Exponential rank and exponential length for Z-stable simple C*-algebras

Abstract: Let

A

be a unital separable simple

c a l Z

-stable C*-algebra which has rational tracial rank at most one and let

u i n U_{0} (A),

the connected component of the unitary group of

A .

We show that, for any

e p s i l o n > 0,

there exists a self-adjoint element

h i n A

such that

|u-exp(ih)|<epsilon.

The lower bound of

| h |

could be as large as one wants. If

u i n C U (A),

the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element

h i n A

such that

|u-exp(ih)| <epsilon and |h|le 2pi.

Examples are given that the bound

2 p i

for

| h |

is the optimal in general. For the Jiang-Su algebra

c a l Z,

we show that, if

u i n U_{0} (c a l Z)

and

e p s i l o n > 0,

there exists a real number

- p i < t l e p i

and a self-adjoint element

h i n c a l Z

with

| h | l e 2 p i

such that

|e^{it}u-exp(ih)|<epsilon.

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