Peirce inner ideals in Jordan\(^*\)-triples
A complex Jordan*-triple \(A\) is said to be anisotropic if \(\{ aaa \}\neq 0\) whenever \(a\) is in \(A\setminus \{0\}\). Let \(A\) be an anisotropic complex Jordan*-triple, and \(J\) be an inner ideal of \(A\). The annihilator \(J^\perp\) and the kernel \(\text{Ker} (J)\) of \(J\) in \(A\) are defined by means of the equalities \(J^\perp:= \{a\in A: \{ JaA \}=0\}\) and \(\text{Ker} (J):= \{a\in A: \{JaJ \}= 0\}\), respectively, so that \(J^\perp\) is an inner ideal of \(A\) contained in \(\text{Ker} (J)\). \(J\) is said to be complemented in \(A\) if \(A= J\oplus \text{Ker} (J)\). The paper under review is devoted to studying complemented inner ideals \(J\) such that \(J^\perp\) is complemented too. In this case, if we put \(A_0 (J):= J^\perp\), \(A_1 (J):= \text{Ker} (J)\cap \text{Ker} (J^\perp)\), and \(A_2 (J): =J\), then we have \(A= A_0 (J) \oplus A_1 (J) \oplus A_2 (J)\). A classical and well-understood example of the situation above is the one when \(J= \{u \{uAu \}u\}\) for a given tripotent \(u\) in \(A\). In this particular setting, it is well-known that the so-called ``Peirce rules'' (i.e., \(\{A_1 (J) A_j (J) A_k (J)\} \subseteq A_{i+k-j} (J)\) when \(i+k-j\) is equal to 0, 1, or 2, and \(\{A_i (J) A_j (J) A_k (J)\}= \{0\}\) otherwise) hold. The authors prove that, in the general context considered here, (identifying pairs of symmetric relations) all except six of the ``generalized'' Peirce rules hold. More precisely, the only possible exceptions arise when \((i, j,k)\) is equal to \((0, 1,1)\), \((1,1, 0)\), \((1,0, 1)\), \((2,1,1)\), \((1,1,2)\), \((1,2,1)\), \((1,1,1)\), \((2,1,0)\), or \((0,1,2)\). Let \(A\) be a JBW*-triple, and \(J\) be an inner ideal of \(A\). Then \(A\) is anisotropic and, by previous results of the authors and \textit{K. McCrimmon} [J. Funct. Anal. 139, 196-224 (1996)], \(J\) is complemented in \(A\) if and only if it is weak*-closed in \(A\). Since \(J^\perp\) is automatically weak*-closed in \(A\) (for every \(J\)), if \(J\) is weak*-closed, then the result in the first paragraph applies. Actually, the authors show that the number of generalized Peirce rules which hold in this case is increased by one (or two is symmetric relations are distinguished). Precisely, the exceptions \((i, j, k)\) equal to \((2,1,0)\), or \((0,1,2)\) now are not allowed. In addition, an example is given of a weak*-closed inner ideal in a JBW*-triple for which all five remaining generalized Peirce rules fail to hold.
scientific article; zbMATH DE number 869799