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Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid F(V)\subseteq V\}.$$

So $A$ is a Banach algebra.

Can we equip $A$ with an involution $*$ such that we get a $C^*$ algebra structure on $A$?

What are K-groups of $A$ as a Banach algebra?

Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$

So $A$ is a Banach algebra.

Can we equip $A$ with an involution $*$ such that we get a $C^*$ algebra structure on $A$?

What are K-groups of $A$ as a Banach algebra?

Let $H$ be an infinite dimensional seperable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid F(V)\subseteq V\}$$

So $A$ is a Banach algebra.

Can we equip $A$ with an involution $*$ such that we get a $C^*$ algebra structure on $A$?

What are K-groups of $A$ as a Banach algebra?

Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid F(V)\subseteq V\}.$$

So $A$ is a Banach algebra.

Can we equip $A$ with an involution $*$ such that we get a $C^*$ algebra structure on $A$?

What are K-groups of $A$ as a Banach algebra?