### Abstract:

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
physics.
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
as 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
Fragoulopoulou.
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 Wenjen, Sya Do-Shin, Brooks, Inoue, Schmüdgen,
Fritzsche, Fragoulopoulou, Phillips, 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 authors.
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.
In chapter 4 we give new definitions and study several properties of locally C*-
and locally JB-algebras. Then we show that a real locally C*-algebra (locally JBalgebra)
is locally isometric to some closed subalgebra of L(HR).
In chapter 5 we study complex and real Abelian locally C*-algebras.
In chapter 6 we study universal enveloping algebras for locally JB-algebras.
In chapter 7 we define and study dual space characterizations of real locally C*
and locally JB-algebras.
In chapter 8 we define barreled real locally C* and locally JB-algebras and study
their representations as unbounded operators acting on dense subspaces of some Hilbert
spaces.
It is beneficial to extend the existing theory to the case of real and Jordan
analogues of complex locally C*-algebras. The present thesis is devoted to study such
analogues, which we call real locally C*- and locally JB-algebras.