Hilbert algebras Tomita–Takesaki theory

the main results of tomita–takesaki theory proved using left , right hilbert algebras.

a left hilbert algebra algebra involution x→x , inner product (,) such that

a right hilbert algebra defined (with involution ♭) left , right reversed in conditions above.

a hilbert algebra left hilbert algebra such in addition ♯ isometry, in other words (x,y) = (y, x).

examples: if m von neumann algebra acting on hilbert space h cyclic separating vector v, put = mv , define (xv)(yv) = xyv , (xv) = x*v. tomita s key discovery makes left hilbert algebra, in particular closure of operator has polar decomposition above. vector v identity of a, unital left hilbert algebra.

if g locally compact group, vector space of continuous complex functions on g compact support right hilbert algebra if multiplication given convolution, , x(g) = x(g)*.