Semigroup Forum

, Volume 68, Issue 1, pp 25–46 | Cite as

On the Relation between the Scalar Moment Problem and the Matrix Moment Problem on *-Semigroups

Research Article


There is a countable cancellative commutative *-semigroup S withzero (in fact, a *-subsemigroup of G × N0 for some abelian group G carrying the inverse involution) such that the answer to the question “if f is a function on S , with values in Md(C) (the square matrices of order d) and such that $\sum^{n}_{j,k=1} \lbrak f(s^*_k s_j)\xi_j, \xi_k \rbrak \ge 0$ for all n in N, s1, . . . , sn in S , and $\xi_1$, . . . , $\xi_n$ in Cd, does it follow that $f(s) = \int_{S^*}\sigma (s) d\mu(\sigma) (s \memb S)$ for some measure $\mu$ (with values in Md(C)+ , the positive semidenite matrices) on the space S of hermitian multiplicative functions on S?” is “yes” if d = 1 but “no” if d = 2 (hence also for d > 2).


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Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Nandrupsvej 7 st. th. DK-2000 Frederiksberg CDenmark

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