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Note that the function f (x) = 1 goes to the identity operator I H Borrowing from Berezin and Lieb their terminology, one names the function f (x) = A f (x) the contravariant (Berezin) or the upper symbol (Lieb) of the operator A f , whereas one names the mean value x|A f |x = A f (x) the covariant or the lower symbol of A f The terminology lower and upper comes from the following inequalities involving symbols of self-adjoint operators Let g ( ) be a real-valued convex function u

a i g ( i )

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ai = 1 (1313)

and let A be a self-adjoint operator on H that has well-de ned lower and upper symbols with respect to the coherent state frame {|x }x X Then the Berezin Lieb inequalities g (A(x)) (dx) u Tr g (A) u

g (A(x)) (dx)

(1314)

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hold depending on suitable properties of A A particular case of these inequalities was proved and applied to quantum spin systems in 7 Now, in agreement with 12, we should make more precise, although still loose, our de nition of a classical observable in terms of lower symbols De nition 131 Given a frame {|x , x X }, a function X x f (x) K, possibly understood in a distributional sense, is a coherent state quantizable classical observable along the map f A f de ned by (1312) (i) if the map X x x|A f |x is a regular function with respect to some additional structure allocated to X (ii) we must get the right classical limit, which means that x|A f |x W f (x) as a certain parameter goes to 0 One can say that, according to this approach, a quantization of the observation set X is in one-to-one correspondence with the choice of a frame in the sense of (1311) This term of frame [61] is more appropriate for designating the total family {|x }x X To a certain extent, the coherent state quantization scheme consists in adopting a certain point of view in dealing with X This frame can be discrete or continuous, depending on the topology additionally allocated to the set X, and it can be overcomplete, of course The validity of a precise frame choice is determined by comparing spectral characteristics of quantum observables A f with experimental data Of course, a quantization scheme associated with a speci c frame is intrinsically limited to all those classical observables for which the expansion (1312) is mathematically justi ed within the theory of operators in Hilbert space (eg, weak convergence) However, it is well known that limitations hold for any quantization scheme

ance goals and objectives This, in turn, forms a basis for determining how risks should be managed Risk assessment is one of management s responsibilities and enables management to act proactively in reducing unwanted surprises Failure to consciously manage risk can result in a lack of confidence that operational, financial, and compliance goals will be achieved A risk is anything that could jeopardize the achievement of an objective Asking the following questions helps to identify risks:

The advantage of the coherent state quantization illustrated here is that it requires a minimal signi cant structure on X, namely, the existence of a measure (dx), together with a -algebra of measurable subsets, and some additional structure to be de ned according to the context The construction of the Hilbert space H (or K) is equivalent to the choice of a class of eligible quantum states, together with a technical condition of continuity A correspondence between classical and quantum observables is then provided through a suitable generalization of the standard coherent states An interesting question to be addressed concerns the emergence or not of noncommutativity through the quantum processing of the set X Suppose the large signal space L2 (X , ) has an orthonormal basis { n (x), n N} that satis es the K required condition N (x) =

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