# F-algebra

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)
How to Cite This Entry:
F-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=F-algebra&oldid=46895
This article was adapted from an original article by C.B. Huijsmans (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article