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let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.

Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\alpha_t$ of morphisms which connect them.

Does this imply that $A$ has no nontrivial idempotent?does this imply that the spectrum of every element is a path connected subset of $\mathbb{C}$?

Does existence of such a homotopicity depend on the $C^*$ norm we are choosing for the algebraic tensor product $A\otimes A$? Namely is it possible that these morphism are homotopic with respect to a given $C^*$ norm and are not homotopic with respect to another $C^*$ norm?

Obviously the answer to each question mentioned above is positive for the commutative case.

Does the matrix algebra satisfy this homotopic property?

let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.

Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\alpha_t$ of morphisms which connect them.

Does this imply that $A$ has no nontrivial idempotent?does this imply that the spectrum of every element is a path connected subset of $\mathbb{C}$?

Does existence of such a homotopicity depend on the $C^*$ norm we are choosing for the algebraic tensor product $A\otimes A$? Namely is it possible that these morphism are homotopic with respect to a given $C^*$ norm and are not homotopic with respect to another $C^*$ norm?

When $A$ is a commutative algebra, the answer to each question mentioned above is affirmative.

Does the matrix algebra satisfy this homotopic property?

let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.

Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\alpha_t$ of morphisms which connect them.

Does this imply that $A$ has no nontrivial idempotent?does this imply that the spectrum of every element is a path connected subset of $\mathbb{C}$?

Does existence of such a homotopicity depend on the $C^*$ norm we are choosing for the algebraic tensor product $A\otimes A$? Namely is it possible that these morphism are homotopic with respect to a given $C^*$ norm and are not homotopic with respect to another $C^*$ norm?

Obviously the answer is positive for the commutative case.

Does the matrix algebra satisfy this homotopic property?

let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.

Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\alpha_t$ of morphisms which connect them.

Does this imply that $A$ has no nontrivial idempotent?does this imply that the spectrum of every element is a path connected subset of $\mathbb{C}$?

Does existence of such a homotopicity depend on the $C^*$ norm we are choosing for the algebraic tensor product $A\otimes A$? Namely is it possible that these morphism are homotopic with respect to a given $C^*$ norm and are not homotopic with respect to another $C^*$ norm?

Obviously the answer to each question mentioned above is positive for the commutative case.

Does the matrix algebra satisfy this homotopic property?

let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.

Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\alpha_t$ of morphisms which connect them.

Does this imply that $A$ has no nontrivial idempotent?does this imply that the spectrum of every element is a path connected subset of $\mathbb{C}$?

Does existence of such a homotopicity depend on the $C^*$ norm we are choosing for the algebraic tensor product $A\otimes A$? Namely is it possible that these morphism are homotopic with respect to a given $C^*$ norm and are not homotopic with respect to another $C^*$ norm?

Obviously the answer is positive for the commutative case.

let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.

Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\alpha_t$ of morphisms which connect them.

Does this imply that $A$ has no nontrivial idempotent?does this imply that the spectrum of every element is a path connected subset of $\mathbb{C}$?

Does existence of such a homotopicity depend on the $C^*$ norm we are choosing for the algebraic tensor product $A\otimes A$? Namely is it possible that these morphism are homotopic with respect to a given $C^*$ norm and are not homotopic with respect to another $C^*$ norm?

Obviously the answer is positive for the commutative case.

Does the matrix algebra satisfy this homotopic property?