zbMATH — the first resource for mathematics

Solution of the contractive projection problem. (English) Zbl 0558.46035
Building on tools developed in an earlier paper [Math. Scand. 52, 279-311 (1983)] the authors show that the range of any contractive projection acting on a \(C^*\)-algebra has a natural structure of Jordan triple system. More generally they prove that the category of \(J^*\)-algebras [in the sense of L. A. Harris, Lecture Nores Math. 364, 13-40 (1974; Zbl 0293.46049)] is stable under the action of norm one projections.
The significance of this result is that it shows that JB\({}^*\)-triples in the sense of W. Kaup [Math. Z. 183, 503-529 (1983; Zbl 0519.32024)] occur naturally in the theory of (associative) operator algebras. Inspired by this result, W. Kaup proved the analogous result for \(JB^*\)-triples using holomorphic methods [Math. Scand. 54, 95-100 (1984)]. This result of Kaup, proved earlier in a geometric form by L. L. Stacho [Acta Sci. Math. 44, 99-124 (1982; Zbl 0505.58008)] has already found an important application in the representation theory of Jordan triple systems [cf. the authors: The Gelfand Naimark theorem for \(JB^*\)-triples, preprint].

46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17A40 Ternary compositions
47C15 Linear operators in \(C^*\)- or von Neumann algebras
17A65 Radical theory (nonassociative rings and algebras)
46L40 Automorphisms of selfadjoint operator algebras
Full Text: DOI
[1] Alfsen, E; Shultz, F.W, State spaces of Jordan algebras, Acta math., 140, 155-190, (1978) · Zbl 0397.46066
[2] Alfsen, E; Hanche-Olsen, H; Shultz, F.W, State spaces of C∗-algebras, Acta math., 144, 267-305, (1980) · Zbl 0458.46047
[3] Alfsen, E; Shultz, F.W; Størmer, E, A Gelfand-Naimark theorem for Jordan algebras, Adv. in math., 28, 11-56, (1978) · Zbl 0397.46065
[4] Arazy, J; Friedman, Y, Contractive projections in C1and C∞, Mem. amer. math. soc., 13, No. 200, (1978)
[5] Choi, M.D; Effros, E, Injectivity and operator spaces, J. funct. anal., 24, 156-209, (1977) · Zbl 0341.46049
[6] Effros, E; Størmer, E, Positive projections and Jordan structure in operator algebras, Math. scand., 45, 127-138, (1979) · Zbl 0455.46059
[7] Friedman, Y; Russo, B, Contractive projections on C0(K), Trans. amer. math. soc., 273, 57-73, (1982) · Zbl 0534.46037
[8] Friedman, Y; Russo, B, Contractive projections on C∗-algebras, (), 615-618 · Zbl 0525.46029
[9] Friedman, Y; Russo, B, Contractive projections on operator triple systems, Math. scand., 52, 279-311, (1983) · Zbl 0547.46048
[10] Harris, L, Bounded symmetric homogeneous domains in infinite dimensional spaces, (), 13-40 · Zbl 0293.46049
[11] Harris, L, A generalization of C∗-algebras, (), 331-361 · Zbl 0476.46054
[12] Loos, O, Jordan pairs, () · Zbl 0301.17003
[13] Takesaki, M, Theory of operator algebras I, (1979), Springer-Verlag Berlin/New York/Heidelberg · Zbl 0990.46034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.