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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?

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

Does the matrix algebra satisfy this homotopic property?

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    $\begingroup$ The article arxiv.org/abs/1609.04001 may be relevant to your question (although I confess I didn't study it closely). $\endgroup$
    – Mark Grant
    Commented Aug 24, 2019 at 11:50
  • $\begingroup$ @MarkGrant Thank you very much for this interesting paper. $\endgroup$
    – Ali Taghavi
    Commented Aug 25, 2019 at 15:02
  • $\begingroup$ The answer of you first two questions is no, I guess. You have to look for a unital $C^*$-Algebra with non-trivial projections. What about, for example, $\mathcal{O}_2$ the Cuntz algebra on 2 generators or even corners $A=p\mathcal{O}_2p$ of it, where $p\in \mathcal{O}_2$ is a nontrivial projection? $\endgroup$
    – Sabrina Gemsa
    Commented Sep 10, 2019 at 22:16
  • $\begingroup$ @SabrinaGemsa Thank you for your comments. Why do the two maps $a\otimes 1$ and $1\otimes a$ are homotopic morphism in the Cuntz algebra? $\endgroup$
    – Ali Taghavi
    Commented Sep 10, 2019 at 22:39

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