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**INTRODUCTION**

The abstract Banach associative symmetrical *-algebras over C, so called C*- algebras, were introduced first in 1943 by Gelfand and Naimark24. In the present time the theory of C*-algebras has become a vast portion of functional analysis having connections and applications in almost all branches of modern mathematics and theoretical physics51,55.

From the 1940’s and the beginning of 1950’s there were numerous attempts made to extend the theory of C*-algebras to a category wider than Banach algebras. For example, in 1952, while working on the theory of locally-multiplicatively-convex algebras s projective limits of projective families of Banach algebras, Arens in the paper8 and Michael in the monograph48 independently for the first time studied projective limits of projective families of functional algebras in the commutative case and projective limits of projective families of operator algebras in the non-commutative case.

In 1971 Inoue in the paper33 explicitly studied topological *-algebras which are topologically -isomorphic to projective limits of projective families of C*-algebras and obtained their basic properties. He as well suggested a name of locally C*-algebras for that category.

For the present state of the theory of locally C*-algebras see the monograph of Fragoulopoulou20.

Also there were many attempts to extend the theory of C*-algebras to nonassociative algebras which are close in properties to associative algebras (in particular, to Jordan algebras). In fact, the real Jordan analogues of C*-algebras, so called JB- algebras, were first introduced in 1978 by Alfsen, Shultz and Størmer in1. One of the main results of the aforementioned paper stated that modulo factorization over a unique Jordan ideal each JB-algebra is isometrically isomorphic to a JC-algebra, i.e. an operator norm closed Jordan subalgebra of the Jordan algebra of all bounded self-adjoint operators with symmetric multiplication acting on a complex Hilbert space.

Projective limits of Banach algebras have been studied sporadically by many authors since 1952, when they were first introduced by Arens8 and Michael48. Projective limits of complex C*-algebras were first mentioned by Arens. They have since been studied under various names by Wenjen75, Sya Do-Shin69, Brooks12, Inoue33, Schmüdgen62, Fritzsche22,23, Fragoulopoulou21, Phillips58, etc.

We will follow Inoue33 in the usage of the name « locally C*-algebras » for these objects. At the same time, in parallel with the theory of complex C*-algebras, a theory of their real and Jordan analogues, namely real C*-algebras and JB-algebras, has been actively developed by various authors9,30,45.

In chapter 2 we present definitions and basic theorems on complex and real C*-algebras, JB-algebras and complex locally C*-algebras to be used further. In chapter 3 we define a real locally Hilbert space HR and an algebra of operators L(HR) (not bounded anymore) acting on HR.

CHAPTER 1. INTRODUCTION

CHAPTER 2. PRELIMINARIES

2.1 C*-algebras and Locally C*-algebras

2.2 Jordan Algebras and JB-algebras

CHAPTER 3. PROPERTIES OF LOCALLY ADMISSIBLE OPERATOR ALGEBRAS

3.1 Real Locally Hilbert Spaces

3.2 Properties of Admissible Operators

3.3 Projective Family of Operators

CHAPTER 4. FUNDAMENTAL DEFINITIONS AND PROPERTIES

4.1 Projective Limits of Real C*-algebras

4.2 Complexification of Real Locally C*-algebras

4.3 Locally Isometry of Locally C*-algebras

4.4 Locally Isometry of Projective Limits of JB-algebras

4.5 Properties of Projective Limits of JB-algebras

CHAPTER 5. ABELIAN COMPLEX, REAL LOCALLY C*- AND LOCALLY JB-ALGEBRAS

5.1 Introduction

5.2 Gelfand-Naimark type Theorem for Abelian Complex Locally C*-algebras

5.3 Arens-Kaplansky type Theorem for Abelian Real Locally C*-algebras .

5.4 Gelfand-Naimark type Theorem for Abelian Locally JB-algebras .

CHAPTER 6. UNIVERSAL ENVELOPING ALGEBRAS FOR LOCALLY JBALGEBRAS

6.1 Representations of Locally JB-algebras as Locally JC-algebras

6.2 Universal Representations of Locally JB-algebras and their Universal

Locally C*-algebra Envelopes

CHAPTER 7. DUAL SPACE CHARACTERIZATIONS OF REAL LOCALLY C*- AND LOCALLY JB-ALGEBRAS

7.1 Preliminary Theorems

7.2 Theorems of Decomposition of Functionals

7.3 Dual Space Characterization for Real C*- and Real Locally C*-algebras

7.4 Dual Space Characterization for JB- and Locally JB-algebras

CHAPTER 8. REPRESENTATIONS OF BARRELED REAL LOCALLY C*- AND BARRELED LOCALLY JB-ALGEBRAS ON REAL AND COMPLEX HILBERT SPACE

8.1 Real and Jordan Lassner Algebras

8.2 Representations of Barreled Real Locally C*- and Locally JB-algebras

REFERENCES CITED

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STRUCTURE AND REPRESENTATION OF REAL LOCALLY C*- AND LOCALLY JB-ALGEBRAS