The Schatten space 𝑆₄ is a 𝑄-algebra

Le Merdy, Christian

doi:10.1090/S0002-9939-98-04545-6

The Schatten space $S_4$ is a $Q$-algebra
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by Christian Le Merdy PDF

Proc. Amer. Math. Soc. 126 (1998), 715-719 Request permission

Abstract:

For any $1 \leq p \leq \infty$, let $S_{p}$ denote the classical $p$-Schatten space of operators on the Hilbert space $\ell _{2}$. It was shown by Varopoulos (for $p \geq 2$) and by Blecher and the author (full result) that for any $1 \leq p \leq \infty , S_{p}$ equipped with the Schur product is an operator algebra. Here we prove that $S_{4}$ (and thus $S_{p}$ for any $2 \leq p \leq 4$) is actually a $Q$-algebra, which means that it is isomorphic to some quotient of a uniform algebra in the Banach algebra sense.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
David P. Blecher, Tensor products of operator spaces. II, Canad. J. Math. 44 (1992), no. 1, 75–90. MR 1152667, DOI 10.4153/CJM-1992-004-5
David P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), no. 1, 15–30. MR 1145913, DOI 10.2140/pjm.1992.153.15
David P. Blecher and Christian Le Merdy, On quotients of function algebras and operator algebra structures on $l_p$, J. Operator Theory 34 (1995), no. 2, 315–346. MR 1373327
David P. Blecher and Vern I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), no. 2, 262–292. MR 1121615, DOI 10.1016/0022-1236(91)90042-4
A. M. Davie, Quotient algebras of uniform algebras, J. London Math. Soc. (2) 7 (1973), 31–40. MR 324427, DOI 10.1112/jlms/s2-7.1.31
Edward G. Effros and Zhong-Jin Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), no. 3, 329–337. MR 1127754, DOI 10.4153/CMB-1991-053-x
Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 829919, DOI 10.1090/cbms/060
Gilles Pisier, The operator Hilbert space $\textrm {OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. MR 1342022, DOI 10.1090/memo/0585
Nicholas Th. Varopoulos, Some remarks on $Q$-algebras, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 4, 1–11 (English, with French summary). MR 338780, DOI 10.5802/aif.432
Nicholas Th. Varopoulos, Sur les quotients des algèbres uniformes, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1344–A1346 (French). MR 293403
N. Th. Varopoulos, A theorem on operator algebras, Math. Scand. 37 (1975), no. 1, 173–182. MR 397436, DOI 10.7146/math.scand.a-11599
J. Wermer, Quotient algebras of uniform algebras, in Symposium on function algebras and rational approximation, University of Michigan, 1969.