# Ideals of Projections According to σ-Algebras and Unbounded Measurements

## Abstract

**:**

## 1. Introduction

**Main results. Ideals of projections and their properties**

**Definition**

**1.**

**ideal**(of projections), if

**Definition**

**2.**

- is called the
**weight**.

**faithful**, if $\varphi \left(A\right)=0$ follows $A=0$;

**semi-finite**if $lin\{A\in {\mathcal{M}}^{+}:\varphi \left(A\right)<+\infty \}$ is an ultra-weakly dense set on $\mathcal{M}$;

**normal**if ${A}_{i}\nearrow A$ $\in {\mathcal{M}}^{+}$ follows $\varphi \left(A\right)={sup}_{i}\varphi \left({A}_{i}\right)$;

**trace**if $\varphi \left({A}^{*}A\right)=\varphi \left(A{A}^{*}\right)$ $\forall A\in \mathcal{M}$.

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Lemma**

**1.**

**Proof.**

## 2. Measures on Ideals

**Definition**

**3.**

**measure**if $\mu \left({\sum}_{i}{P}_{i}\right)={\sum}_{i}\mu \left({P}_{i}\right)$, ${P}_{i}\in \mathfrak{M}$.

**finite**(=

**bounded**) if $sup\left\{\mu \right(P):P\in \mathfrak{M}\}<+\infty $,

**infinite**(=

**unbounded**) if $sup\left\{\mu \right(P):P\in \mathfrak{M}\}=+\infty $,

**regular**if there is weight $\varphi $, such that $\mu \left(P\right)=\varphi \left(P\right)$ for all $P\in \mathfrak{M}$.

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

**Definition**

**4.**

**Theorem**

**2.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Corollary**

**2.**

**Lemma**

**5.**

**Proof.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

**Proof.**

**Bilinear forms and measures**Let $T\ge 0$ be a self-adjoining operator that is associated with $\mathcal{M}$. Let ${\mathfrak{M}}_{T}=\{P\in \Pi :\tau \left(TP\right)<+\infty \}$. Note: (a) the function $\tau (T\xb7)$ is understood in the sense of article [12]; (b) ${\mathfrak{M}}_{I}={\mathfrak{M}}_{\tau}$. The set ${\mathfrak{M}}_{T}$ is the ideal of projections and $\mu :P\in {\mathfrak{M}}_{T}\to \tau \left(TP\right)$ is a closed measure.

**Proposition**

**3.**

**Proposition**

**4.**

**Proof.**

**Definition**

**5.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

## 3. $\mathbf{\sigma}$-Finite Measure

**Definition**

**6.**

**Remark**

**3.**

**Proof.**

**Proposition**

**5.**

**Lemma**

**6**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Definition**

**7.**

**Proposition**

**6.**

**Proof.**

**Corollary**

**4.**

**Proposition**

**7.**

**Proof.**

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Matvejchuk, M.
Ideals of Projections According to *σ*-Algebras and Unbounded Measurements. *Axioms* **2023**, *12*, 167.
https://doi.org/10.3390/axioms12020167

**AMA Style**

Matvejchuk M.
Ideals of Projections According to *σ*-Algebras and Unbounded Measurements. *Axioms*. 2023; 12(2):167.
https://doi.org/10.3390/axioms12020167

**Chicago/Turabian Style**

Matvejchuk, Marjan.
2023. "Ideals of Projections According to *σ*-Algebras and Unbounded Measurements" *Axioms* 12, no. 2: 167.
https://doi.org/10.3390/axioms12020167