# Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $$C^{*}$$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $$\mathbb{C}$$.

What is an example of a non commutative path connected $$C^*$$ algebra? What is an example of a simple path connected $$C^{*}$$ algebra? In particular does $$C^{*}_{red} F_{2}$$ satisfiy this property?

As another question: Is the tensor product of two path connected algebra, a path connected algebra?(For spatial norm).

• Oh you meant simple as in simple not as in easy... – Johannes Hahn Feb 7 '15 at 0:31
• Well $\mathbb{C}$ is simple. But I assume you want something less trivial. A stab in the dark: One could imagine putting some group action on $C(X)$ such that none of its non-trivial ideals are $G$-invariant and hope that $C(X)\rtimes G$ is simple. – Johannes Hahn Feb 7 '15 at 0:37
• The spectrum of a nontrivial projection (i.e., nonzero and not the unit) is disconnected. Thus, a unital C*-algebra with the property you consider cannot contain any nontrivial projections. Conversely, if a C*-algebra contains a normal element with disconnected spectrum, then it contains a nontrivial projection. Thus, in a unital, simple C*-algebras without nontrivial projections (such as the Jiang-Su algebra), at least the spectrum of every normal element is connected. – Hannes Thiel Feb 25 '15 at 18:46
• Hannes' observation can be improved: all elements of a C*-algebra with no non-trivial projections have connected spectrum. Indeed given an element with disconnected spectrum the holomorphic functional calculus gives a non-trivial idempotent. This will be similar to some self-adjoint idempotent (see Blackadar's K-Theory book) which will be non-trivial. – Sam Evington Mar 4 '15 at 15:13
• Example of a connected C∗ algebra : See B.E. Blackadar. A simple unital projectionless C∗-algebra. J. Operator Theory, 5:63–71, 1981. By the comments of Hannes Thiel and Sam Evington this algebra is an example of a connected C* algebra. But is it also path connected (remark by Ali Taghavi) ? – jjcale Nov 7 '15 at 16:29