www. kokanduni.uz denotes the left-annihilator of S.
Definition 2.3. A Rickart *-ring is a *-ring A such that, for each
x A
,
({ })
R x gA =
with
g a projection (note that such a projection is unique). It follows that
({ })
( ({ *}))*
(
)*
L x R x hA Ah =
=
=
for a suitable projection
h . A (complex) C*-algebra that is a
Rickart *-ring will be called a Rickart C*-algebra
The connection of a rickart real c*-algebra with its enveloping rickart (complex) c*-algebra: In this section, we consider the connection of a Rickart real C*-algebra with its
enveloping Rickart (complex) C*-algebra. There is example of a Rickart real C*-algebra for
which the enveloping C*-algebra (i.e., its complexification) is not a Rickart (complex) C*-
algebra.
By applying the scheme of proof of [7, Proposition 4.2.3] we obtain:
Theorem 3.1. The complex C*-algebra B+iB is not a Rickart C*-algebra.
Now let us consider the converse problem: if A is a real C*-algebra and A+iA is a Rickart
C*-algebra is A necessarily a Rickart real C*-algebra?
The following result gives a positive answer to this problem, which is the main result of
this section.
Theorem 3.2. Let A be a real C*-algebra and let
c A A iA = +
be its complexification.
Suppose that
c A is a Rickart C*-algebra. Then A is a Rickart real C*-algebra.
Main results. In this section, we will show that there is a Rickart real C*-algebra whose
projection lattice is not complete. We will consider the connection between Rickart real C*-
algebra and real AW*-algebra.
