F-algebra

A real vector space $ A $ that is simultaneously a lattice is called a vector lattice (or Riesz space) whenever $ x \leq y $( $ \leq $ is the lattice order) implies $ x + z \leq y + z $ for all $ z \in A $ and $ ax \leq ay $ for all positive real numbers $ a $. If $ A $ is also an algebra and $ zx \leq yz $ and $ xz \leq yz $ for all $ z \in A ^ {+} $, the positive cone of $ A $, then $ A $ is called an $ l $- algebra (a lattice-ordered algebra, Riesz algebra).

A Riesz algebra $ A $ is called an $ f $- algebra ( $ f $ for "function" ) whenever

$$ \inf ( x,y ) = 0 \Rightarrow \inf ( zx,y ) = \inf ( xz,y ) = 0 , \forall z \in A ^ {+} . $$

This notion was introduced by G. Birkhoff and R.S. Pierce in 1956.

An important example of an $ f $- algebra is $ A = C ( X ) $, the space of continuous functions (cf. Continuous functions, space of) on some topological space $ X $. Other examples are spaces of Baire functions, measurable functions and essentially bounded functions. $ f $- 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 $ f $- algebra. A linear operator $ T $ on some vector lattice $ A $ is called an orthomorphism whenever $ T $ is the difference of two positive orthomorphisms; a positive orthomorphism $ S $ on $ A $ leaves the positive cone of $ A $ invariant and satisfies $ \inf ( Sx,y ) = 0 $ whenever $ \inf ( x,y ) = 0 $. The space $ { \mathop{\rm Orth} } ( A ) $ of all orthomorphisms of $ A $ is an important example of an $ f $- algebra in the theory of vector lattices.

A vector lattice $ A $ is termed Archimedean if $ 0 \leq nx \leq y $( $ n = 1,2, \dots $) implies $ x = 0 $. Archimedean $ f $- algebras are automatically commutative and associative. An Archimedean $ f $- algebra with unit element is semi-prime (i.e., the only nilpotent element is $ 0 $). The latter two properties are nice examples of the interplay between order properties and algebraic properties in an $ f $- algebra. Many properties of $ C ( X ) $ are inherited by an $ f $- algebra $ A $ with a unit element (under some additional completeness condition), such as the existence of the square root of a positive element (if $ x \in A ^ {+} $, then there exists a unique $ y \in A ^ {+} $ such that $ y ^ {( 2 ) } = x $) and the existence of an inverse: if $ e $ is the unit element of $ A $ and $ e \leq x $, then $ x ^ {- 1 } $ exists in $ A $.

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)