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Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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Is the Loewner maximum uniquely defined?

Given 2 (symmetric) PSD matrices $A,B$, is the following set $S_{A,B}$ non-empty? $$ S_{A,B} = \{ C: C\succeq A, C\succeq B, \text{ and }\forall D, D\succeq A, D\succeq B \implies D\succeq C \} $$ If …
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