TOPICS
Search

Left Hilbert Algebra


Let A be an involutive algebra over the field C of complex numbers with involution xi|->xi^♯. Then A is a left Hilbert algebra if A has an inner product <·,·> satisfying:

1. For all xi in A, pi_l(xi):eta|->xieta is bounded on A.

2. <xieta,zeta>=<eta,xi^♯zeta>.

3. The involution xi|->xi^♯ is closable.

4. The linear span A^2 of products xieta, xi,eta in A, is a dense subalgebra of A.

Left Hilbert algebras are historically known as generalized Hilbert algebras (Takesaki 1970).

A basic result in functional analysis says that if the involution map ♯:xi|->xi^♯ on a left Hilbert algebra A is an antilinear isometry with respect to the inner product <·,·>, then A is also a right Hilbert algebra with respect to the involution ♯. The converse also holds.


See also

Hilbert Algebra, Hilbert Space, Inner Product Space, Involutive 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

Explore with Wolfram|Alpha

References

Nelson, B. "Tomita-Takesaki Theory." http://www.math.ucla.edu/~bnelson6/Tomita-Takesaki%20Theory.pdf.Takesaki, M. Tomita's Theory of Modular Hilbert Algebras and its Applications. Berlin: Springer-Verlag, 1970.

Cite this as:

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

Subject classifications