Advances in Operator Theory

Semicontinuity and closed faces of $C^*$-algebras

Lawrence G. Brown

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785-795.] defined three concepts of semicontinuity for self-adjoint elements of $A^{**}$, the enveloping von Neumann algebra of a $C^*$-algebra $A$. We give the basic properties of the analogous concepts for elements of $pA^{**}p$, where $p$ is a closed projection in $A^{**}$. In other words, in place of affine functionals on $Q$, the quasi-state space of $A$, we consider functionals on $F(p)$, the closed face of $Q$ suppported by $p$. We prove an interpolation theorem: If $h \geq k$, where $h$ is lower semicontinuous on $F(p)$ and $k$ upper semicontinuous, then there is a continuous affine functional $x$ on $F(p)$ such that $k \leq x \leq h$. We also prove an interpolation-extension theorem: Now $h$ and $k$ are given on $Q$, $x$ is given on $F(p)$ between $h_{|F(p)}$ and $k_{|F(p)}$, and we seek to extend $x$ to $\widetilde x$ on $Q$ so that $k \leq \widetilde x \leq h$. We give a characterization of $pM(A)_{{\mathrm{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.

Article information

Adv. Oper. Theory Volume 3, Number 1 (2018), 17-41.

Received: 1 November 2016
Accepted: 4 March 2017
First available in Project Euclid: 5 December 2017

Permanent link to this document

Digital Object Identifier

Primary: 46L05: General theory of $C^*$-algebras

operator algebras semicontinuity closed projection operator convex


Brown, Lawrence G. Semicontinuity and closed faces of $C^*$-algebras. Adv. Oper. Theory 3 (2018), no. 1, 17--41. doi:10.22034/aot.1611-1048.

Export citation


  • C. A. Akemann, The general stone–weierstrass problem, J. Funct. Anal. 4 (1969), 277–294.
  • C. A. Akemann, A Gelfand representation theory for $C^*$-algebras, Pac. J. Math 33 (1970), 543–550.
  • C. A. Akemann and G. K. Pedersen, $C^*$-algebra theory, Duke Math. J. 40 (1973), 785–795.
  • C. A. Akemann, G. K. Pedersen, and J. Tomiyama, Multipliers of $C^*$-algebras, J. Funct. Anal. 13 (1973), 277–301.
  • L. G. Brown, Semicontinuity and multipliers of $C^*$-algebras, Canad. J. Math. 40 (1988), 865–988.
  • L. G. Brown, Nearly relatively compact projections in operator algebras, Banach J. Math. Anal. (to appear), arXiv no. 1406.3651.
  • L. G. Brown, MASA's and certain type $I$ closed faces of $C^*$-algebras, in “Group representations, Ergodic Theory, and Mathematical Physics: A Tributy to George W. Mackey,” (R. Doran, C. Moore and R. Zimmer, editors), Contemporary Mathematics 449 (2008), 69–98.
  • L. G. Brown, Convergence of functions of self-adjoint operators and applications, Publ. Mat. 60 (2016), no. 2, 551–564.
  • F. Combes, Sur les face dúne $C^*$-algèbre, Bull. Sci. Math. 93 (1969), 37–62.
  • F. Combes, Quelques propriétés des $C^*$-algèbres, Bull. Sci. Math. 94 (1970), 165–192.
  • Ch. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (1957), 42–44.
  • Ch. Davis, Notions generalizing convexity for functions defined on spaces of matrices, 1963 Proc. Sympos. Pure Math., Vol. VII pp. 187–201 Amer. Math. Soc., Providence, R.I.
  • E. G. Effros, Order ideals in a $C^*$-algebra and its dual, Duke Math. J. 30 (1963), 391–412.
  • F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1982), 229–241.
  • G. K. Pedersen, $C^*$-algebras and their automorphism groups, Academic Press, London–New York, 1979.
  • G. K. Pedersen, Applications of weak$^*$ semicontinuity in $C^*$-algebra theory, Duke Math. J. 39 (1972), 431–450.
  • G. K. Pedersen, $SAW^*$–algebras and corona $C^*$-algebras, contributions to non–commutative topology, J. Operator Theory 15 (1986), 15–32.
  • S. Wassermann, Exact $C^*$-algebras and related topics, Lecture Note Series 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994.