Homomorphic conditional expectations as noncommutative retractions

Abstract

Let $A$ be a $C^*$-algebra and $\mathcal{E}: A \to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$\mathcal{E}(x)^* \mathcal{E}(x) \leq \mathcal{E}(x^* x),$$ implies that $$\Vert \mathcal{E}(x)\Vert ^2 \leq \Vert \mathcal{E}(x^* x)\Vert.$$ In this note we show that $\mathcal{E}$ is homomorphic (in the sense that $\mathcal{E}(xy) = \mathcal{E}(x)\mathcal{E}(y)$ for every $x, y$ in $A$) if and only if $$\Vert \mathcal{E}(x)\Vert^2 = \Vert \mathcal{E}(x^*x)\Vert,$$ for every $x$ in $A$. We also prove that a homomorphic conditional expectation on a commutative $C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of $X$. One may therefore consider homomorphic conditional expectations as noncommutative retractions.

Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 396-408.

Dates
Accepted: 6 June 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431716

Digital Object Identifier
doi:10.22034/aot.1705-1161

Mathematical Reviews number (MathSciNet)
MR3730035

Zentralblatt MATH identifier
1385.46055

Citation

Pluta, Robert; Russo, Bernard. Homomorphic conditional expectations as noncommutative retractions. Adv. Oper. Theory 2 (2017), no. 4, 396--408. doi:10.22034/aot.1705-1161. https://projecteuclid.org/euclid.aot/1512431716

References

• T. Barton and R. M. Timoney, Weak*-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), no. 2, 177–191.
• B. Blackadar, Operator algebras. Theory of $C^*$-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, 122, Springer-Verlag, Berlin, 2006.
• M. D. Choi, A Schwarz inequality for positive linear maps on C$^*$-algebras, Illinois J. Math. 18 (1974), 565–574.
• C.-H. Chu, Jordan structures in geometry and analysis, Cambridge Tracts in Mathematics, 190. Cambridge University Press, Cambridge, 2012.
• P. Civin and B. Yood, Lie and Jordan structures in Banach algebras, Pacific J. Math. 15 (1965), 775–797.
• H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, 24. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.
• E. G. Effros and E. Størmer, Positive projections and Jordan structure in operator algebras, Math. Scand. 45 (1979), no. 1, 127–138.
• Y. Friedman and B. Russo, Conditional expectation without order, Pacific J. Math. 115 (1984), no. 2, 351–360.
• Y. Friedman and B. Russo, Solution of the contractive projection problem, J. Funct. Anal. 60 (1985), no. 1, 56–79.
• Y. Friedman and B. Russo, Structure of the predual of a $JBW^\ast$-triple, J. Reine Angew. Math. 356 (1985), 67–89.
• Y. Kato, Some theorems on projections of von Neumann algebras, Math. Japon. 21 (1976), no. 4, 367–370.
• A. M. Peralta, Positive definite hermitian mappings associated with tripotent elements, Expo. Math. 33 (2015), no. 2, 252–258.
• R. Pluta, Ranges of bimodule projections and conditional expectations, Cambridge Scholars Publishing, Newcastle upon Tyne, 2013.
• C. 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.
• A. G. Robertson and M. A. Youngson, Positive projections with contractive complements on Jordan algebras, J. London Math. Soc. (2) 25 (1982), no. 2, 365–374.
• M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481.
• J. Tomiyama, On the projection of norm one in $W^*$-algebras, Proc. Japan Acad. 33 (1957), 608–612.