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Hilbert Algebra


There are at least two distinct (though related) notions of the term Hilbert algebra in functional analysis.

In some literature, a linear manifold A of a (not necessarily separable) Hilbert space H=(H,<·,·>) is a Hilbert algebra if the following conditions are satisfied:

1. A is dense in H.

2. A is a ring so that, for any a,b in A, there is defined an element ab in A such that (ab)c=a(bc), a(b+c)=ab+ac, (a+b)c=ac+bc, and (alphaa)b=a(alphab)=alphaab for any complex number alpha in C.

3. For any a in A, there exists an adjoint element a^* in A such that <ab,c>=<b,a^*c>, and <ba,c>=<b,ca^*>.

4. For any a in A, there exists a positive number alpha_(a) such that ax<=alpha_(a)x for all x in A.

5. For every a in A, there exists a unique bounded linear operator T_(a) on H such that T_(a)x=ax for all x in A. Moreover, if T_(x)f=0 for an element f in H and for all x in A, then f=0.

At least one author defines a Hilbert algebra to be a quasi-Hilbert algebra

 U=(U,<·,·>,H,*, ^ , v )

for which x^ ^ =x for all x in U (Dixmier 1981).


See also

Hilbert Space, Inner Product Space, Left Hilbert Algebra, Linear Manifold, Modular Hilbert Algebra, Quasi-Hilbert Algebra, Right Hilbert Algebra, Ring, Subspace, Unimodular Hilbert Algebra, Vector Space, von Neumann Algebra

This entry contributed by Christopher Stover

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References

Dixmier, J. Von Neumann Algebras. Amsterdam, Netherlands: North-Holland, 1981.Nakano, H. "Hilbert Algebras." Tôhoku Math. J., 2, 4-23, 1950.

Cite this as:

Stover, Christopher. "Hilbert Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HilbertAlgebra.html

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