# Distributions and partial differential equations on by Khrennikov

By Khrennikov

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Nauk SSSR, 313, No. 2, 325-329 (1990). 69. A. Yu. Khrennikov, "Representation of second quantization over non-Archimedean number fields," DokI. Akad. Nauk SSSR, 314, 1380-1384 (1990). 70. A. Yu. Khrennikov, "Quantum mechanics over Galois extensions of number fields," Dokl. Akad. Nauk SSSR, 315,860-864 (1990). 71. A. Yu. Khrennikov, "Trotter's formula for the heat and Schr5dinger equations on non-Archimedean superspace," Sib. Mat. , 32, No. 5, 155-165 (1991). 72. A. Yu. Khrennikov, "Wiener-Feynman process on superspace," Teor.

OL# *L U. u) = ~ The convolution of a distribution # and of a test function ~ is well-defined by the equality ~ . ~ ( ~ , o ) = f u(d~dr - u,e - ~), Let us check, for instance, that the mapping * : E(C~ 'm) x E'(C~ 'm) --+ E(C~ 'm) is continuous. Let 0 " 0L~ ,,m ), # 6 E'(C~'m). 7, # = ~x" Oz" 00~ 5(x, O)~,~, and with q~ 6 ER(C A 848 this iI~IIR I 0" = E I1~11R~ < ~ for any R. Consequently, q~, #(x, 0) = E < R~[][~[]]R. a~ 0 ~ 0~ ~(x, 0)poZ; note that Therefore, II1~*~111, < IIt~1II. ~ limekiln" = IIl~lBIRll,llR.

Then IIj(A~ = inf z=z~ 1E M ]l( ~0 -[- z~ -[- zlll = z~ inf zIEBi (11 ~ + z~ + Ilzlll) inf IIA~ + z~ = IIj(;~~ z~ . Thus, the quotient algebra B / M ~- Bo/Mo is isomorphic to the field C ( here we use a familiar property of maximal ideals in commutative Banach algebras). Consequently, the maximal ideal M has codimension one, and the identity element e of the algebra B does not belong to the ideal. It remains to set b e = e and b IM = 0. By this theorem, there is a one-to-one correspondence between the body projections in a CSA and the maximal ideals in the CSA.