Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-12T06:29:04.916Z Has data issue: false hasContentIssue false
  • English
  • Français

Tensor products of compact convex sets and Banach algebras

Published online by Cambridge University Press:  24 October 2008

C. J. K. Batty
C. J. K. Batty
Affiliation:
Mathematical Institute, Oxford
Get access Rights & Permissions [Opens in a new window]

Extract

Alfsen and Andersen(2) defined the centre of the complete order-unit space A(K) associated with a compact convex set K to be the set of functions in A(K) which multiply with A(K) pointwise on the extreme boundary of K, thereby generalizing the concept of centres of C*-algebras. It is therefore possible to extend this definition to include the space A (K; B) of continuous affine functions of K into a Banach algebra B. Such spaces arise in the theory of weak tensor products E ⊗λB of B with a Banach space E, which may be embedded in A(K; B) where K is the unit ball of E* in the weak* topology. Andersen and Atkinson(4) considered multipliers in A(K; B) and showed that if B is unital, then the multipliers are precisely those functions which are continuous in the facial topology on the extreme boundary. It is shown here that this result extends to non-unital Banach algebras with trivial left annihilator.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 3 , May 1978 , pp. 419 - 427
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Alfsen, E. M.Compact convex sets and boundary integrals (Berlin: Springer, 1971).CrossRefGoogle Scholar
(2)Alfsen, E. M. and Andersen, T. B.On the concept of centre in A(K). J. London Math. Soc. (2) 4 (1972), 411–417.CrossRefGoogle Scholar
(3)Andersen, T. B.On multipliers and order-bounded operators in C*-algebras. Proc. Amer. Math. Soc. 25 (1970), 896–899.Google Scholar
(4)Andersen, T. B. and Atkinson, H. R.On Banach algebra valued facially continuous functions. J. London Math. Soc. (2) 9 (1974), 381–384.CrossRefGoogle Scholar
(5)Archbold, R. J.On the centre of a tensor product of C*-algebras. J. London Math. Soc. (2) 10 (1975), 257–262.CrossRefGoogle Scholar
(6)Batty, C. J. K.On relative commutants in tensor products of C*-algebras. Math. Z. 161 (1976), 215–218.CrossRefGoogle Scholar
(7)Choi, M. D. and Effros, E. G.Nuclear C*-algebras and the approximation property. Amer. J. Math. (in the Press).Google Scholar
(8)Chu, C.-H.On standard elements and tensor products of compact convex sets. J. London Math. Soc. (2) 14 (1976), 71–78.CrossRefGoogle Scholar
(9)Grothendieck, A.Produits tensoriels et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955).Google Scholar
(10)Kelley, J. L. and Namioka, I.Linear topological spaces (Princeton: van Nostrand, 1963).CrossRefGoogle Scholar
(11)Namioka, I and Phelps, R. R.Tensor products of compact convex sets. Pacific J. Math. 31 (1969), 469–480.CrossRefGoogle Scholar
(12)Sakai, S.C*-algebras and W*-algebras (Berlin: Springer, 1971).Google Scholar
(13)Schatten, R.A theory of cross-spaces, Ann. of Math. Studies 26 (Princeton, 1950).Google Scholar
(14)Vincent-Smith, G. F.The centre of the tensor product of A(K)-spaces and C*-algebras. Quart. J. Math. (Oxford) 28 (1977), 87–91.CrossRefGoogle Scholar
(15)Wickstead, A. W.The centraliser of E ⊗λ F. Pacific J. Math. 65 (1976), 563–571.CrossRefGoogle Scholar
(16)Wils, W.The ideal centre of partially ordered vector spaces. Acta Math. 127 (1971), 41–77.CrossRefGoogle Scholar