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Spectral algebras. (English) Zbl 0790.46038

Summary: Spectral algebras are a class of abstract complex algebras which share many of the good properties of Banach algebras. In the commutative case they are precisely the class of abstract algebras having a full Gelfand theory. Any irreducible representation of the spectral algebra is strictly dense. Spectral algebras are defined and characterized in terms of spectral pseudo-norms and spectral subalgebras. Spectral algebras, spectral subalgebras and spectral pseudo-norms are shown to occur frequently in analysis.
It is also shown that when the spectral radius is finite valued, if it is either subadditive or submultiplicative, then it has both properties and that this occurs exactly for algebras which are spectral algebras and commutative modulo their Jacobson radicals.

MSC:

46H05 General theory of topological algebras
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[1] B.A. Barnes, Modular annihilator algebras , Canad. J. Math. 18 (1966), 566-578, MR 33 2681. · Zbl 0156.04003
[2] N. Bourbaki, Theories Spectrales , 1 and 2, Elem. Math. Fasc. 23 , Hermann, Paris, 1967, MR 35 4725. · Zbl 0152.32603
[3] S.B. Cleveland, Homomorphisms of non-commutative \(^*\)-algebras , Pacific J. Math. 13 (1963), 1097-1109, MR 28 1500. · Zbl 0205.42203
[4] H.G. Dales, Norming nil algebras , Proc. Amer. Math. Soc. 83 (1981), 71-74, MR 82j 46069. · Zbl 0474.46041
[5] R. Fuster and A. Marquina, Geometric series in incomplete normed algebras , Amer. Math. Monthly 91 (1984), 49-51, MR 85g 46059. JSTOR: · Zbl 0553.46032
[6] I.M. Gelfand, Normierte Ringe , Mat. Sbornik (N.S.) 9 (51) (1941), 3-24, MR 3 51. · JFM 67.0406.02
[7] N. Jacobson, The radical and semisimplicity for arbitrary rings , Amer. J. Math. 67 (1945), 300-320, MR 7 2. JSTOR: · Zbl 0060.07305
[8] ——–, Structure theory of simple rings without finiteness assumptions , Trans. Amer. Math. Soc. 57 (1945), 228-245, MR 6 200. JSTOR: · Zbl 0060.07401
[9] I. Kaplansky, Topological rings , Amer. J. Math. 69 (1947), 153-183, MR 8 434. JSTOR: · Zbl 0034.16604
[10] ——–, Locally compact rings , Amer. J. Math. 70 (1948), 447-459, MR 9 562. JSTOR: · Zbl 0036.02203
[11] ——–, Normed algebras , Duke Math. J. 16 (1949), 399-418, MR 11 115. · Zbl 0033.18701
[12] C. Le Page, Sur quelques conditions entrainant la commutativité dans les algèbres de Banach , C.R. Acad. Sci. Paris Ser. A-B 265 (1967), 235-237, MR 37 1999. · Zbl 0158.14102
[13] V. Mascioni, Some characterizations of complex \(Q\)-algebras , El. Math. 42 (1987), 10-14, MR 88k 46053. · Zbl 0713.46032
[14] E.A. Michael, Locally multiplicatively-convex topological algebras , Mem. Amer. Math. Soc. No. 11 (1952), 79 pp, MR 14 482. · Zbl 0047.35502
[15] Angel Rodriquez Palacios, The uniqueness of the complete norm topology in complete normed nonassociative algebras , J. Funct. Anal. 60 (1985), 1-15, MR 86f 46053. · Zbl 0602.46055
[16] T.W. Palmer, Hermitian Banach \(^*\)-algebras , Bull. Amer. Math. Soc. 78 (1972), 522-524, MR 45 7481. · Zbl 0255.46045
[17] Vlastimil Ptak, On the spectral radius in Banach algebras with involution , Bull. London Math. Soc. (2) (1970), 327-334, MR 43 932. · Zbl 0209.44403
[18] Vlastimil Ptak and Jaroslav Zemanek, On uniform continuity of the spectral radius in Banach algebras , Manuscripta Math. 20 (1977), 177-189, MR 56 1065. · Zbl 0347.46051
[19] D.A. Raikov, To the theory of normed rings with involution , C.R. (Doklady) Acad. Sci. URSS (N.S.) 54 (1946), 387-390, MR 8 469. · Zbl 0060.27102
[20] C.E. Rickart, The uniqueness of norm problem in Banach algebras , Ann. of Math. (2) 51 (1950), 615-628, MR 11 670. JSTOR: · Zbl 0037.20003
[21] ——–, On spectral permanence for certain Banach algebras , Proc. Amer. Math. Soc. 4 (1953), 191-196, MR 14 660. JSTOR: · Zbl 0051.09106
[22] ——–, An elementary proof of the fundamental theorem in the theory of Banach algebras , Michigan Math. J. 5 (1958), 75-78, MR 20 4786. · Zbl 0084.33502
[23] ——–, General theory of Banach algebras , van Nostand, Princeton, 1960. (Reprint: Krieger, Huntington, NY, 1974), MR 22 A 5903.
[24] M.A. Rieffel, Square-integrable representations of Hilbert algebras , J. Funct. Anal. 3 (1969), 265-300, MR 39 6094. · Zbl 0174.44902
[25] M. Rosenbloom, On a theorem of Fuglede and Putnam , J. London Math. Soc. 33 (1958), 376-377, MR 20 6037. · Zbl 0081.11902
[26] G.E. Šilov, On the extension of maximal ideals , C.R. (Doklady) Acad. Sci. URSS (N.S.) 29 (1940), 83-84, MR 2 314. · JFM 66.0108.03
[27] A. Wilansky, Letter to the editor , Amer. Math. Monthly 91 (1984), 531. · Zbl 0109.24501
[28] B. Yood, Homomorphisms on normed algebras , Pacific J. Math. 8 (1958), 373-381, MR 21 2924. · Zbl 0084.33601
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