Unital multiplications on a Hilbert space

Froelich, John

doi:10.1090/S0002-9939-1993-1116259-8

Unital multiplications on a Hilbert space
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by John Froelich PDF

Proc. Amer. Math. Soc. 117 (1993), 757-759 Request permission

Abstract:

We provide a new proof of two theorems of Ingelstam that identify the Hilbert spaces with a unital multiplication satisfying $||xy|| \leqslant ||x|| ||y||$ and $||1|| = 1$.

References

H. F. Bohnenblust and S. Karlin, Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. (2) 62 (1955), 217–229. MR 71733, DOI 10.2307/1969676
John Froelich, Strictly cyclic operator algebras, Trans. Amer. Math. Soc. 325 (1991), no. 1, 73–86. MR 989575, DOI 10.1090/S0002-9947-1991-0989575-0
Lars Ingelstam, A vertex property for Banach algebras with identity, Math. Scand. 11 (1962), 22–32. MR 146679, DOI 10.7146/math.scand.a-10646
Lars Ingelstam, Hilbert algebras with identity, Bull. Amer. Math. Soc. 69 (1963), 794–796. MR 154145, DOI 10.1090/S0002-9904-1963-11035-6
Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717–726. MR 310664
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
M. F. Smiley, Real Hilbert algebras with identity, Proc. Amer. Math. Soc. 16 (1965), 440–441. MR 176316, DOI 10.1090/S0002-9939-1965-0176316-2