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

## Authors

- First Online:

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DOI: 10.1007/s00233-002-0003-7

- Cite this article as:
- Bisgaard, T. Semigroup Forum (2004) 68: 25. doi:10.1007/s00233-002-0003-7

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## Abstract

There is a countable cancellative commutative *-semigroup S withzero (in
fact, a *-subsemigroup of G × N_{0} 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 M_{d}(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, s_{1}, . . . , s_{n} in S , and $\xi_1$, . . . , $\xi_n$ in
C^{d}, does it follow that $f(s) = \int_{S^*}\sigma (s) d\mu(\sigma) (s \memb S)$ for some measure $\mu$
(with values in M_{d}(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).