International Journal of Mathematics

Let U₀, U₁, …, U_n be a (finite or infinite) sequence of semigroups of isometries which act on the same separable Hilbert space H, n = 1, 2, …, ∞. {U_j} is said to be orthogonal if for all i ≠ j we have

It is well known that locally defined positive definite functions on Lie groups G generally do not extend to positive definite functions which are defined on the whole group. We introduce two stronger positivity concepts for locally defined functions, and show that they are equivalent to extendability. We then apply this to the case when G is the Heisenberg group. When an additional symmetry is imposed, we obtain a complete spectral analysis of the (locally defined) positive definite functions. The methods of proof are based on unitary dilation techniques (i.e., carefully chosen extensions of some underlying Hilbert space associated to the problem), and on spectral theory for noncommuting operators.

This paper concerns the structure of continuous spatial semigroups of *-endomorphisms of . These are semigroups of *-endomorphisms of for which there exist a strongly continuous one parameter semigroup of intertwining isometries. The nature of the intertwining semigroups and the associated boundary representation is analyzed.