Illinois Journal of Mathematics

Unique pseudo-expectations for $C^{*}$-inclusions

David R. Pitts and Vrej Zarikian

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Given an inclusion $\mathcal{D}\subseteq\mathcal{C}$ of unital $C^{*}$-algebras (with common unit), a unital completely positive linear map $\Phi$ of $\mathcal{C}$ into the injective envelope $I(\mathcal{D})$ of $\mathcal{D}$ which extends the inclusion of $\mathcal{D}$ into $I(\mathcal{D})$ is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ of all pseudo-expectations is a convex set, and when $\mathcal{D}$ is Abelian, we prove a Krein–Milman type theorem showing that $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ can be recovered from its set of extreme points. In general, $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ is not a singleton. However, there are large and natural classes of inclusions (e.g., when $\mathcal{D}$ is a regular MASA in $\mathcal{C}$) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For general inclusions of $C^{*}$-algebras with $\mathcal{D}$ Abelian, we give a characterization of the unique pseudo-expectation property in terms of order structure; and when $\mathcal{C}$ is Abelian, we are able to give a topological description of the unique pseudo-expectation property.

As applications, we show that if an inclusion $\mathcal{D}\subseteq\mathcal{C}$ has a unique pseudo-expectation $\Phi$ which is also faithful, then the$C^{*}$-envelope of any operator space $\mathcal{X}$ with $\mathcal{D}\subseteq\mathcal{X}\subseteq\mathcal{C}$ is the $C^{*}$-subalgebra of $\mathcal{C}$ generated by $\mathcal{X}$; we also show that for many interesting classes of $C^{*}$-inclusions, having a faithful unique pseudo-expectation implies that $\mathcal{D}$ norms $\mathcal{C}$, although this is not true in general.

Article information

Illinois J. Math., Volume 59, Number 2 (2015), 449-483.

Received: 12 August 2015
Revised: 23 December 2015
First available in Project Euclid: 5 May 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras 46L07: Operator spaces and completely bounded maps [See also 47L25] 46L10: General theory of von Neumann algebras
Secondary: 46M10: Projective and injective objects [See also 46A22]


Pitts, David R.; Zarikian, Vrej. Unique pseudo-expectations for $C^{*}$-inclusions. Illinois J. Math. 59 (2015), no. 2, 449--483. doi:10.1215/ijm/1462450709.

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  • C. A. Akemann and D. Sherman, Conditional expectations onto maximal Abelian $*$-subalgebras, J. Operator Theory 68 (2012), no. 2, 597–607.
  • C. Akemann, S. Wassermann and N. Weaver, Pure states on free group $C^*$-algebras, Glasg. Math. J. 52 (2010), no. 1, 151–154.
  • R. J. Archbold, J. W. Bunce and K. D. Gregson, Extensions of states of $C^*$-algebras. II, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 1–2, 113–122.
  • E. Blanchard, Tensor products of $C(X)$-algebras over $C(X)$. Recent advances in operator algebras (Orléans, 1992), Astérisque 232 (1995), 81–92.
  • N. P. Brown and N. Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008.
  • L. J. Bunce and C.-H. Chu, Unique extension of pure states of $C^*$-algebras, J. Operator Theory 39 (1998), no. 2, 319–338.
  • M. D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly 90 (1983), no. 5, 301–312.
  • F. Combes and C. Delaroche, Groupe modulaire d'une espérance conditionnele dans une algèbre de von Neumann, Bull. Soc. Math. France 103 (1975), no. 4, 385–426. (French).
  • R. S. Doran and A. V. Belfi, Characterizations of $C^*$-algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 101, Marcel Decker, Inc., New York, NY, 1986.
  • E. G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.
  • T. Giordano and J. A. Mingo, Tensor products of $C^*$-algebras over Abelian subalgebras, J. Lond. Math. Soc. (2) 55 (1997), no. 1, 170–180.
  • D. Hadwin and V. I. Paulsen, Injectivity and projectivity in analysis and topology, Sci. China Math. 54 (2011), no. 11, 2347–2359.
  • M. Hamana, Injective envelopes of $C^*$-algebras, J. Math. Soc. Japan 31 (1979), no. 1, 181–197.
  • M. Hamana, Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 773–785.
  • M. Hamana, Regular embeddings of $C^*$-algebras in monotone complete $C^*$-algebras, J. Math. Soc. Japan 33 (1981), no. 1, 159–183.
  • M. Hamana, The centre of the regular monotone completion of a $C^*$-algebra, J. Lond. Math. Soc. (2) 26 (1982), no. 3, 522–530.
  • R. V. Kadison, Operator algebras with a faithful weakly-closed representation, Ann. of Math. (2) 64 (1956), 175–181.
  • R. V. Kadison, Diagonalizing matrices, Amer. J. Math. 106 (1984), no. 6, 1451–1468.
  • R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I. Elementary theory, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.
  • R. V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383–400.
  • S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding $C^*$-algebras, Tokyo J. Math. 13 (1990), no. 2, 251–257.
  • A. Kumjian, On $C^*$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008.
  • S. S. Kutateladze, The Krein–Milman theorem and its converse, Sibirsk. Mat. Zh. 21 (1980), no. 1, 130–138. (Russian).
  • B. Magajna, The minimal operator module of a Banach module, Proc. Edinb. Math. Soc. (2) 42 (1999), no. 1, 191–208.
  • D. Maharam, On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA 28 (1942), 108–111.
  • A. W. Marcus, D. A. Spielman and N. Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem, Ann. of Math. (2) 182 (2015), 327–350.
  • V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002.
  • D. R. Pitts, Structure for regular inclusions, preprint, 2012; available at \arxivurlarXiv:1202.6413v2 [math.OA].
  • F. Pop, A. M. Sinclair and R. R. Smith, Norming $C^*$-algebras by $C^*$-subalgebras, J. Funct. Anal. 175 (2000), no. 1, 168–196.
  • S. Popa, A $II_1$ factor approach to the Kadison–Singer problem, Comm. Math. Phys. 332 (2014), no. 1, 379–414.
  • S. Popa and S. Vaes, Paving over arbitrary MASAs in von Neumann algebras, Anal. PDE 8 (2015), no. 4, 1001–1023.
  • J. Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63.
  • W. Rudin, Well-distributed measurable sets, Amer. Math. Monthly 90 (1983), no. 1, 41–42.
  • A. M. Sinclair and R. R. Smith, Finite von Neumann algebras and masas, London Mathematical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008.
  • E. Stormer, Positive linear maps of operator algebras, Springer Monographs in Mathematics, Springer, Heidelberg, 2013.
  • M. Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979.
  • J. Tomiyama, On the projection of norm one in $W^*$-algebras. III, Tôhoku Math. J. (2) 11 (1959), 125–129.
  • J. Tomiyama, On the tensor products of von Neumann algebras, Pacific J. Math. 30 (1969), 263–270.