New Deformations of Convolution Algebras and Fourier Algebras on Locally Compact Groups

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some ûnitely generated discrete groups with connection to the growth rate of the group.

On the deformed photon-added and photon-subtracted states

In this paper, we introduce the deformed algebra whose number operator is expressed in terms of the product of the creation operator and annihilation operator. We give some examples for these kinds of deformed algebras. For Arik–Coon's q-oscillator algebra, we discuss, especially, the photon-added states and the photon-subtracted states and construct their associated generation functions.

ON THE ASSOCIATIVE ALGEBRA STRUCTURES CLOSEST TO ALGEBRA STRUCTURES

We shall define the associative algebra structure closest to a given algebra structure. When we face a new multiplication, being caused by noise and so on, it must be useful to compute with the closest associative multiplication to such a perturbed one. In this paper we shall give a procedure to find the closest associative structure and demonstrate our strategy for the 2-dimensional algebras over the field of real numbers. Then we trace the points of the sets of structure constants of the deformed algebras.