Given any $C^*$-algebra $A$ (not necessarily unital),can we construct a nonzero finite dimensional $C^*$-subalgebra of $A$?
2026-03-26 18:49:07.1774550947
finite dimensional $C^*$ subalgebra
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No. Every finite dimensional $C^*$-algebra is a von Neumann algebra, because it is a Banach dual space. $W^*$ algebras contain many projections -- in particular the range projections of their elements -- but there are $C^*$-algebras with no projections.