Consider the multiplication operator MB in 
$L^2(mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in 
$L^2(mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with 
$M_{z^2}$ or making 
$M_{z^2}$ complex symmetric by introducing a new class of conjugations in 
$L^2(mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.
We establish a new improvement of the classical Lp-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one-dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.
In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.
Let Σ be a σ-algebra of subsets of a set Ω and 
$B(Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let 
$tau(B(Sigma),ca(Sigma))$ denote the natural Mackey topology on 
$B(Sigma)$. It is shown that a linear operator T from 
$B(Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space 
$(B(Sigma),tau(B(Sigma),ca(Sigma)))$ and the Banach space E. We derive a formula for the trace of a Bochner representable operator 
$T:B({cal B} o)rightarrow B({cal B} o)$ generated by a function 
$fin L^1({cal B} o, C(Omega))$, where Ω is a compact Hausdorff space.
Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labelled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the spherical growth series of a Dyer group D with respect to the standard generating set. We give a recursive formula for the spherical growth series of D in terms of the spherical growth series of standard parabolic subgroups. As an application we obtain the rationality of the spherical growth series of a Dyer group. Furthermore, we show that the spherical growth series of D is closely related to the Euler characteristic of D.
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