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C t (V) is xt = X rv . 46 In V the topology is defined by the following N -depending seminorms: E V -+ 1 ¢lln == IINn¢ll , n E No , while the topology TO in £t(V) is introduced by the seminorms ¢ where kENo and f E C, the set of all the positive, bounded and continuous functions on lR+, which are decreasing faster than any inverse power of x : £t(V)[TO] is a complete *-algebra. Let further £(V, V') be the set of all continuous maps from V into V', with their topologies (in V' this is the strong dual topology), and let T denotes the t opology defined by the seminorms X E £(V, V') -+ IIXII!

Therefore, defining K,f := Logcf where Logcf denotes the principal determination of the logarithm of cf one has cf =: e"'! 2) we know that the kernel qo(f, g) is CPDH because q(f, g) is PDH hence a fortiori CPDH. (ii)B(iii). 2). This is equivalent to say that k(f, g) is an infinitely divisible PDH kernel. 2), it has the form (69) for some CPDH kernel qo. 1), and up to replacing qo by an equivalent CPD hermitian kernel, Vfa E S the kernel 30 is PDH. )vf, e-qo(fo,g)v g) = and (iv) follows by choosing "'f := qo(fo, f) (iv) B (i).

Symmetric Hilbert spaces and related topics, Lect. Notes 33 Math. 261, Springer, Berlin, 1972. 18. : Four dimensional symplectic Lie algebras, Beitrage Algebra Geom. 47 (2006), no. 2, 419-434. 19. R. , Positive definite kernels continuous tensor products and central limit theorems of probability theory, Springer Lecture Notes in Mathematics no. 272, 1972. 34 MODULAR STRUCTURES AND LANDAU LEVELS F. it/~ bagarell We review some recent results concerning Landau levels and TomitaTakesaki modular theory.