F-algebra
A real vector space that is simultaneously a lattice is called a vector lattice (or Riesz space) whenever
(
is the lattice order) implies
for all
and
for all positive real numbers
. If
is also an algebra and
and
for all
, the positive cone of
, then
is called an
-algebra (a lattice-ordered algebra, Riesz algebra).
A Riesz algebra is called an
-algebra (
for "function" ) whenever
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This notion was introduced by G. Birkhoff and R.S. Pierce in 1956.
An important example of an -algebra is
, the space of continuous functions (cf. Continuous functions, space of) on some topological space
. Other examples are spaces of Baire functions, measurable functions and essentially bounded functions.
-Algebras play an important role in operator theory. The second commutant of a commuting subset of bounded Hermitian operators on some Hilbert space is an
-algebra. A linear operator
on some vector lattice
is called an orthomorphism whenever
is the difference of two positive orthomorphisms; a positive orthomorphism
on
leaves the positive cone of
invariant and satisfies
whenever
. The space
of all orthomorphisms of
is an important example of an
-algebra in the theory of vector lattices.
A vector lattice is termed Archimedean if
(
) implies
. Archimedean
-algebras are automatically commutative and associative. An Archimedean
-algebra with unit element is semi-prime (i.e., the only nilpotent element is
). The latter two properties are nice examples of the interplay between order properties and algebraic properties in an
-algebra. Many properties of
are inherited by an
-algebra
with a unit element (under some additional completeness condition), such as the existence of the square root of a positive element (if
, then there exists a unique
such that
) and the existence of an inverse: if
is the unit element of
and
, then
exists in
.
References
[a1] | G. Birkhoff, R.S. Pierce, "Lattice-ordered rings" An. Acad. Brasil. Ci. , 28 (1956) pp. 41–69 |
[a2] | A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983) |
F-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=F-algebra&oldid=46895