Friday, 1 June 2007

examples - description of functions of conditionally negative type on a group

Recall that a kernel conditionaly of negative type on a set X is a map psi:XtimesXrightarrowmathbbR with the following properties:



1) psi(x,x)=0



2) psi(y,x)=psi(x,y)



3) for any elements x1,...xn and all real numbers c1,...,cn, with c1+...+cn=0, the following inequality holds:
sumi=1nsumj=1ncicjpsi(xi,xj)leq0.



Let G be a discrete group.
Recall that a function GrightarrowmathbbR is conditionally of
negative type if the kernel psi, defined by psi(g,h)=psi(h1g) is conditionaly of negative type.




Does there exist class of discrete groups which admit an explicit description of functions which are conditionaly of negative type?

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