Starting with the normal symbold of M. Pflaum, we generalize the machinery of global symbol calculi from the familiar setting of a compact Riemannian manifold to the b-calculus on a compact manifold-with-boundary, endowed with an exact b-metric. We define a notion of b-linearization, crucial to the symbol calculus. and show that an exact b-metric can be used to construct a b-linearization. Using the b-linearization, we define the global symbol of a b-pseudodifferential operator as a fiberwise Fourier transform over the b-tangent bundle. We prove that the global symbol is truly a symbol, of the same order of its operator, and define a quantization map with which one can recover the operator (modulo b-smoothing operators.) We compute the global symbol for a b-Laplacian, and give a formula for the b-trace in terms of the global symbol.
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