On the Lie ideals of \(C^\ast\)-algebras (Q2835241)

This paper contains an introduction and the four sections 1. From pure algebra to \(C^\ast\)-algebras; 2. Nilpotents and polynomials; 3. Finite sums and sums of products; 4. Similarity invariance and the span of \(N_2\). Recall that a Lie ideal of a \(C^\ast\)-algebra \(A\) is a subspace \(L\) of \(A\) such that \([L, A]\subset L\), where \([L, A]\) is generated by additive commutators \([x, y]=xy-yx\) for elements \(x \in L\) and \(y \in A\). Recall also that the Pedersen ideal of a \(C^\ast\)-algebra is the smallest dense two-sided ideal of the \(C^\ast\)-algebra. Considered in Section 1 are several properties involving commutator ideals such as \([L, L]\) and \([A, A]\), Lie ideals, Pedersen ideals, and (associated) closed two-sided ideals. Namely, an ideal theory is obtained. Recall that the set of nilpotent elements of a \(C^\ast\)-algebra \(A\) of order \(k\) is denoted by \(N_k\). Then the closed linear subspace (denoted by \([N_k]\) here) generated by \(N_k\) is a Lie ideal of \(A\). Considered in Section 2 are several properties for \([N_k]\) as well as the Lie ideal \([f(A)]\), where \(f(A)\) consists of polynomials \(f(x_1, \dots, x_n)\) with \(x_1, \dots, x_n \in A\), where \(f\) is a polynomial with a finite number \(n\) of variables with coefficients in \(\mathbb C\). It is obtained and shown as a conclusion of Theorem 3.6 that there is a finite number \(n\) such that every element of a unital \(C^\ast\)-algebra with the closure of \([P, A]\) equal to \(A\), with \(P\) the set of projections of \(A\), can be expressed as a linear combination of \(n\) projections and \(n\) products of two projections. As another conclusion, there is a finite number \(k\) such that every commutator \([x, y]\) for \(x, y \in A\) can be expressed as a linear combination of \(k\) projections. It is obtained and shown as a conclusion of Theorem 4.2 that, if a \(C^\ast\)-algebra \(A\) has no one-dimensional representations, then the linear subspace generated by \(SN_2^c\) is equal to the commutator \([\mathrm{Pd}(A), \mathrm{Pd}(A)]\) of the Pedersen ideal \(\mathrm{Pd}(A)\) of \(A\), where \(N_2^c\) is a certain dense subset of \(N_2\) and \(SN_2^c\) is a certain union with respect to \(N_2^c\). If, in addition, \(A\) is unital, then the linear subspace generated by \(N_2\) is \([A, A]\). Furthermore, in the unital case, there is a finite number \(k\) such that any commutator \([x, y]\) in \(A\) is a sum of at most \(k\) square zero elements.