contrast with idempotents.
In this sequel to arXiv1407.4089 by the second author, we extend to multi-dimensional (or infinite-dimensional) settings the Goldie equation arising in the general regular variation of `General regular variation, Popa groups and quantifier weakening, J. Math. Anal. Appl. 483 (2020) 123610, 31 pp. (arXiv1901.05996). The theory focusses on extension of the treatment there of Popa groups, permitting a characterization of Popa homomorphisms (in two complementary theorems, 4A and 4B below). This in turn enables a characterization of the (real-valued) solutions of the multivariate Goldie equation, to be presented in the further sequel arXiv1910.05817. The Popa groups here contribute to a structure theorem describing Banach-algebra value
In this the sequel to arXiv1910.05816, we derive a necessary and sufficient condition characterizing which real-valued continuous solutions of a multivariate Goldie functional equation express homomorphy between the multivariate Popa groups defined and characterized in the earlier work. This enables us to deduce that all (real-valued) continuous solutions are homomorphisms between such groups. We use this result also to characterize as Popa homomorphisms smooth solutions of a related more general equation, also of Levi-Civita type. A key result here (Theorem 2) on purely radial behaviour is generalized in arXiv2105.07794 to a Banach-algebra setting involving radial tilting behaviour.
The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator. The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.
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