Geometry of star-algebras and sequence spaces (Q2725636)

Let \(A\) be a complex \(B^*\)-algebra with regular norm. An element \(h\) of \(A\) is called Hermitian if the numerical range of \(h\), NEWLINE\[NEWLINEV_h(h)\equiv \{f(hx); f\in A', \|f\|=1, f(x)=\|x\|= 1\}\subset\mathbb{R}NEWLINE\]NEWLINE and an element \(a\) of \(A\) is called self-adjoint if \(a^*= a\). The authors show that the two notions of Hermitian and self-adjoint elements are equivalent. Using this result, they extend the Vidav-Palmer theorem [A unital Banach algebra with an involution is a \(C^*\)-algebra if and only if \(A= H(A)+ iH(A)\), where \(H(A)\) is the set of all Hermitian elements] to the case when \(A\) has no identity but has regular norm. Furthermore, the state preserving operators on a unital \(C^*\)-algebra are characterized. An operator \(T\) on a \(C^*\)-algebra \(A\) is state preserving if \(T^* D(A,1)\subset D(A,1)\), where \(D(A,1)\) is the state space of \(A\). They obtain the following result:NEWLINENEWLINENEWLINELet \(T\) be a bounded linear operator on a \(C^*\)-algebra \(A\). Then \(T\) is state preserving if and only if \(Ta\in H(A)\) and \(T1= 1\) whenever \(a\in H(A)\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00019].