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Adelic analysis and functional analysis on the finite Adele ring

Cho I.

Opuscula Mathematica

2018

Vol. 38, no. 2

139--185

In this paper, we study operator theory on the *-algebra Mp, consisting of all measurable functions on the finite Adele ring Aq, in extended free-probabilistic sense. Even though our *-algebra Mp is commutative, our Adelic-analytic data and properties on Mv are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on Aq, we construct the suitable Hilbert-space representation, and study a C*-algebra M-p generated by M-p under representation. In particular, we focus on operator-theoretic properties of certain generating operators on Mp.

Semicircular elements induced by p-adic number fields

Cho I., Jorgensen P. E. T.

Opuscula Mathematica

2017

Vol. 37, no. 5

665--703

In this paper, we study semicircular-like elements, and semicircular elements induced by p-adic analysis, for each prime p. Starting from a p-adic number field Qp, we construct a Banach *-algebra [formula], for a fixed prime p, and show the generating elements Qpj of [formula] form weighted-semicircular elements, and the corresponding scalar-multiples Θpj of Qpj become semicircular elements, for all j ∈ Z. The main result of this paper is the very construction of suitable linear functionals [formula] on [formula], making Qpj be weighted-semicircular, for all j ∈ Z.

The Order on Projections in C*-Algebras of Real Rank Zero

Bice T.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 1

37--58

We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the canonical homomorphism to the Calkin algebra preserves least upper bounds of countable collections of projections, namely that this occurs precisely when the span of the corresponding subspaces is closed.