On \(q\)-normal operators and the quantum complex plane (Q2862127)

The paper contributes to non-commutative real algebra. The methods are mostly from functional analysis. An important tool are \(q\)-normal operators on Hilbert spaces, where \(0 < q \in \mathbb{R}\). The notion generalizes normal operators. After the discussion of \(q\)-normal operators the authors turn to the study of the \(\ast\)-algebra \(A = \mathbb{C}\langle x,x^{\ast}\rangle\), where \(x,x^{\ast}\) are variables, \(xx^{\ast} = qx^{\ast}x\) and the involution is given by \((\lambda x)^{\ast} = \overline{\lambda}x^{\ast}\). In \(A\) there is a notion of positivity, which is defined using representations produced with \(q\)-normal operators. There are also sums of squares in \(A\). These are the sums of elements \(a^{\ast}a\) with \(a \in A\). The set of positive elements is denoted by \(A_+\), the subset of sums of squares by \(\Sigma A^2\). The auhors show how one can decide whether an element \(a \in A\) belongs to \(A_+\) or to \(\Sigma A^2\). They exhibit a construction producing plenty of elements in \(A_+ \setminus \Sigma A^2\). In commutative real algebra the moment problem is known to be closely related to Positivstellensätze. The notion of a \(q\)-moment functional is introduced and it is shown that a functional \(F\) on \(A\) is a \(q\)-moment functional if and only if \(F(a) \geq 0\) for all \(a \in A_+\).