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Self-Adjoint Element

Let A be a C^*-algebra. An element a in A is called self-adjoint if a^*=a.

For example, the real functions of the C^*-algebra of C([a,b]) of continuous complex-valued functions on [a,b] are the self-adjoint elements.

Each a in A can be expressed (uniquely) in the form h+ik, where h=(a+a^*)/2 and k=(a-a^*)/(2i) are self-adjoint elements of A.

This entry contributed by Mohammad Sal Moslehian

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References

Kadison, R. V. and Ringrose, J. R. Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary Theory. Providence, RI: Amer. Math. Soc., 1997.Murphy, G. J. C-*-Algebras and Operator Theory. New York: Academic Press, 1990.

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Self-Adjoint Element

Cite this as:

Moslehian, Mohammad Sal. "Self-Adjoint Element." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Self-AdjointElement.html

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