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edited Jul 21 2011 at 12:58

I do not know about general C^*-algebras, but the statement is true for complex matrices.

Uniqueness: Assume b = xax. Then a^1/2ba^1/2 = a^1/2xaxa^1/2 = (a^1/2xa^1/2)², which implies that a^1/2xa^1/2 = (a^1/2ba^1/2)^1/2. Since a is invertible, x must be a^−1/2(a^1/2ba^1/2)^1/2a^−1/2.

Existence: It is easy to check that x = a^−1/2(a^1/2ba^1/2)^1/2a^{−1/2$x=\sqrt{a}^{-1}\sqrt{\sqrt{a}\hspace{.15cm}b\sqrt{a}\quad}\sqrt{a}^{-1}$
satisfies the condition.}

I do not think that anything in this argument depends on the fact that we are considering matrices, but let me avoid claiming things about the subject which I do not know well.

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answered Jul 20 2011 at 19:46

Tsuyoshi Ito
1,349●6●15

I do not know about general C^*-algebras, but the statement is true for complex matrices.

Uniqueness: Assume b = xax. Then a^1/2ba^1/2 = a^1/2xaxa^1/2 = (a^1/2xa^1/2)², which implies that a^1/2xa^1/2 = (a^1/2ba^1/2)^1/2. Since a is invertible, x must be a^−1/2(a^1/2ba^1/2)^1/2a^−1/2.

Existence: It is easy to check that x = a^−1/2(a^1/2ba^1/2)^1/2a^−1/2 satisfies the condition.

I do not think that anything in this argument depends on the fact that we are considering matrices, but let me avoid claiming things about the subject which I do not know well.