Abstract
We introduce a new product of two test functions denoted by f square g ( where f and g in the Schwartz space S(R)). Based on the space of entire functions with theta-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product f square g, such operators give us a new representation of the centerless Virasoro-Zamolodchikov-omega(infinity) *-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function f as any Hermite function. Replacing the classical pointwise product f . g of two test functions f and g by f square g, we prove the existence of new *-Lie algebras as counterpart of the classical powers of white noise *-Lie algebra, the renormalized higher powers of white noise (RHPWN) *-Lie algebra and the second quantized centerless Virasoro-Zamolodchikov-omega(infinity) *-Lie algebra.