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We investigate the algebraic properties of the quantum counterpart of theclassical canonical transformations using the symbol-calculus approach toquantum mechanics. In this framework we construct a set of pseudo-differentialoperators which act on the symbols of operators, i.e., on functions definedover phase-space. They act as operatorial left- and right- multiplication andform a $W_{\infty}\times W_{\infty}$- algebra which contracts to its diagonalsubalgebra in the classical limit. We also describe the Gel'fand-Naimark-Segal(GNS) construction in this language and show that the GNS representation-space(a doubled Hilbert space) is closely related to the algebra of functions overphase-space equipped with the star-product of the symbol-calculus.Comment: TeX file with phyzzs macro, 23 pages, no figure

QUANTUM-DEFORMED CANONICAL TRANSFORMATIONS, W∞ ALGEBRAS AND UNITARY TRANSFORMATIONS

QUANTUM-DEFORMED CANONICAL TRANSFORMATIONS, W_∞ ALGEBRAS AND UNITARY TRANSFORMATIONS