By Richard V. Kadison

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**Extra info for A Representation Theory for Commutative Topological Algebra**

**Example text**

I) If ~ is infinite dimensional and everywhere defined, then and A is uniquely given by Moreover where D = ~ B for some A 89 ~e is bounded and , their norms are equivalent A(x,y) = ~,B-ly) is self adjoint and B+ _ aS B -I ~=~+~ ~_ for all , with x and y. B = B+ G B_ i a > 0 , and f(x)dx= Ve~XmI2 ~ i j B+~fe~IX21~- f(xl,X2idx2~dx I , where f(x1'x2) = f(xs*x2) ' ~ B + is the closure of D(B+) c ~ + 2 in the norm IXlB = (x,B+x) , and the integrals on the right + - m hand side are ordinary normalized integrals as defined in section 2.

Sense that it has a bounded the range of However easely B , and that is if that A(x,y) D* . e. by restricting D B . is D* , is the dual space of just the restriction D . A(x,y) B case A is non degenerate continuous D = ~ inverse , since =(x,B-ly) . D Hence is in B -I contains A is in this case unique and real. Since fixed A(x,y) x , A(x,y) hence in D* . 8), a left inverse D(B) c D* on of of D B , considered into D* on D, which as a map from D . We now define the Fresnel integral with respect to the form A.

Dt n . 5 n where on ax = Z aixi , for some bounded complex measures ~ and i=I Since e iax is a generalized eigenfunction for H o , Rn . 3) that (DO $(x,t) = Z n=O (-i) n I'''I 0_