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Quasi-Hilbert Algebra


Let U=(U,<··>) be a T2 associative inner product space over the field C of complex numbers with completion H, and assume that U comes with an antilinear involution xi|->xi^* and a bijective linear mapping xi|->xi^ ^ with inverse xi|->xi^ v . U is said to be a quasi-Hilbert algebra if the following axioms are satisfied:

1. <x,y>=<y^*,x^*> for all x,y in U.

2. <xy,z>=<y,(x^*)^ ^ z> for all x,y,z in U.

3. For each x in U, the map y|->xy is continuous.

4. The collection of all products xy of elements x,y in U is dense in U.

5. If a,b are elements of H such that <a,xy>=<b,x^ ^ y^ ^ > for every x,y in U, there exists a sequence {x_n} in U such that x_n->b and x_n^ ^ ->a.

When all components of a quasi-Hilbert algebra need to be explicitly acknowledged, one may write U=(U,<·,·>,H,*, ^ , v ).


See also

Hilbert Algebra, Hilbert Space, Inner Product Space, Involutive Algebra, Left Hilbert Algebra, Linear Manifold, Modular 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.

Cite this as:

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

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